{"id":58941,"date":"2025-10-27T14:54:05","date_gmt":"2025-10-27T09:24:05","guid":{"rendered":"https:\/\/therightmentor.com\/?p=58941"},"modified":"2025-10-28T10:02:03","modified_gmt":"2025-10-28T04:32:03","slug":"quadratic-equations","status":"publish","type":"post","link":"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/quadratic-equations\/","title":{"rendered":"Quadratic Equations | Notes"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_82_2 ez-toc-wrap-center counter-hierarchy ez-toc-counter ez-toc-light-blue ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title ez-toc-toggle\" style=\"cursor:pointer\">Click For Contents <\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 eztoc-toggle-hide-by-default' ><li class='ez-toc-page-1 ez-toc-heading-level-1'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/quadratic-equations\/#Quadratic_Equations\" >Quadratic Equations<\/a><ul class='ez-toc-list-level-2' ><li class='ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/quadratic-equations\/#%F0%9F%94%B9_Definition\" >\ud83d\udd39 Definition<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/quadratic-equations\/#ax2bxc0ax2_bx_c_0\" >ax2+bx+c=0ax^2 + bx + c = 0<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/quadratic-equations\/#%F0%9F%94%B9_Standard_Form\" >\ud83d\udd39 Standard Form<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/quadratic-equations\/#ax2bxc0ax2_bx_c_0-2\" >ax2+bx+c=0ax^2 + bx + c = 0<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/quadratic-equations\/#%F0%9F%94%B9_Methods_of_Solving_a_Quadratic_Equation\" >\ud83d\udd39 Methods of Solving a Quadratic Equation<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/quadratic-equations\/#%F0%9F%A7%AD_Method_1_Factorisation_Splitting_the_Middle_Term\" >\ud83e\udded Method 1: Factorisation (Splitting the Middle Term)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/quadratic-equations\/#%E2%9C%85_Stepwise_Procedure\" >\u2705 Stepwise Procedure<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/quadratic-equations\/#ax2bxc0ax2_bx_c_0-3\" >ax2+bx+c=0ax^2 + bx + c = 0<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/quadratic-equations\/#Coefficient_of_x2%C3%97Constant_terma%C3%97ctextCoefficient_of_x2_times_textConstant_term_a_times_c\" >(Coefficient\u00a0of\u00a0x2)\u00d7(Constant\u00a0term)=a\u00d7c(\\text{Coefficient of } x^2) \\times (\\text{Constant term}) = a \\times c<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-11\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/quadratic-equations\/#mnbm%C3%97na%C3%97cm_n_b_quad_m_times_n_a_times_c\" >m+n=b,m\u00d7n=a\u00d7cm + n = b, \\quad m \\times n = a \\times c<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-12\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/quadratic-equations\/#%F0%9F%A7%A9_Rule_Box_Splitting_the_Middle_Term\" >\ud83e\udde9 Rule Box: Splitting the Middle Term<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-13\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/quadratic-equations\/#mnCoefficient_of_xm_n_textCoefficient_of_x\" >m+n=Coefficient\u00a0of\u00a0xm + n = \\text{Coefficient of } x<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-14\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/quadratic-equations\/#and_also_m%C3%97nCoefficient_of_x2%C3%97Constant_termm_times_n_textCoefficient_of_x2_times_textConstant_termThen_rewrite_bxbx_as_mxnxmx_nx\" >and also \nm\u00d7n=(Coefficient\u00a0of\u00a0x2)\u00d7(Constant\u00a0term)m \\times n = (\\text{Coefficient of } x^2) \\times (\\text{Constant term})Then rewrite bxbx as mx+nxmx + nx,<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-15\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/quadratic-equations\/#Now_group_and_factorise\" >Now group and factorise.<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-16\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/quadratic-equations\/#%E2%9C%B3%EF%B8%8F_Example_1_Easy\" >\u2733\ufe0f Example 1 (Easy)<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-17\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/quadratic-equations\/#ax2bxc0On_comparision_we_get\" >ax2+bx+c=0On comparision we get:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-18\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/quadratic-equations\/#x22x3x60x2_2x_3x_6_0\" >x2+2x+3x+6=0x^2 + 2x + 3x + 6 = 0<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-19\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/quadratic-equations\/#xx23x20xx_2_3x_2_0_x2x30x_2x_3_0Step_6\" >x(x+2)+3(x+2)=0x(x + 2) + 3(x + 2) = 0 \n(x+2)(x+3)=0(x + 2)(x + 3) = 0Step 6:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-20\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/quadratic-equations\/#x20%E2%87%92x%E2%88%922andx30%E2%87%92x%E2%88%923x_2_0_Rightarrow_x_-2_quad_textand_quad_x_3_0_Rightarrow_x_-3\" >x+2=0\u21d2x=\u22122andx+3=0\u21d2x=\u22123x + 2 = 0 \\Rightarrow x = -2 \\quad \\text{and} \\quad x + 3 = 0 \\Rightarrow x = -3<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-21\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/quadratic-equations\/#%E2%9C%B3%EF%B8%8F_Example_2_Moderate\" >\u2733\ufe0f Example 2 (Moderate)<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-22\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/quadratic-equations\/#ax2bxc0On_comparision_we_get-2\" >ax2+bx+c=0On comparision we get:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-23\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/quadratic-equations\/#2x24xx202x2_4x_x_2_0\" >2x2+4x+x+2=02x^2 + 4x + x + 2 = 0<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-24\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/quadratic-equations\/#2xx21x202xx_2_1x_2_0_x22x10x_22x_1_0Step_6\" >2x(x+2)+1(x+2)=02x(x + 2) + 1(x + 2) = 0 \n(x+2)(2x+1)=0(x + 2)(2x + 1) = 0Step 6:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-25\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/quadratic-equations\/#x%E2%88%922orx%E2%88%9212x_-2_quad_textor_quad_x_-frac12\" >x=\u22122orx=\u221212x = -2 \\quad \\text{or} \\quad x = -\\frac{1}{2}<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-26\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/quadratic-equations\/#%E2%9C%B3%EF%B8%8F_Example_3_Hard\" >\u2733\ufe0f Example 3 (Hard)<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-27\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/quadratic-equations\/#ax2bxc0On_comparision_we_get-3\" >ax2+bx+c=0On comparision we get:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-28\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/quadratic-equations\/#3x2%E2%88%929xx%E2%88%92303x2_%E2%80%93_9x_x_%E2%80%93_3_0\" >3x2\u22129x+x\u22123=03x^2 &#8211; 9x + x &#8211; 3 = 0<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-29\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/quadratic-equations\/#3xx%E2%88%9231x%E2%88%92303xx_%E2%80%93_3_1x_%E2%80%93_3_0\" >3x(x\u22123)+1(x\u22123)=03x(x &#8211; 3) + 1(x &#8211; 3) = 0<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-30\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/quadratic-equations\/#x%E2%88%9233x10x_%E2%80%93_33x_1_0\" >(x\u22123)(3x+1)=0(x &#8211; 3)(3x + 1) = 0<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-31\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/quadratic-equations\/#x3x%E2%88%9213x_3_quad_x_-frac13\" >x=3,x=\u221213x = 3, \\quad x = -\\frac{1}{3}<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-32\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/quadratic-equations\/#Method_2_Completing_the_Square\" >Method 2: Completing the Square<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-33\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/quadratic-equations\/#%F0%9F%A7%AD_Concept\" >\ud83e\udded Concept<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-34\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/quadratic-equations\/#%E2%9C%85_Step-by-Step_Procedure\" >\u2705 Step-by-Step Procedure<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-35\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/quadratic-equations\/#%F0%9F%A7%A9_Rule_Box_Completing_the_Square\" >\ud83e\udde9 Rule Box: Completing the Square<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-36\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/quadratic-equations\/#%E2%9C%B3%EF%B8%8F_Example_1_Easy-2\" >\u2733\ufe0f Example 1 (Easy)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-37\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/quadratic-equations\/#%E2%9C%B3%EF%B8%8F_Example_2_Moderate-2\" >\u2733\ufe0f Example 2 (Moderate)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-38\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/quadratic-equations\/#%E2%9C%B3%EF%B8%8F_Example_3_Hard-2\" >\u2733\ufe0f Example 3 (Hard)<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-39\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/quadratic-equations\/#%F0%9F%94%B9_Method_3_Quadratic_Formula\" >\ud83d\udd39 Method 3: Quadratic Formula<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-40\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/quadratic-equations\/#%F0%9F%A7%AD_Concept-2\" >\ud83e\udded Concept<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-41\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/quadratic-equations\/#%E2%9C%85_Step-by-Step_Procedure-2\" >\u2705 Step-by-Step Procedure<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-42\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/quadratic-equations\/#%F0%9F%A7%A9_Rule_Box_Quadratic_Formula\" >\ud83e\udde9 Rule Box: Quadratic Formula<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-43\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/quadratic-equations\/#%E2%9C%B3%EF%B8%8F_Example_1_Easy-3\" >\u2733\ufe0f Example 1 (Easy)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-44\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/quadratic-equations\/#%E2%9C%B3%EF%B8%8F_Example_2_Moderate-3\" >\u2733\ufe0f Example 2 (Moderate)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-45\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/quadratic-equations\/#%E2%9C%B3%EF%B8%8F_Example_3_Hard-3\" >\u2733\ufe0f Example 3 (Hard)<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-46\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/quadratic-equations\/#6%EF%B8%8F%E2%83%A3_Nature_of_Roots\" >6\ufe0f\u20e3 Nature of Roots<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-47\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/quadratic-equations\/#%F0%9F%94%B9_7%EF%B8%8F%E2%83%A3_Summary_of_Formulas\" >\ud83d\udd39 7\ufe0f\u20e3 Summary of Formulas<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-48\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/quadratic-equations\/#%F0%9F%94%B9_8%EF%B8%8F%E2%83%A3_Practice_Questions_Worksheet\" >\ud83d\udd39 8\ufe0f\u20e3 Practice Questions (Worksheet)<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-49\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/quadratic-equations\/#Q1_Solve_by_factorisation\" >Q1. Solve by factorisation:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-50\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/quadratic-equations\/#Q2_Solve_by_splitting_the_middle_term\" >Q2. Solve by splitting the middle term:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-51\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/quadratic-equations\/#Q3_Find_the_discriminant_and_nature_of_roots\" >Q3. Find the discriminant and nature of roots:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-52\" href=\"https:\/\/therightmentor.com\/trm-course\/home\/courses\/uncategorized\/quadratic-equations\/#Q4_Find_the_sum_and_product_of_the_roots\" >Q4. Find the sum and product of the roots:<\/a><\/li><\/ul><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<h1 data-start=\"182\" data-end=\"219\"><span class=\"ez-toc-section\" id=\"Quadratic_Equations\"><\/span><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"187\" data-end=\"219\">Quadratic Equations<\/strong><\/span><span class=\"ez-toc-section-end\"><\/span><\/h1>\n<hr data-start=\"221\" data-end=\"224\" \/>\n<h2 data-start=\"226\" data-end=\"250\"><span class=\"ez-toc-section\" id=\"%F0%9F%94%B9_Definition\"><\/span><span style=\"font-size: 24px; color: #000000;\">\ud83d\udd39 <strong data-start=\"232\" data-end=\"250\">Definition<\/strong><\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p data-start=\"252\" data-end=\"322\"><span style=\"font-size: 24px; color: #000000;\">A <strong data-start=\"254\" data-end=\"276\">quadratic equation<\/strong> in one variable is an equation of the form:<\/span><\/p>\n<h2 data-start=\"430\" data-end=\"471\"><span class=\"ez-toc-section\" id=\"ax2bxc0ax2_bx_c_0\"><\/span><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"436\" data-end=\"471\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mi>b<\/mi><mi>x<\/mi><mo>+<\/mo><mi>c<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">ax^2 + bx + c = 0<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/strong><\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p data-start=\"349\" data-end=\"356\"><span style=\"font-size: 24px; color: #000000;\">where<\/span><\/p>\n<ul data-start=\"357\" data-end=\"411\">\n<li data-start=\"357\" data-end=\"394\"><span style=\"font-size: 24px; color: #000000;\">x is a variable<\/span><\/li>\n<li data-start=\"357\" data-end=\"394\">\n<p data-start=\"359\" data-end=\"394\"><span style=\"font-size: 24px; color: #000000;\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><mo separator=\"true\">,<\/mo><mi>b<\/mi><mo separator=\"true\">,<\/mo><mi>c<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a, b, c<\/annotation><\/semantics><\/math><\/span><\/span> are real numbers, and<\/span><\/p>\n<\/li>\n<li data-start=\"395\" data-end=\"411\">\n<p data-start=\"397\" data-end=\"411\"><span style=\"font-size: 24px; color: #000000;\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><mo mathvariant=\"normal\">\u2260<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">a \\ne 0<\/annotation><\/semantics><\/math><\/span><\/span>.<\/span><\/p>\n<\/li>\n<\/ul>\n<hr data-start=\"413\" data-end=\"416\" \/>\n<h2 data-start=\"418\" data-end=\"445\"><span class=\"ez-toc-section\" id=\"%F0%9F%94%B9_Standard_Form\"><\/span><span style=\"font-size: 24px; color: #000000;\">\ud83d\udd39 <strong data-start=\"424\" data-end=\"445\">Standard Form<\/strong><\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h2 data-start=\"430\" data-end=\"471\"><span class=\"ez-toc-section\" id=\"ax2bxc0ax2_bx_c_0-2\"><\/span><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"436\" data-end=\"471\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mi>b<\/mi><mi>x<\/mi><mo>+<\/mo><mi>c<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">ax^2 + bx + c = 0<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/strong><\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p data-start=\"472\" data-end=\"481\"><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"472\" data-end=\"481\">Here:<\/strong><\/span><\/p>\n<ul data-start=\"483\" data-end=\"566\">\n<li data-start=\"483\" data-end=\"513\">\n<p data-start=\"485\" data-end=\"513\"><span style=\"font-size: 24px; color: #000000;\">Coefficient of <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>=<\/mo><mi>a<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">x^2 = a<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/span><\/p>\n<\/li>\n<li data-start=\"514\" data-end=\"542\">\n<p data-start=\"516\" data-end=\"542\"><span style=\"font-size: 24px; color: #000000;\">Coefficient of <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mi>b<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">x = b<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/span><\/p>\n<\/li>\n<li data-start=\"543\" data-end=\"566\">\n<p data-start=\"545\" data-end=\"566\"><span style=\"font-size: 24px; color: #000000;\">Constant term <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo>=<\/mo><mi>c<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">= c<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/span><\/p>\n<\/li>\n<\/ul>\n<hr data-start=\"568\" data-end=\"571\" \/>\n<h2 data-start=\"573\" data-end=\"626\"><span class=\"ez-toc-section\" id=\"%F0%9F%94%B9_Methods_of_Solving_a_Quadratic_Equation\"><\/span><span style=\"font-size: 24px; color: #000000;\">\ud83d\udd39 <strong data-start=\"579\" data-end=\"626\">Methods of Solving a Quadratic Equation<\/strong><\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 data-start=\"628\" data-end=\"690\"><span class=\"ez-toc-section\" id=\"%F0%9F%A7%AD_Method_1_Factorisation_Splitting_the_Middle_Term\"><\/span><span style=\"font-size: 24px; color: #000000;\">\ud83e\udded <strong data-start=\"635\" data-end=\"690\">Method 1: Factorisation (Splitting the Middle Term)<\/strong><\/span><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<hr data-start=\"692\" data-end=\"695\" \/>\n<h3 data-start=\"697\" data-end=\"725\"><span class=\"ez-toc-section\" id=\"%E2%9C%85_Stepwise_Procedure\"><\/span><span style=\"font-size: 24px; color: #000000;\">\u2705 <strong data-start=\"703\" data-end=\"725\">Stepwise Procedure<\/strong><\/span><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p data-start=\"727\" data-end=\"777\"><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"727\" data-end=\"738\">Step 1:<\/strong> Write the equation in standard form:<\/span><\/p>\n<h2 data-start=\"430\" data-end=\"471\"><span class=\"ez-toc-section\" id=\"ax2bxc0ax2_bx_c_0-3\"><\/span><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"436\" data-end=\"471\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mi>b<\/mi><mi>x<\/mi><mo>+<\/mo><mi>c<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">ax^2 + bx + c = 0<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/strong><\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p data-start=\"803\" data-end=\"839\"><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"803\" data-end=\"814\">Step 2:<\/strong> Identify coefficients.<\/span><\/p>\n<ul data-start=\"840\" data-end=\"923\">\n<li data-start=\"840\" data-end=\"870\"><span style=\"font-size: 24px; color: #000000;\">x=variable<\/span><\/li>\n<li data-start=\"840\" data-end=\"870\">\n<p data-start=\"842\" data-end=\"870\"><span style=\"font-size: 24px; color: #000000;\">Coefficient of <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>=<\/mo><mi>a<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">x^2 = a<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/span><\/p>\n<\/li>\n<li data-start=\"871\" data-end=\"899\">\n<p data-start=\"873\" data-end=\"899\"><span style=\"font-size: 24px; color: #000000;\">Coefficient of <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mi>b<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">x = b<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/span><\/p>\n<\/li>\n<li data-start=\"900\" data-end=\"923\">\n<p data-start=\"902\" data-end=\"923\"><span style=\"font-size: 24px; color: #000000;\">Constant term <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo>=<\/mo><mi>c<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">= c<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/span><\/p>\n<\/li>\n<\/ul>\n<p data-start=\"925\" data-end=\"947\"><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"925\" data-end=\"936\">Step 3:<\/strong> Multiply<\/span><\/p>\n<h2 data-start=\"430\" data-end=\"471\"><span class=\"ez-toc-section\" id=\"Coefficient_of_x2%C3%97Constant_terma%C3%97ctextCoefficient_of_x2_times_textConstant_term_a_times_c\"><\/span><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"436\" data-end=\"471\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo stretchy=\"false\">(<\/mo><mtext>Coefficient\u00a0of\u00a0<\/mtext><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo stretchy=\"false\">)<\/mo><mo>\u00d7<\/mo><mo stretchy=\"false\">(<\/mo><mtext>Constant\u00a0term<\/mtext><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>a<\/mi><mo>\u00d7<\/mo><mi>c<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">(\\text{Coefficient of } x^2) \\times (\\text{Constant term}) = a \\times c<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/strong><\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p data-start=\"1027\" data-end=\"1083\"><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"1027\" data-end=\"1038\">Step 4:<\/strong> Find two numbers <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>m<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">m<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">m<\/span><\/span><\/span><\/span> and <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>n<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">n<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">n<\/span><\/span><\/span><\/span> such that<\/span><\/p>\n<h2 data-start=\"430\" data-end=\"471\"><span class=\"ez-toc-section\" id=\"mnbm%C3%97na%C3%97cm_n_b_quad_m_times_n_a_times_c\"><\/span><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"436\" data-end=\"471\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>m<\/mi><mo>+<\/mo><mi>n<\/mi><mo>=<\/mo><mi>b<\/mi><mo separator=\"true\">,<\/mo><mspace width=\"1em\"><\/mspace><mi>m<\/mi><mo>\u00d7<\/mo><mi>n<\/mi><mo>=<\/mo><mi>a<\/mi><mo>\u00d7<\/mo><mi>c<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">m + n = b, \\quad m \\times n = a \\times c<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/strong><\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p data-start=\"1132\" data-end=\"1192\"><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"1132\" data-end=\"1143\">Step 5:<\/strong> Rewrite the middle term <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>b<\/mi><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">bx<\/annotation><\/semantics><\/math><\/span><\/span> as <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>m<\/mi><mi>x<\/mi><mo>+<\/mo><mi>n<\/mi><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">mx + nx<\/annotation><\/semantics><\/math><\/span><\/span>.<\/span><\/p>\n<p data-start=\"1194\" data-end=\"1238\"><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"1194\" data-end=\"1205\">Step 6:<\/strong> Group and factorise the terms.<\/span><\/p>\n<p data-start=\"1240\" data-end=\"1307\"><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"1240\" data-end=\"1251\">Step 7:<\/strong> Equate each factor to zero and find the value of <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">x<\/annotation><\/semantics><\/math><\/span><\/span>.<\/span><\/p>\n<hr data-start=\"1309\" data-end=\"1312\" \/>\n<h3 data-start=\"1314\" data-end=\"1360\"><span class=\"ez-toc-section\" id=\"%F0%9F%A7%A9_Rule_Box_Splitting_the_Middle_Term\"><\/span><span style=\"font-size: 24px; color: #000000;\">\ud83e\udde9 <strong data-start=\"1321\" data-end=\"1360\">Rule Box: Splitting the Middle Term<\/strong><\/span><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p data-start=\"1362\" data-end=\"1406\"><span style=\"font-size: 24px; color: #000000;\">Find two numbers <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>m<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">m<\/annotation><\/semantics><\/math><\/span><\/span> and <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>n<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">n<\/annotation><\/semantics><\/math><\/span><\/span> such that<\/span><\/p>\n<h2 data-start=\"430\" data-end=\"471\"><span class=\"ez-toc-section\" id=\"mnCoefficient_of_xm_n_textCoefficient_of_x\"><\/span><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"436\" data-end=\"471\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>m<\/mi><mo>+<\/mo><mi>n<\/mi><mo>=<\/mo><mtext>Coefficient\u00a0of\u00a0<\/mtext><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">m + n = \\text{Coefficient of } x<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/strong><\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h2 data-start=\"430\" data-end=\"471\"><span class=\"ez-toc-section\" id=\"and_also_m%C3%97nCoefficient_of_x2%C3%97Constant_termm_times_n_textCoefficient_of_x2_times_textConstant_termThen_rewrite_bxbx_as_mxnxmx_nx\"><\/span><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"436\" data-end=\"471\"><span class=\"katex-display\"><span class=\"katex\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><annotation encoding=\"application\/x-tex\"><\/annotation><\/semantics><\/math>and also<br \/>\n<\/span><\/span><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>m<\/mi><mo>\u00d7<\/mo><mi>n<\/mi><mo>=<\/mo><mo stretchy=\"false\">(<\/mo><mtext>Coefficient\u00a0of\u00a0<\/mtext><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo stretchy=\"false\">)<\/mo><mo>\u00d7<\/mo><mo stretchy=\"false\">(<\/mo><mtext>Constant\u00a0term<\/mtext><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">m \\times n = (\\text{Coefficient of } x^2) \\times (\\text{Constant term})<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/strong>Then rewrite <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>b<\/mi><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">bx<\/annotation><\/semantics><\/math><\/span><\/span> as <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>m<\/mi><mi>x<\/mi><mo>+<\/mo><mi>n<\/mi><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">mx + nx<\/annotation><\/semantics><\/math><\/span><\/span>,<\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h2 data-start=\"430\" data-end=\"471\"><span class=\"ez-toc-section\" id=\"Now_group_and_factorise\"><\/span><span style=\"font-size: 24px; color: #000000;\">Now group and factorise.<\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<hr data-start=\"1586\" data-end=\"1589\" \/>\n<h3 data-start=\"1591\" data-end=\"1618\"><span class=\"ez-toc-section\" id=\"%E2%9C%B3%EF%B8%8F_Example_1_Easy\"><\/span><span style=\"font-size: 24px; color: #000000;\">\u2733\ufe0f <strong data-start=\"1598\" data-end=\"1618\">Example 1 (Easy)<\/strong><\/span><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p data-start=\"1620\" data-end=\"1646\"><span style=\"font-size: 24px; color: #000000;\">Solve <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>5<\/mn><mi>x<\/mi><mo>+<\/mo><mn>6<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x^2 + 5x + 6 = 0<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/p>\n<p data-start=\"1648\" data-end=\"1702\"><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"1648\" data-end=\"1659\">Step 1:<\/strong> Compare the Coefficients of the given qudratic equation with the standard quadratic equation<\/span><\/p>\n<h2 data-start=\"430\" data-end=\"471\"><span class=\"ez-toc-section\" id=\"ax2bxc0On_comparision_we_get\"><\/span><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"436\" data-end=\"471\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mi>b<\/mi><mi>x<\/mi><mo>+<\/mo><mi>c<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow><\/semantics><\/math><\/span><\/span><\/span><\/strong><\/span><span class=\"katex\" style=\"font-size: 24px; color: #000000;\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>On comparision we get:<\/mi><\/mrow><\/semantics><\/math><\/span><\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p data-start=\"1648\" data-end=\"1702\"><span class=\"katex\" style=\"font-size: 24px; color: #000000;\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><mo>=<\/mo><mn>1<\/mn><mo separator=\"true\">,<\/mo><mi>b<\/mi><mo>=<\/mo><mn>5<\/mn><mo separator=\"true\">,<\/mo><mi>c<\/mi><mo>=<\/mo><mn>6<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">a = 1, b = 5, c = 6<\/annotation><\/semantics><\/math><\/span><\/span><\/p>\n<p data-start=\"1648\" data-end=\"1702\"><span style=\"font-size: 24px; color: #000000;\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><annotation encoding=\"application\/x-tex\"><\/annotation><\/semantics><\/math><strong data-start=\"1704\" data-end=\"1715\">Step 2:<\/strong> Multiply <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><mo>\u00d7<\/mo><mi>c<\/mi><mo>=<\/mo><mn>1<\/mn><mo>\u00d7<\/mo><mn>6<\/mn><mo>=<\/mo><mn>6<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">a \\times c = 1 \\times 6 = 6<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/span><\/p>\n<p data-start=\"1758\" data-end=\"1845\"><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"1758\" data-end=\"1769\">Step 3:<\/strong> Find <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>m<\/mi><mo separator=\"true\">,<\/mo><mi>n<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">m, n<\/annotation><\/semantics><\/math><\/span><\/span> such that:<\/span><br data-start=\"1784\" data-end=\"1787\" \/><span class=\"katex\" style=\"font-size: 24px; color: #000000;\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>m<\/mi><mo>+<\/mo><mi>n<\/mi><mo>=<\/mo><mn>5<\/mn><mo separator=\"true\">,<\/mo><mspace width=\"1em\"><\/mspace><mi>m<\/mi><mo>\u00d7<\/mo><mi>n<\/mi><mo>=<\/mo><mn>6<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">m + n = 5, \\quad m \\times n = 6<\/annotation><\/semantics><\/math><\/span><\/span><br data-start=\"1822\" data-end=\"1825\" \/><span style=\"font-size: 24px; color: #000000;\">So, <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>m<\/mi><mo>=<\/mo><mn>2<\/mn><mo separator=\"true\">,<\/mo><mi>n<\/mi><mo>=<\/mo><mn>3<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">m = 2, n = 3<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/span><\/p>\n<p data-start=\"1847\" data-end=\"1881\"><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"1847\" data-end=\"1858\">Step 4:<\/strong> Rewrite middle term:<\/span><\/p>\n<h2 data-start=\"430\" data-end=\"471\"><span class=\"ez-toc-section\" id=\"x22x3x60x2_2x_3x_6_0\"><\/span><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"436\" data-end=\"471\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>6<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x^2 + 2x + 3x + 6 = 0<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/strong><\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p data-start=\"1911\" data-end=\"1945\"><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"1911\" data-end=\"1922\">Step 5:<\/strong> Group and factorise:<\/span><\/p>\n<h2 data-start=\"430\" data-end=\"471\"><span class=\"ez-toc-section\" id=\"xx23x20xx_2_3x_2_0_x2x30x_2x_3_0Step_6\"><\/span><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"436\" data-end=\"471\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>3<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x(x + 2) + 3(x + 2) = 0<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/span><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">(x + 2)(x + 3) = 0<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/strong><strong data-start=\"2003\" data-end=\"2014\">Step 6:<\/strong><\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h2 data-start=\"430\" data-end=\"471\"><span class=\"ez-toc-section\" id=\"x20%E2%87%92x%E2%88%922andx30%E2%87%92x%E2%88%923x_2_0_Rightarrow_x_-2_quad_textand_quad_x_3_0_Rightarrow_x_-3\"><\/span><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"436\" data-end=\"471\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo>=<\/mo><mn>0<\/mn><mo>\u21d2<\/mo><mi>x<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mn>2<\/mn><mspace width=\"1em\"><\/mspace><mtext>and<\/mtext><mspace width=\"1em\"><\/mspace><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mo>=<\/mo><mn>0<\/mn><mo>\u21d2<\/mo><mi>x<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mn>3<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x + 2 = 0 \\Rightarrow x = -2 \\quad \\text{and} \\quad x + 3 = 0 \\Rightarrow x = -3<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/span><\/strong><\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p data-start=\"2109\" data-end=\"2136\"><span style=\"font-size: 24px; color: #000000;\">\u2705 <strong data-start=\"2111\" data-end=\"2121\">Roots:<\/strong> <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mn>2<\/mn><mo separator=\"true\">,<\/mo><mo>\u2212<\/mo><mn>3<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x = -2, -3<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/span><\/p>\n<hr data-start=\"2138\" data-end=\"2141\" \/>\n<h3 data-start=\"2143\" data-end=\"2174\"><span class=\"ez-toc-section\" id=\"%E2%9C%B3%EF%B8%8F_Example_2_Moderate\"><\/span><span style=\"font-size: 24px; color: #000000;\">\u2733\ufe0f <strong data-start=\"2150\" data-end=\"2174\">Example 2 (Moderate)<\/strong><\/span><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p data-start=\"2176\" data-end=\"2203\"><span style=\"font-size: 24px; color: #000000;\">Solve <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>2<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>5<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">2x^2 + 5x + 2 = 0<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/span><\/p>\n<p data-start=\"2205\" data-end=\"2240\"><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"2205\" data-end=\"2216\">Step 1:<\/strong><\/span><\/p>\n<p data-start=\"1648\" data-end=\"1702\"><span style=\"font-size: 24px; color: #000000;\">Compare the Coefficients of the given qudratic equation with the standard quadratic equation<\/span><\/p>\n<h2 data-start=\"430\" data-end=\"471\"><span class=\"ez-toc-section\" id=\"ax2bxc0On_comparision_we_get-2\"><\/span><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"436\" data-end=\"471\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mi>b<\/mi><mi>x<\/mi><mo>+<\/mo><mi>c<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow><\/semantics><\/math><\/span><\/span><\/span><\/strong><\/span><span class=\"katex\" style=\"font-size: 24px; color: #000000;\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>On comparision we get:<\/mi><\/mrow><\/semantics><\/math><\/span><\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p data-start=\"2205\" data-end=\"2240\"><span style=\"font-size: 24px; color: #000000;\"> <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><mo>=<\/mo><mn>2<\/mn><mo separator=\"true\">,<\/mo><mi>b<\/mi><mo>=<\/mo><mn>5<\/mn><mo separator=\"true\">,<\/mo><mi>c<\/mi><mo>=<\/mo><mn>2<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">a = 2, b = 5, c = 2<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/span><\/p>\n<p data-start=\"2242\" data-end=\"2272\"><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"2242\" data-end=\"2253\">Step 2:<\/strong> <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><mo>\u00d7<\/mo><mi>c<\/mi><mo>=<\/mo><mn>4<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">a \\times c = 4<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/span><\/p>\n<p data-start=\"2274\" data-end=\"2357\"><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"2274\" data-end=\"2285\">Step 3:<\/strong> Find <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>m<\/mi><mo separator=\"true\">,<\/mo><mi>n<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">m, n<\/annotation><\/semantics><\/math><\/span><\/span> such that:<\/span><br data-start=\"2300\" data-end=\"2303\" \/><span class=\"katex\" style=\"font-size: 24px; color: #000000;\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>m<\/mi><mo>+<\/mo><mi>n<\/mi><mo>=<\/mo><mn>5<\/mn><mo separator=\"true\">,<\/mo><mi>m<\/mi><mo>\u00d7<\/mo><mi>n<\/mi><mo>=<\/mo><mn>4<\/mn><\/mrow><\/semantics><\/math><\/span><\/span><\/p>\n<p data-start=\"2274\" data-end=\"2357\"><span class=\"katex\" style=\"font-size: 24px; color: #000000;\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo>\u21d2<\/mo><mi>m<\/mi><mo>=<\/mo><mn>4<\/mn><mo separator=\"true\">,<\/mo><mi>n<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">m + n = 5, m \\times n = 4 \\Rightarrow m = 4, n = 1<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/p>\n<p data-start=\"2359\" data-end=\"2393\"><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"2359\" data-end=\"2370\">Step 4:<\/strong> Rewrite middle term:<\/span><\/p>\n<h2 data-start=\"430\" data-end=\"471\"><span class=\"ez-toc-section\" id=\"2x24xx202x2_4x_x_2_0\"><\/span><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"436\" data-end=\"471\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>2<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">2x^2 + 4x + x + 2 = 0<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/strong><\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p data-start=\"2423\" data-end=\"2457\"><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"2423\" data-end=\"2434\">Step 5:<\/strong> Group and factorise:<\/span><\/p>\n<h2 data-start=\"430\" data-end=\"471\"><span class=\"ez-toc-section\" id=\"2xx21x202xx_2_1x_2_0_x22x10x_22x_1_0Step_6\"><\/span><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"436\" data-end=\"471\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>2<\/mn><mi>x<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">2x(x + 2) + 1(x + 2) = 0<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/span><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">(x + 2)(2x + 1) = 0<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/strong><strong data-start=\"2517\" data-end=\"2528\">Step 6:<\/strong><\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h2 data-start=\"430\" data-end=\"471\"><span class=\"ez-toc-section\" id=\"x%E2%88%922orx%E2%88%9212x_-2_quad_textor_quad_x_-frac12\"><\/span><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"436\" data-end=\"471\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mn>2<\/mn><mspace width=\"1em\"><\/mspace><mtext>or<\/mtext><mspace width=\"1em\"><\/mspace><mi>x<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">x = -2 \\quad \\text{or} \\quad x = -\\frac{1}{2}<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/span><\/strong><\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p data-start=\"2584\" data-end=\"2621\"><span style=\"font-size: 24px; color: #000000;\">\u2705 <strong data-start=\"2586\" data-end=\"2596\">Roots:<\/strong> <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mn>2<\/mn><mo separator=\"true\">,<\/mo><mo>\u2212<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">x = -2, -\\frac{1}{2}<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<hr data-start=\"2623\" data-end=\"2626\" \/>\n<h3 data-start=\"2628\" data-end=\"2655\"><span class=\"ez-toc-section\" id=\"%E2%9C%B3%EF%B8%8F_Example_3_Hard\"><\/span><span style=\"font-size: 24px; color: #000000;\">\u2733\ufe0f <strong data-start=\"2635\" data-end=\"2655\">Example 3 (Hard)<\/strong><\/span><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p data-start=\"2657\" data-end=\"2684\"><span style=\"font-size: 24px; color: #000000;\">Solve <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>3<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>8<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">3x^2 &#8211; 8x &#8211; 3 = 0<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/span><\/p>\n<p data-start=\"2686\" data-end=\"2723\"><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"2686\" data-end=\"2697\">Step 1:<\/strong><\/span><\/p>\n<p data-start=\"1648\" data-end=\"1702\"><span style=\"font-size: 24px; color: #000000;\">Compare the Coefficients of the given qudratic equation with the standard quadratic equation<\/span><\/p>\n<h2 data-start=\"430\" data-end=\"471\"><span class=\"ez-toc-section\" id=\"ax2bxc0On_comparision_we_get-3\"><\/span><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"436\" data-end=\"471\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mi>b<\/mi><mi>x<\/mi><mo>+<\/mo><mi>c<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow><\/semantics><\/math><\/span><\/span><\/span><\/strong><\/span><span class=\"katex\" style=\"font-size: 24px; color: #000000;\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>On comparision we get:<\/mi><\/mrow><\/semantics><\/math><\/span><\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p data-start=\"2686\" data-end=\"2723\"><span style=\"font-size: 24px; color: #000000;\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><mo>=<\/mo><mn>3<\/mn><mo separator=\"true\">,<\/mo><mi>b<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mn>8<\/mn><mo separator=\"true\">,<\/mo><mi>c<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mn>3<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">a = 3, b = -8, c = -3<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/span><\/p>\n<p data-start=\"2725\" data-end=\"2756\"><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"2725\" data-end=\"2736\">Step 2:<\/strong> <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><mo>\u00d7<\/mo><mi>c<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mn>9<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">a \\times c = -9<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/span><\/p>\n<p data-start=\"2758\" data-end=\"2838\"><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"2758\" data-end=\"2769\">Step 3:<\/strong> Find <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>m<\/mi><mo separator=\"true\">,<\/mo><mi>n<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">m, n<\/annotation><\/semantics><\/math><\/span><\/span> such that:<\/span><br data-start=\"2784\" data-end=\"2787\" \/><span class=\"katex\" style=\"font-size: 24px; color: #000000;\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>m<\/mi><mo>+<\/mo><mi>n<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mn>8<\/mn><mo separator=\"true\">,<\/mo><mi>m<\/mi><mo>\u00d7<\/mo><mi>n<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mn>9<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">m + n = -8, m \\times n = -9<\/annotation><\/semantics><\/math><\/span><\/span><br data-start=\"2818\" data-end=\"2821\" \/><span class=\"katex\" style=\"font-size: 24px; color: #000000;\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>m<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mn>9<\/mn><mo separator=\"true\">,<\/mo><mi>n<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">m = -9, n = 1<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/p>\n<p data-start=\"2840\" data-end=\"2874\"><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"2840\" data-end=\"2851\">Step 4:<\/strong> Rewrite middle term:<\/span><\/p>\n<h2 data-start=\"430\" data-end=\"471\"><span class=\"ez-toc-section\" id=\"3x2%E2%88%929xx%E2%88%92303x2_%E2%80%93_9x_x_%E2%80%93_3_0\"><\/span><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"436\" data-end=\"471\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>3<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>9<\/mn><mi>x<\/mi><mo>+<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">3x^2 &#8211; 9x + x &#8211; 3 = 0<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/strong><\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p data-start=\"2904\" data-end=\"2938\"><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"2904\" data-end=\"2915\">Step 5:<\/strong> Group and factorise:<\/span><\/p>\n<h2 data-start=\"430\" data-end=\"471\"><span class=\"ez-toc-section\" id=\"3xx%E2%88%9231x%E2%88%92303xx_%E2%80%93_3_1x_%E2%80%93_3_0\"><\/span><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"436\" data-end=\"471\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>3<\/mn><mi>x<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">3x(x &#8211; 3) + 1(x &#8211; 3) = 0<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/strong><\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h2 data-start=\"430\" data-end=\"471\"><span class=\"ez-toc-section\" id=\"x%E2%88%9233x10x_%E2%80%93_33x_1_0\"><\/span><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"436\" data-end=\"471\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><annotation encoding=\"application\/x-tex\"><\/annotation><\/semantics><\/math><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">(x &#8211; 3)(3x + 1) = 0<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/strong><\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p data-start=\"2998\" data-end=\"3011\"><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"2998\" data-end=\"3009\">Step 6:<\/strong><\/span><\/p>\n<h2 data-start=\"430\" data-end=\"471\"><span class=\"ez-toc-section\" id=\"x3x%E2%88%9213x_3_quad_x_-frac13\"><\/span><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"436\" data-end=\"471\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mn>3<\/mn><mo separator=\"true\">,<\/mo><mspace width=\"1em\"><\/mspace><mi>x<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mfrac><mn>1<\/mn><mn>3<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">x = 3, \\quad x = -\\frac{1}{3}<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/strong><\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p data-start=\"3049\" data-end=\"3085\"><span style=\"font-size: 24px; color: #000000;\">\u2705 <strong data-start=\"3051\" data-end=\"3061\">Roots:<\/strong> <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mn>3<\/mn><mo separator=\"true\">,<\/mo><mo>\u2212<\/mo><mfrac><mn>1<\/mn><mn>3<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">x = 3, -\\frac{1}{3}<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<h2 data-start=\"430\" data-end=\"471\"><span class=\"ez-toc-section\" id=\"Method_2_Completing_the_Square\"><\/span><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"436\" data-end=\"471\">Method 2: Completing the Square<\/strong><\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<hr data-start=\"473\" data-end=\"476\" \/>\n<h3 data-start=\"478\" data-end=\"496\"><span class=\"ez-toc-section\" id=\"%F0%9F%A7%AD_Concept\"><\/span><span style=\"font-size: 24px; color: #000000;\">\ud83e\udded <strong data-start=\"485\" data-end=\"496\">Concept<\/strong><\/span><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p data-start=\"498\" data-end=\"520\"><span style=\"font-size: 24px; color: #000000;\">A quadratic equation<\/span><\/p>\n<p><span class=\"katex-display\" style=\"font-size: 24px; color: #000000;\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mi>b<\/mi><mi>x<\/mi><mo>+<\/mo><mi>c<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">ax^2 + bx + c = 0<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/p>\n<p data-start=\"545\" data-end=\"614\"><span style=\"font-size: 24px; color: #000000;\">can be solved by expressing it as a <strong data-start=\"581\" data-end=\"599\">perfect square<\/strong> of a binomial.<\/span><\/p>\n<hr data-start=\"616\" data-end=\"619\" \/>\n<h3 data-start=\"621\" data-end=\"653\"><span class=\"ez-toc-section\" id=\"%E2%9C%85_Step-by-Step_Procedure\"><\/span><span style=\"font-size: 24px; color: #000000;\">\u2705 <strong data-start=\"627\" data-end=\"653\">Step-by-Step Procedure<\/strong><\/span><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p data-start=\"655\" data-end=\"699\"><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"655\" data-end=\"666\">Step 1:<\/strong> Bring the equation to the form<\/span><\/p>\n<p><span class=\"katex-display\" style=\"font-size: 24px; color: #000000;\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mi>b<\/mi><mi>x<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mi>c<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">ax^2 + bx = -c<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/p>\n<p data-start=\"722\" data-end=\"808\"><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"722\" data-end=\"733\">Step 2:<\/strong> Divide each term by <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><\/span><\/span><\/span> to make the coefficient of <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>x<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">x^2<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> equal to 1.<\/span><\/p>\n<p><span style=\"font-size: 24px; color: #000000;\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mfrac><mi>b<\/mi><mi>a<\/mi><\/mfrac><mi>x<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mfrac><mi>c<\/mi><mi>a<\/mi><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\"><\/annotation><\/semantics><\/math><\/span><\/span><\/span><strong data-start=\"851\" data-end=\"862\">Step 3:<\/strong> Add <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mrow><mo fence=\"true\">(<\/mo><mfrac><mi>b<\/mi><mrow><mn>2<\/mn><mi>a<\/mi><\/mrow><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\left(\\frac{b}{2a}\\right)^2<\/annotation><\/semantics><\/math><\/span><\/span>to both sides to make a perfect square.<\/span><\/p>\n<p data-start=\"940\" data-end=\"980\"><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"940\" data-end=\"951\">Step 4:<\/strong> Write the LHS as a square:<\/span><\/p>\n<p><span style=\"font-size: 24px; color: #000000;\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mrow><mo fence=\"true\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mfrac><mi>b<\/mi><mrow><mn>2<\/mn><mi>a<\/mi><\/mrow><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mn>2<\/mn><\/msup><mo>=<\/mo><mfrac><mrow><msup><mi>b<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><\/mrow><mrow><mn>4<\/mn><msup><mi>a<\/mi><mn>2<\/mn><\/msup><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\"><\/annotation><\/semantics><\/math><\/span><\/span><\/span><strong data-start=\"1045\" data-end=\"1056\">Step 5:<\/strong> Take square root on both sides.<\/span><\/p>\n<p data-start=\"1090\" data-end=\"1118\"><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"1090\" data-end=\"1101\">Step 6:<\/strong> Solve for <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">x<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><\/span><\/span><\/span>.<\/span><\/p>\n<hr data-start=\"1120\" data-end=\"1123\" \/>\n<h3 data-start=\"1125\" data-end=\"1167\"><span class=\"ez-toc-section\" id=\"%F0%9F%A7%A9_Rule_Box_Completing_the_Square\"><\/span><span style=\"font-size: 24px; color: #000000;\">\ud83e\udde9 <strong data-start=\"1132\" data-end=\"1167\">Rule Box: Completing the Square<\/strong><\/span><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p data-start=\"1169\" data-end=\"1260\"><span style=\"font-size: 24px; color: #000000;\">To complete the square for <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mi>p<\/mi><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">x^2 + px<\/annotation><\/semantics><\/math><\/span><\/span>,<\/span><br data-start=\"1209\" data-end=\"1212\" \/><span style=\"font-size: 24px; color: #000000;\">add and subtract <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mrow><mo fence=\"true\">(<\/mo><mfrac><mi>p<\/mi><mn>2<\/mn><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\left(\\frac{p}{2}\\right)^2<\/annotation><\/semantics><\/math><\/span><\/span>.<\/span><\/p>\n<p><span class=\"katex-display\" style=\"font-size: 24px; color: #000000;\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mi>p<\/mi><mi>x<\/mi><mo>+<\/mo><msup><mrow><mo fence=\"true\">(<\/mo><mfrac><mi>p<\/mi><mn>2<\/mn><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mn>2<\/mn><\/msup><mo>=<\/mo><msup><mrow><mo fence=\"true\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mfrac><mi>p<\/mi><mn>2<\/mn><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">x^2 + px + \\left(\\frac{p}{2}\\right)^2 = \\left(x + \\frac{p}{2}\\right)^2<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/span><\/p>\n<hr data-start=\"1340\" data-end=\"1343\" \/>\n<h3 data-start=\"1345\" data-end=\"1374\"><span class=\"ez-toc-section\" id=\"%E2%9C%B3%EF%B8%8F_Example_1_Easy-2\"><\/span><span style=\"font-size: 24px; color: #000000;\">\u2733\ufe0f <strong data-start=\"1352\" data-end=\"1372\">Example 1 (Easy)<\/strong><\/span><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p data-start=\"1375\" data-end=\"1401\"><span style=\"font-size: 24px; color: #000000;\">Solve <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>6<\/mn><mi>x<\/mi><mo>+<\/mo><mn>5<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x^2 + 6x + 5 = 0<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/span><\/p>\n<p data-start=\"1403\" data-end=\"1438\"><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"1403\" data-end=\"1414\">Step 1:<\/strong> Move constant to RHS:<\/span><\/p>\n<p><span class=\"katex-display\" style=\"font-size: 24px; color: #000000;\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>6<\/mn><mi>x<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mn>5<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x^2 + 6x = -5<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/p>\n<p data-start=\"1460\" data-end=\"1527\"><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"1460\" data-end=\"1471\">Step 2:<\/strong> Add <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mrow><mo fence=\"true\">(<\/mo><mfrac><mn>6<\/mn><mn>2<\/mn><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mn>2<\/mn><\/msup><mo>=<\/mo><mn>9<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\left(\\frac{6}{2}\\right)^2 = 9<\/annotation><\/semantics><\/math><\/span><\/span> to both sides:<\/span><\/p>\n<p><span class=\"katex-display\" style=\"font-size: 24px; color: #000000;\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>6<\/mn><mi>x<\/mi><mo>+<\/mo><mn>9<\/mn><mo>=<\/mo><mo>\u2212<\/mo><mn>5<\/mn><mo>+<\/mo><mn>9<\/mn><\/mrow><\/semantics><\/math><\/span><\/span><\/span><\/p>\n<p data-start=\"1557\" data-end=\"1595\"><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"1557\" data-end=\"1568\">Step 3:<\/strong> Write as perfect square:<\/span><\/p>\n<p><span class=\"katex-display\" style=\"font-size: 24px; color: #000000;\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><mo>=<\/mo><mn>4<\/mn><\/mrow><\/semantics><\/math><\/span><\/span><\/span><\/p>\n<p data-start=\"1617\" data-end=\"1649\"><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"1617\" data-end=\"1628\">Step 4:<\/strong> Take square roots:<\/span><\/p>\n<p><span class=\"katex-display\" style=\"font-size: 24px; color: #000000;\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mo>=<\/mo><mo>\u00b1<\/mo><mn>2<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x + 3 = \\pm 2<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/p>\n<p data-start=\"1671\" data-end=\"1684\"><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"1671\" data-end=\"1682\">Step 5:<\/strong><\/span><\/p>\n<p><span class=\"katex-display\" style=\"font-size: 24px; color: #000000;\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mn>3<\/mn><mo>\u00b1<\/mo><mn>2<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x = -3 \\pm 2<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/p>\n<p data-start=\"1705\" data-end=\"1732\"><span style=\"font-size: 24px; color: #000000;\">\u2705 <strong data-start=\"1707\" data-end=\"1717\">Roots:<\/strong> <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mn>1<\/mn><mo separator=\"true\">,<\/mo><mo>\u2212<\/mo><mn>5<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x = -1, -5<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/span><\/p>\n<hr data-start=\"1734\" data-end=\"1737\" \/>\n<h3 data-start=\"1739\" data-end=\"1772\"><span class=\"ez-toc-section\" id=\"%E2%9C%B3%EF%B8%8F_Example_2_Moderate-2\"><\/span><span style=\"font-size: 24px; color: #000000;\">\u2733\ufe0f <strong data-start=\"1746\" data-end=\"1770\">Example 2 (Moderate)<\/strong><\/span><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p data-start=\"1773\" data-end=\"1800\"><span style=\"font-size: 24px; color: #000000;\">Solve <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>2<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">2x^2 + 3x &#8211; 2 = 0<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/p>\n<p data-start=\"1802\" data-end=\"1828\"><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"1802\" data-end=\"1813\">Step 1:<\/strong> Divide by 2:<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mfrac><mn>3<\/mn><mn>2<\/mn><\/mfrac><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x^2 + \\frac{3}{2}x &#8211; 1 = 0<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/span><\/p>\n<p data-start=\"1863\" data-end=\"1891\"><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"1863\" data-end=\"1874\">Step 2:<\/strong> Move constant:<\/span><\/p>\n<p><span class=\"katex-display\" style=\"font-size: 24px; color: #000000;\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mfrac><mn>3<\/mn><mn>2<\/mn><\/mfrac><mi>x<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><\/semantics><\/math><\/span><\/span><\/span><\/p>\n<p data-start=\"1922\" data-end=\"1986\"><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"1922\" data-end=\"1933\">Step 3:<\/strong> Add <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mrow><mo fence=\"true\">(<\/mo><mfrac><mn>3<\/mn><mn>4<\/mn><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mn>2<\/mn><\/msup><mo>=<\/mo><mfrac><mn>9<\/mn><mn>16<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\left(\\frac{3}{4}\\right)^2 = \\frac{9}{16}<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>:<\/span><\/p>\n<p><span class=\"katex-display\" style=\"font-size: 24px; color: #000000;\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mfrac><mn>3<\/mn><mn>2<\/mn><\/mfrac><mi>x<\/mi><mo>+<\/mo><mfrac><mn>9<\/mn><mn>16<\/mn><\/mfrac><mo>=<\/mo><mn>1<\/mn><mo>+<\/mo><mfrac><mn>9<\/mn><mn>16<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">x^2 + \\frac{3}{2}x + \\frac{9}{16} = 1 + \\frac{9}{16}<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/p>\n<p data-start=\"2047\" data-end=\"2077\"><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"2047\" data-end=\"2058\">Step 4:<\/strong> Write as square:<\/span><\/p>\n<p><span class=\"katex-display\" style=\"font-size: 24px; color: #000000;\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mrow><mo fence=\"true\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mfrac><mn>3<\/mn><mn>4<\/mn><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mn>2<\/mn><\/msup><mo>=<\/mo><mfrac><mn>25<\/mn><mn>16<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\left(x + \\frac{3}{4}\\right)^2 = \\frac{25}{16}<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/p>\n<p data-start=\"2132\" data-end=\"2164\"><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"2132\" data-end=\"2143\">Step 5:<\/strong> Take square roots:<\/span><\/p>\n<p><span class=\"katex-display\" style=\"font-size: 24px; color: #000000;\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>+<\/mo><mfrac><mn>3<\/mn><mn>4<\/mn><\/mfrac><mo>=<\/mo><mo>\u00b1<\/mo><mfrac><mn>5<\/mn><mn>4<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">x + \\frac{3}{4} = \\pm \\frac{5}{4}<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/p>\n<p data-start=\"2206\" data-end=\"2219\"><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"2206\" data-end=\"2217\">Step 6:<\/strong><\/span><\/p>\n<p><span class=\"katex-display\" style=\"font-size: 24px; color: #000000;\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mfrac><mrow><mo>\u2212<\/mo><mn>3<\/mn><mo>\u00b1<\/mo><mn>5<\/mn><\/mrow><mn>4<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">x = \\frac{-3 \\pm 5}{4}<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/p>\n<p data-start=\"2250\" data-end=\"2287\"><span style=\"font-size: 24px; color: #000000;\">\u2705 <strong data-start=\"2252\" data-end=\"2262\">Roots:<\/strong> <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mo separator=\"true\">,<\/mo><mo>\u2212<\/mo><mn>2<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x = \\tfrac{1}{2}, -2<\/annotation><\/semantics><\/math><\/span>.<\/span><\/span><\/p>\n<hr data-start=\"2289\" data-end=\"2292\" \/>\n<h3 data-start=\"2294\" data-end=\"2323\"><span class=\"ez-toc-section\" id=\"%E2%9C%B3%EF%B8%8F_Example_3_Hard-2\"><\/span><span style=\"font-size: 24px; color: #000000;\">\u2733\ufe0f <strong data-start=\"2301\" data-end=\"2321\">Example 3 (Hard)<\/strong><\/span><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p data-start=\"2324\" data-end=\"2352\"><span style=\"font-size: 24px; color: #000000;\">Solve <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>3<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>10<\/mn><mi>x<\/mi><mo>+<\/mo><mn>7<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">3x^2 &#8211; 10x + 7 = 0<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/span><\/p>\n<p data-start=\"2354\" data-end=\"2380\"><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"2354\" data-end=\"2365\">Step 1:<\/strong> Divide by 3:<\/span><\/p>\n<p><span class=\"katex-display\" style=\"font-size: 24px; color: #000000;\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mfrac><mn>10<\/mn><mn>3<\/mn><\/mfrac><mi>x<\/mi><mo>+<\/mo><mfrac><mn>7<\/mn><mn>3<\/mn><\/mfrac><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x^2 &#8211; \\frac{10}{3}x + \\frac{7}{3} = 0<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/p>\n<p data-start=\"2426\" data-end=\"2454\"><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"2426\" data-end=\"2437\">Step 2:<\/strong> Move constant:<\/span><\/p>\n<p><span class=\"katex-display\" style=\"font-size: 24px; color: #000000;\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mfrac><mn>10<\/mn><mn>3<\/mn><\/mfrac><mi>x<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mfrac><mn>7<\/mn><mn>3<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">x^2 &#8211; \\frac{10}{3}x = -\\frac{7}{3}<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/p>\n<p data-start=\"2497\" data-end=\"2563\"><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"2497\" data-end=\"2508\">Step 3:<\/strong> Add <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mrow><mo fence=\"true\">(<\/mo><mfrac><mrow><mo>\u2212<\/mo><mn>10<\/mn><\/mrow><mn>6<\/mn><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mn>2<\/mn><\/msup><mo>=<\/mo><mfrac><mn>25<\/mn><mn>9<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\left(\\frac{-10}{6}\\right)^2 = \\frac{25}{9}<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>:<\/span><\/p>\n<p><span class=\"katex-display\" style=\"font-size: 24px; color: #000000;\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mfrac><mn>10<\/mn><mn>3<\/mn><\/mfrac><mi>x<\/mi><mo>+<\/mo><mfrac><mn>25<\/mn><mn>9<\/mn><\/mfrac><mo>=<\/mo><mo>\u2212<\/mo><mfrac><mn>7<\/mn><mn>3<\/mn><\/mfrac><mo>+<\/mo><mfrac><mn>25<\/mn><mn>9<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">x^2 &#8211; \\frac{10}{3}x + \\frac{25}{9} = -\\frac{7}{3} + \\frac{25}{9}<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/p>\n<p data-start=\"2636\" data-end=\"2663\"><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"2636\" data-end=\"2647\">Step 4:<\/strong> Simplify RHS:<\/span><\/p>\n<p><span style=\"font-size: 24px; color: #000000;\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo>\u2212<\/mo><mfrac><mn>7<\/mn><mn>3<\/mn><\/mfrac><mo>+<\/mo><mfrac><mn>25<\/mn><mn>9<\/mn><\/mfrac><mo>=<\/mo><mfrac><mn>4<\/mn><mn>9<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">-\\frac{7}{3} + \\frac{25}{9} = \\frac{4}{9}<\/annotation><\/semantics><\/math><\/span><\/span><\/span><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mrow><mo fence=\"true\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mfrac><mn>5<\/mn><mn>3<\/mn><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mn>2<\/mn><\/msup><mo>=<\/mo><mfrac><mn>4<\/mn><mn>9<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\left(x &#8211; \\frac{5}{3}\\right)^2 = \\frac{4}{9}<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/span><\/p>\n<p data-start=\"2765\" data-end=\"2778\"><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"2765\" data-end=\"2776\">Step 5:<\/strong><\/span><\/p>\n<p><span class=\"katex-display\" style=\"font-size: 24px; color: #000000;\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>\u2212<\/mo><mfrac><mn>5<\/mn><mn>3<\/mn><\/mfrac><mo>=<\/mo><mo>\u00b1<\/mo><mfrac><mn>2<\/mn><mn>3<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">x &#8211; \\frac{5}{3} = \\pm \\frac{2}{3}<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/p>\n<p data-start=\"2820\" data-end=\"2833\"><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"2820\" data-end=\"2831\">Step 6:<\/strong><\/span><\/p>\n<p><span class=\"katex-display\" style=\"font-size: 24px; color: #000000;\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mfrac><mrow><mn>5<\/mn><mo>\u00b1<\/mo><mn>2<\/mn><\/mrow><mn>3<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">x = \\frac{5 \\pm 2}{3}<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/p>\n<p data-start=\"2863\" data-end=\"2898\"><span style=\"font-size: 24px; color: #000000;\">\u2705 <strong data-start=\"2865\" data-end=\"2875\">Roots:<\/strong> <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mfrac><mn>7<\/mn><mn>3<\/mn><\/mfrac><mo separator=\"true\">,<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x = \\frac{7}{3}, 1<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/span><\/p>\n<hr data-start=\"2900\" data-end=\"2903\" \/>\n<h2 data-start=\"2905\" data-end=\"2942\"><span class=\"ez-toc-section\" id=\"%F0%9F%94%B9_Method_3_Quadratic_Formula\"><\/span><span style=\"font-size: 24px; color: #000000;\">\ud83d\udd39 <strong data-start=\"2911\" data-end=\"2942\">Method 3: Quadratic Formula<\/strong><\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<hr data-start=\"2944\" data-end=\"2947\" \/>\n<h3 data-start=\"2949\" data-end=\"2967\"><span class=\"ez-toc-section\" id=\"%F0%9F%A7%AD_Concept-2\"><\/span><span style=\"font-size: 24px; color: #000000;\">\ud83e\udded <strong data-start=\"2956\" data-end=\"2967\">Concept<\/strong><\/span><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p data-start=\"2969\" data-end=\"2997\"><span style=\"font-size: 24px; color: #000000;\">For any quadratic equation<\/span><\/p>\n<p><span class=\"katex-display\" style=\"font-size: 24px; color: #000000;\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mi>b<\/mi><mi>x<\/mi><mo>+<\/mo><mi>c<\/mi><mo>=<\/mo><mn>0<\/mn><mo separator=\"true\">,<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">ax^2 + bx + c = 0,<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/p>\n<p data-start=\"3023\" data-end=\"3074\"><span style=\"font-size: 24px; color: #000000;\">the solution is given by the <strong data-start=\"3052\" data-end=\"3073\">quadratic formula<\/strong>:<\/span><\/p>\n<p><span class=\"katex-display\" style=\"font-size: 24px; color: #000000;\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mfrac><mrow><mo>\u2212<\/mo><mi>b<\/mi><mo>\u00b1<\/mo><msqrt><mrow><msup><mi>b<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><\/mrow><\/msqrt><\/mrow><mrow><mn>2<\/mn><mi>a<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">x = \\frac{-b \\pm \\sqrt{b^2 &#8211; 4ac}}{2a}<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/p>\n<p data-start=\"3122\" data-end=\"3129\"><span style=\"font-size: 24px; color: #000000;\">Here,<\/span><\/p>\n<p><span class=\"katex-display\" style=\"font-size: 24px; color: #000000;\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>D<\/mi><mo>=<\/mo><msup><mi>b<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">D = b^2 &#8211; 4ac<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/p>\n<p data-start=\"3150\" data-end=\"3181\"><span style=\"font-size: 24px; color: #000000;\">s called the <strong data-start=\"3164\" data-end=\"3180\">Discriminant<\/strong>.<\/span><\/p>\n<hr data-start=\"3183\" data-end=\"3186\" \/>\n<h3 data-start=\"3188\" data-end=\"3220\"><span class=\"ez-toc-section\" id=\"%E2%9C%85_Step-by-Step_Procedure-2\"><\/span><span style=\"font-size: 24px; color: #000000;\">\u2705 <strong data-start=\"3194\" data-end=\"3220\">Step-by-Step Procedure<\/strong><\/span><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p data-start=\"3222\" data-end=\"3351\"><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"3222\" data-end=\"3233\">Step 1:<\/strong> Identify <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><mo separator=\"true\">,<\/mo><mi>b<\/mi><mo separator=\"true\">,<\/mo><mi>c<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a, b, c<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><span class=\"mpunct\">,<\/span><span class=\"mord mathnormal\">b<\/span><span class=\"mpunct\">,<\/span><span class=\"mord mathnormal\">c<\/span><\/span><\/span><\/span>.<\/span><br data-start=\"3255\" data-end=\"3258\" \/><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"3258\" data-end=\"3269\">Step 2:<\/strong> Compute discriminant <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>D<\/mi><mo>=<\/mo><msup><mi>b<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">D = b^2 &#8211; 4ac<\/annotation><\/semantics><\/math><\/span><\/span>.<\/span><br data-start=\"3309\" data-end=\"3312\" \/><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"3312\" data-end=\"3323\">Step 3:<\/strong> Substitute in the formula<\/span><\/p>\n<p><span class=\"katex-display\" style=\"font-size: 24px; color: #000000;\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mfrac><mrow><mo>\u2212<\/mo><mi>b<\/mi><mo>\u00b1<\/mo><msqrt><mi>D<\/mi><\/msqrt><\/mrow><mrow><mn>2<\/mn><mi>a<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">x = \\frac{-b \\pm \\sqrt{D}}{2a}<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/p>\n<p data-start=\"3389\" data-end=\"3427\"><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"3389\" data-end=\"3400\">Step 4:<\/strong> Simplify to get the roots.<\/span><\/p>\n<hr data-start=\"3429\" data-end=\"3432\" \/>\n<h3 data-start=\"3434\" data-end=\"3472\"><span class=\"ez-toc-section\" id=\"%F0%9F%A7%A9_Rule_Box_Quadratic_Formula\"><\/span><span style=\"font-size: 24px; color: #000000;\">\ud83e\udde9 <strong data-start=\"3441\" data-end=\"3472\">Rule Box: Quadratic Formula<\/strong><\/span><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><span style=\"font-size: 24px; color: #000000;\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mfrac><mrow><mo>\u2212<\/mo><mi>b<\/mi><mo>\u00b1<\/mo><msqrt><mrow><msup><mi>b<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><\/mrow><\/msqrt><\/mrow><mrow><mn>2<\/mn><mi>a<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">x = \\frac{-b \\pm \\sqrt{b^2 &#8211; 4ac}}{2a}<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>Let <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>D<\/mi><mo>=<\/mo><msup><mi>b<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">D = b^2 &#8211; 4ac<\/annotation><\/semantics><\/math><\/span><\/span>:<\/span><\/p>\n<div class=\"_tableContainer_1rjym_1\">\n<div class=\"group _tableWrapper_1rjym_13 flex w-fit flex-col-reverse\" tabindex=\"-1\">\n<table class=\"w-fit min-w-(--thread-content-width)\" data-start=\"3544\" data-end=\"3723\">\n<thead data-start=\"3544\" data-end=\"3575\">\n<tr data-start=\"3544\" data-end=\"3575\">\n<th data-start=\"3544\" data-end=\"3556\" data-col-size=\"sm\"><span style=\"font-size: 24px; color: #000000;\">Condition<\/span><\/th>\n<th data-start=\"3556\" data-end=\"3575\" data-col-size=\"sm\"><span style=\"font-size: 24px; color: #000000;\">Nature of Roots<\/span><\/th>\n<\/tr>\n<\/thead>\n<tbody data-start=\"3609\" data-end=\"3723\">\n<tr data-start=\"3609\" data-end=\"3648\">\n<td data-start=\"3609\" data-end=\"3621\" data-col-size=\"sm\"><span class=\"katex\" style=\"font-size: 24px; color: #000000;\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>D<\/mi><mo>&gt;<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">D &gt; 0<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/td>\n<td data-start=\"3621\" data-end=\"3648\" data-col-size=\"sm\"><span style=\"font-size: 24px; color: #000000;\">Two distinct real roots<\/span><\/td>\n<\/tr>\n<tr data-start=\"3649\" data-end=\"3681\">\n<td data-start=\"3649\" data-end=\"3661\" data-col-size=\"sm\"><span class=\"katex\" style=\"font-size: 24px; color: #000000;\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>D<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">D = 0<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/td>\n<td data-start=\"3661\" data-end=\"3681\" data-col-size=\"sm\"><span style=\"font-size: 24px; color: #000000;\">Equal real roots<\/span><\/td>\n<\/tr>\n<tr data-start=\"3682\" data-end=\"3723\">\n<td data-start=\"3682\" data-end=\"3694\" data-col-size=\"sm\"><span class=\"katex\" style=\"font-size: 24px; color: #000000;\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>D<\/mi><mo>&lt;<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">D &lt; 0<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/td>\n<td data-start=\"3694\" data-end=\"3723\" data-col-size=\"sm\"><span style=\"font-size: 24px; color: #000000;\">No real roots (imaginary)<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<hr data-start=\"3725\" data-end=\"3728\" \/>\n<h3 data-start=\"3730\" data-end=\"3759\"><span class=\"ez-toc-section\" id=\"%E2%9C%B3%EF%B8%8F_Example_1_Easy-3\"><\/span><span style=\"font-size: 24px; color: #000000;\">\u2733\ufe0f <strong data-start=\"3737\" data-end=\"3757\">Example 1 (Easy)<\/strong><\/span><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p data-start=\"3760\" data-end=\"3786\"><span style=\"font-size: 24px; color: #000000;\">Solve <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>5<\/mn><mi>x<\/mi><mo>+<\/mo><mn>6<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x^2 &#8211; 5x + 6 = 0<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/span><\/p>\n<p data-start=\"3788\" data-end=\"3824\"><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"3788\" data-end=\"3799\">Step 1:<\/strong> <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><mo>=<\/mo><mn>1<\/mn><mo separator=\"true\">,<\/mo><mi>b<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mn>5<\/mn><mo separator=\"true\">,<\/mo><mi>c<\/mi><mo>=<\/mo><mn>6<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">a = 1, b = -5, c = 6<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/span><\/p>\n<p data-start=\"3826\" data-end=\"3876\"><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"3826\" data-end=\"3837\">Step 2:<\/strong> <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>D<\/mi><mo>=<\/mo><mo stretchy=\"false\">(<\/mo><mo>\u2212<\/mo><mn>5<\/mn><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>4<\/mn><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mn>6<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>25<\/mn><mo>\u2212<\/mo><mn>24<\/mn><mo>=<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">D = (-5)^2 &#8211; 4(1)(6) = 25 &#8211; 24 = 1<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/span><\/p>\n<p data-start=\"3878\" data-end=\"3891\"><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"3878\" data-end=\"3889\">Step 3:<\/strong><\/span><\/p>\n<p><span class=\"katex-display\" style=\"font-size: 24px; color: #000000;\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mfrac><mrow><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mo>\u2212<\/mo><mn>5<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u00b1<\/mo><msqrt><mn>1<\/mn><\/msqrt><\/mrow><mrow><mn>2<\/mn><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mn>5<\/mn><mo>\u00b1<\/mo><mn>1<\/mn><\/mrow><mn>2<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">x = \\frac{-(-5) \\pm \\sqrt{1}}{2(1)} = \\frac{5 \\pm 1}{2}<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/p>\n<p data-start=\"3955\" data-end=\"3968\"><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"3955\" data-end=\"3966\">Step 4:<\/strong><\/span><\/p>\n<p><span class=\"katex-display\" style=\"font-size: 24px; color: #000000;\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mn>3<\/mn><mtext>\u00a0or\u00a0<\/mtext><mn>2<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x = 3 \\text{ or } 2<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/p>\n<p data-start=\"3996\" data-end=\"4021\"><span style=\"font-size: 24px; color: #000000;\">\u2705 <strong data-start=\"3998\" data-end=\"4008\">Roots:<\/strong> <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mn>2<\/mn><mo separator=\"true\">,<\/mo><mn>3<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x = 2, 3<\/annotation><\/semantics><\/math><\/span>.<\/span><\/span><\/p>\n<hr data-start=\"4023\" data-end=\"4026\" \/>\n<h3 data-start=\"4028\" data-end=\"4061\"><span class=\"ez-toc-section\" id=\"%E2%9C%B3%EF%B8%8F_Example_2_Moderate-3\"><\/span><span style=\"font-size: 24px; color: #000000;\">\u2733\ufe0f <strong data-start=\"4035\" data-end=\"4059\">Example 2 (Moderate)<\/strong><\/span><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p data-start=\"4062\" data-end=\"4089\"><span style=\"font-size: 24px; color: #000000;\">Solve <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>2<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">2x^2 + 3x &#8211; 2 = 0<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/span><\/p>\n<p data-start=\"4091\" data-end=\"4127\"><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"4091\" data-end=\"4102\">Step 1:<\/strong> <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><mo>=<\/mo><mn>2<\/mn><mo separator=\"true\">,<\/mo><mi>b<\/mi><mo>=<\/mo><mn>3<\/mn><mo separator=\"true\">,<\/mo><mi>c<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mn>2<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">a = 2, b = 3, c = -2<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/span><\/p>\n<p data-start=\"4129\" data-end=\"4177\"><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"4129\" data-end=\"4140\">Step 2:<\/strong> <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>D<\/mi><mo>=<\/mo><msup><mn>3<\/mn><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>4<\/mn><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mo>\u2212<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>9<\/mn><mo>+<\/mo><mn>16<\/mn><mo>=<\/mo><mn>25<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">D = 3^2 &#8211; 4(2)(-2) = 9 + 16 = 25<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/span><\/p>\n<p data-start=\"4179\" data-end=\"4192\"><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"4179\" data-end=\"4190\">Step 3:<\/strong><\/span><\/p>\n<p><span class=\"katex-display\" style=\"font-size: 24px; color: #000000;\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mfrac><mrow><mo>\u2212<\/mo><mn>3<\/mn><mo>\u00b1<\/mo><msqrt><mn>25<\/mn><\/msqrt><\/mrow><mn>4<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">x = \\frac{-3 \\pm \\sqrt{25}}{4}<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/p>\n<p data-start=\"4231\" data-end=\"4244\"><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"4231\" data-end=\"4242\">Step 4:<\/strong><\/span><\/p>\n<p><span class=\"katex-display\" style=\"font-size: 24px; color: #000000;\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mfrac><mrow><mo>\u2212<\/mo><mn>3<\/mn><mo>\u00b1<\/mo><mn>5<\/mn><\/mrow><mn>4<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">x = \\frac{-3 \\pm 5}{4}<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/p>\n<p data-start=\"4275\" data-end=\"4288\"><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"4275\" data-end=\"4286\">Step 5:<\/strong><\/span><\/p>\n<p><span class=\"katex-display\" style=\"font-size: 24px; color: #000000;\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mo separator=\"true\">,<\/mo><mo>\u2212<\/mo><mn>2<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x = \\frac{1}{2}, -2<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/p>\n<p data-start=\"4316\" data-end=\"4353\"><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"4318\" data-end=\"4328\">Roots:<\/strong> <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mo separator=\"true\">,<\/mo><mo>\u2212<\/mo><mn>2<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x = \\tfrac{1}{2}, -2<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/span><\/p>\n<hr data-start=\"4355\" data-end=\"4358\" \/>\n<h3 data-start=\"4360\" data-end=\"4389\"><span class=\"ez-toc-section\" id=\"%E2%9C%B3%EF%B8%8F_Example_3_Hard-3\"><\/span><span style=\"font-size: 24px; color: #000000;\">\u2733\ufe0f <strong data-start=\"4367\" data-end=\"4387\">Example 3 (Hard)<\/strong><\/span><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p data-start=\"4390\" data-end=\"4417\"><span style=\"font-size: 24px; color: #000000;\">Solve <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>3<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>5<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">3x^2 &#8211; 4x + 5 = 0<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/span><\/p>\n<p data-start=\"4419\" data-end=\"4455\"><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"4419\" data-end=\"4430\">Step 1:<\/strong> <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><mo>=<\/mo><mn>3<\/mn><mo separator=\"true\">,<\/mo><mi>b<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mn>4<\/mn><mo separator=\"true\">,<\/mo><mi>c<\/mi><mo>=<\/mo><mn>5<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">a = 3, b = -4, c = 5<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/span><\/p>\n<p data-start=\"4457\" data-end=\"4509\"><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"4457\" data-end=\"4468\">Step 2:<\/strong> <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>D<\/mi><mo>=<\/mo><mo stretchy=\"false\">(<\/mo><mo>\u2212<\/mo><mn>4<\/mn><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>4<\/mn><mo stretchy=\"false\">(<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mn>5<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>16<\/mn><mo>\u2212<\/mo><mn>60<\/mn><mo>=<\/mo><mo>\u2212<\/mo><mn>44<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">D = (-4)^2 &#8211; 4(3)(5) = 16 &#8211; 60 = -44<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/span><\/p>\n<p data-start=\"4511\" data-end=\"4562\"><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"4511\" data-end=\"4522\">Step 3:<\/strong><\/span><br data-start=\"4522\" data-end=\"4525\" \/><span style=\"font-size: 24px; color: #000000;\">Since <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>D<\/mi><mo>&lt;<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">D &lt; 0<\/annotation><\/semantics><\/math><\/span><\/span>, roots are imaginary.<\/span><\/p>\n<p><span class=\"katex-display\" style=\"font-size: 24px; color: #000000;\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mfrac><mrow><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mo>\u2212<\/mo><mn>4<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u00b1<\/mo><msqrt><mrow><mo>\u2212<\/mo><mn>44<\/mn><\/mrow><\/msqrt><\/mrow><mrow><mn>2<\/mn><mo stretchy=\"false\">(<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mn>4<\/mn><mo>\u00b1<\/mo><mi>i<\/mi><msqrt><mn>44<\/mn><\/msqrt><\/mrow><mn>6<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">x = \\frac{-(-4) \\pm \\sqrt{-44}}{2(3)} = \\frac{4 \\pm i\\sqrt{44}}{6}<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/p>\n<p data-start=\"4638\" data-end=\"4647\"><span style=\"font-size: 24px; color: #000000;\">Simplify:<\/span><\/p>\n<p><span class=\"katex-display\" style=\"font-size: 24px; color: #000000;\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mfrac><mrow><mn>2<\/mn><mo>\u00b1<\/mo><mi>i<\/mi><msqrt><mn>11<\/mn><\/msqrt><\/mrow><mn>3<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">x = \\frac{2 \\pm i\\sqrt{11}}{3}<\/annotation><\/semantics><\/math><\/span><\/span><\/span><\/p>\n<p data-start=\"4686\" data-end=\"4735\"><span style=\"font-size: 24px; color: #000000;\">\u2705 <strong data-start=\"4688\" data-end=\"4708\">Nature of Roots:<\/strong> No real roots (imaginary).<\/span><\/p>\n<h2 data-start=\"3431\" data-end=\"3460\"><span class=\"ez-toc-section\" id=\"6%EF%B8%8F%E2%83%A3_Nature_of_Roots\"><\/span><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"3437\" data-end=\"3460\">6\ufe0f\u20e3 Nature of Roots<\/strong><\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p data-start=\"3462\" data-end=\"3483\"><span style=\"font-size: 24px; color: #000000;\">Let <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>D<\/mi><mo>=<\/mo><msup><mi>b<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">D = b^2 &#8211; 4ac<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/span><\/p>\n<div class=\"_tableContainer_1rjym_1\">\n<div class=\"group _tableWrapper_1rjym_13 flex w-fit flex-col-reverse\" tabindex=\"-1\">\n<table class=\"w-fit min-w-(--thread-content-width)\" data-start=\"3485\" data-end=\"3668\">\n<thead data-start=\"3485\" data-end=\"3516\">\n<tr data-start=\"3485\" data-end=\"3516\">\n<th data-start=\"3485\" data-end=\"3497\" data-col-size=\"sm\"><span style=\"font-size: 24px; color: #000000;\">Condition<\/span><\/th>\n<th data-start=\"3497\" data-end=\"3516\" data-col-size=\"sm\"><span style=\"font-size: 24px; color: #000000;\">Nature of Roots<\/span><\/th>\n<\/tr>\n<\/thead>\n<tbody data-start=\"3550\" data-end=\"3668\">\n<tr data-start=\"3550\" data-end=\"3589\">\n<td data-start=\"3550\" data-end=\"3562\" data-col-size=\"sm\"><span class=\"katex\" style=\"font-size: 24px; color: #000000;\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>D<\/mi><mo>&gt;<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">D &gt; 0<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/td>\n<td data-start=\"3562\" data-end=\"3589\" data-col-size=\"sm\"><span style=\"font-size: 24px; color: #000000;\">Two distinct real roots<\/span><\/td>\n<\/tr>\n<tr data-start=\"3590\" data-end=\"3626\">\n<td data-start=\"3590\" data-end=\"3602\" data-col-size=\"sm\"><span class=\"katex\" style=\"font-size: 24px; color: #000000;\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>D<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">D = 0<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/td>\n<td data-start=\"3602\" data-end=\"3626\" data-col-size=\"sm\"><span style=\"font-size: 24px; color: #000000;\">Two equal real roots<\/span><\/td>\n<\/tr>\n<tr data-start=\"3627\" data-end=\"3668\">\n<td data-start=\"3627\" data-end=\"3639\" data-col-size=\"sm\"><span class=\"katex\" style=\"font-size: 24px; color: #000000;\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>D<\/mi><mo>&lt;<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">D &lt; 0<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/td>\n<td data-start=\"3639\" data-end=\"3668\" data-col-size=\"sm\"><span style=\"font-size: 24px; color: #000000;\">No real roots (imaginary)<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<hr data-start=\"3670\" data-end=\"3673\" \/>\n<h2 data-start=\"3675\" data-end=\"3708\"><span class=\"ez-toc-section\" id=\"%F0%9F%94%B9_7%EF%B8%8F%E2%83%A3_Summary_of_Formulas\"><\/span><span style=\"font-size: 24px; color: #000000;\">\ud83d\udd39 <strong data-start=\"3681\" data-end=\"3708\">7\ufe0f\u20e3 Summary of Formulas<\/strong><\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<ol data-start=\"3710\" data-end=\"3960\">\n<li data-start=\"3710\" data-end=\"3751\">\n<p data-start=\"3713\" data-end=\"3751\"><span style=\"font-size: 24px; color: #000000;\">Standard Form: <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mi>b<\/mi><mi>x<\/mi><mo>+<\/mo><mi>c<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">ax^2 + bx + c = 0<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/span><\/p>\n<\/li>\n<li data-start=\"3752\" data-end=\"3805\">\n<p data-start=\"3755\" data-end=\"3805\"><span style=\"font-size: 24px; color: #000000;\">Product of roots: <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b1<\/mi><mi>\u03b2<\/mi><mo>=<\/mo><mfrac><mi>c<\/mi><mi>a<\/mi><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\alpha \\beta = \\frac{c}{a}<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<\/li>\n<li data-start=\"3806\" data-end=\"3858\">\n<p data-start=\"3809\" data-end=\"3858\"><span style=\"font-size: 24px; color: #000000;\">Sum of roots: <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b1<\/mi><mo>+<\/mo><mi>\u03b2<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mfrac><mi>b<\/mi><mi>a<\/mi><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\alpha + \\beta = -\\frac{b}{a}<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<\/li>\n<li data-start=\"3859\" data-end=\"3925\">\n<p data-start=\"3862\" data-end=\"3925\"><span style=\"font-size: 24px; color: #000000;\">Quadratic Formula: <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mfrac><mrow><mo>\u2212<\/mo><mi>b<\/mi><mo>\u00b1<\/mo><msqrt><mrow><msup><mi>b<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><\/mrow><\/msqrt><\/mrow><mrow><mn>2<\/mn><mi>a<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">x = \\frac{-b \\pm \\sqrt{b^2 &#8211; 4ac}}{2a}<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord sqrt mtight\"><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<\/li>\n<li data-start=\"3926\" data-end=\"3960\">\n<p data-start=\"3929\" data-end=\"3960\"><span style=\"font-size: 24px; color: #000000;\">Discriminant: <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>D<\/mi><mo>=<\/mo><msup><mi>b<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">D = b^2 &#8211; 4ac<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/span><\/p>\n<\/li>\n<\/ol>\n<hr data-start=\"3962\" data-end=\"3965\" \/>\n<h2 data-start=\"3967\" data-end=\"4011\"><span class=\"ez-toc-section\" id=\"%F0%9F%94%B9_8%EF%B8%8F%E2%83%A3_Practice_Questions_Worksheet\"><\/span><span style=\"font-size: 24px; color: #000000;\">\ud83d\udd39 <strong data-start=\"3973\" data-end=\"4011\">8\ufe0f\u20e3 Practice Questions (Worksheet)<\/strong><\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p data-start=\"4013\" data-end=\"4052\"><span style=\"font-size: 24px; color: #000000;\">Each question has 3 variants (a, b, c).<\/span><\/p>\n<hr data-start=\"4054\" data-end=\"4057\" \/>\n<h3 data-start=\"4059\" data-end=\"4094\"><span class=\"ez-toc-section\" id=\"Q1_Solve_by_factorisation\"><\/span><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"4063\" data-end=\"4094\">Q1. Solve by factorisation:<\/strong><\/span><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p data-start=\"4096\" data-end=\"4178\"><span style=\"font-size: 24px; color: #000000;\">(a) <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>7<\/mn><mi>x<\/mi><mo>+<\/mo><mn>10<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x^2 + 7x + 10 = 0<\/annotation><\/semantics><\/math><\/span><\/span><\/span><br data-start=\"4121\" data-end=\"4124\" \/><span style=\"font-size: 24px; color: #000000;\">(b) <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>9<\/mn><mi>x<\/mi><mo>+<\/mo><mn>14<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x^2 + 9x + 14 = 0<\/annotation><\/semantics><\/math><\/span><\/span><\/span><br data-start=\"4149\" data-end=\"4152\" \/><span style=\"font-size: 24px; color: #000000;\">(c) <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>11<\/mn><mi>x<\/mi><mo>+<\/mo><mn>24<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x^2 + 11x + 24 = 0<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/span><\/p>\n<hr data-start=\"4180\" data-end=\"4183\" \/>\n<h3 data-start=\"4185\" data-end=\"4232\"><span class=\"ez-toc-section\" id=\"Q2_Solve_by_splitting_the_middle_term\"><\/span><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"4189\" data-end=\"4232\">Q2. Solve by splitting the middle term:<\/strong><\/span><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p data-start=\"4234\" data-end=\"4315\"><span style=\"font-size: 24px; color: #000000;\">(a) <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>2<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>7<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">2x^2 + 7x + 3 = 0<\/annotation><\/semantics><\/math><\/span><\/span><\/span><br data-start=\"4259\" data-end=\"4262\" \/><span style=\"font-size: 24px; color: #000000;\">(b) <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>3<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>8<\/mn><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">3x^2 + 8x + 4 = 0<\/annotation><\/semantics><\/math><\/span><\/span><\/span><br data-start=\"4287\" data-end=\"4290\" \/><span style=\"font-size: 24px; color: #000000;\">(c) <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>4<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>9<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">4x^2 + 9x + 2 = 0<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/span><\/p>\n<hr data-start=\"4317\" data-end=\"4320\" \/>\n<h3 data-start=\"4322\" data-end=\"4376\"><span class=\"ez-toc-section\" id=\"Q3_Find_the_discriminant_and_nature_of_roots\"><\/span><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"4326\" data-end=\"4376\">Q3. Find the discriminant and nature of roots:<\/strong><\/span><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p data-start=\"4378\" data-end=\"4458\"><span style=\"font-size: 24px; color: #000000;\">(a) <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x^2 &#8211; 4x + 4 = 0<\/annotation><\/semantics><\/math><\/span><\/span><\/span><br data-start=\"4402\" data-end=\"4405\" \/><span style=\"font-size: 24px; color: #000000;\">(b) <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>6<\/mn><mi>x<\/mi><mo>+<\/mo><mn>9<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x^2 &#8211; 6x + 9 = 0<\/annotation><\/semantics><\/math><\/span><\/span><\/span><br data-start=\"4429\" data-end=\"4432\" \/><span style=\"font-size: 24px; color: #000000;\">(c) <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>10<\/mn><mi>x<\/mi><mo>+<\/mo><mn>16<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x^2 &#8211; 10x + 16 = 0<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/span><\/p>\n<hr data-start=\"4460\" data-end=\"4463\" \/>\n<h3 data-start=\"4465\" data-end=\"4515\"><span class=\"ez-toc-section\" id=\"Q4_Find_the_sum_and_product_of_the_roots\"><\/span><span style=\"font-size: 24px; color: #000000;\"><strong data-start=\"4469\" data-end=\"4515\">Q4. Find the sum and product of the roots:<\/strong><\/span><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p data-start=\"4517\" data-end=\"4598\"><span style=\"font-size: 24px; color: #000000;\">(a) <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>2<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>5<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">2x^2 &#8211; 5x + 3 = 0<\/annotation><\/semantics><\/math><\/span><\/span><\/span><br data-start=\"4542\" data-end=\"4545\" \/><span style=\"font-size: 24px; color: #000000;\">(b) <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>3<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>7<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">3x^2 + 7x + 2 = 0<\/annotation><\/semantics><\/math><\/span><\/span><\/span><br data-start=\"4570\" data-end=\"4573\" \/><span style=\"font-size: 24px; color: #000000;\">(c) <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>5<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>9<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">5x^2 &#8211; 9x &#8211; 2 = 0<\/annotation><\/semantics><\/math><\/span><br \/>\n<\/span><\/span><\/p>\n\n    <div class=\"xs_social_share_widget xs_share_url after_content \t\tmain_content  wslu-style-1 wslu-share-box-shaped wslu-fill-colored wslu-none wslu-share-horizontal wslu-theme-font-no wslu-main_content\">\n\n\t\t\n        <ul>\n\t\t\t        <\/ul>\n    <\/div> \n","protected":false},"excerpt":{"rendered":"<div class='heateorSssClear'><\/div><div  class='heateor_sss_sharing_container heateor_sss_horizontal_sharing' 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