{"id":5688,"date":"2026-07-01T10:39:48","date_gmt":"2026-07-01T05:09:48","guid":{"rendered":"https:\/\/mymockmate.com\/notes\/?p=5688"},"modified":"2026-07-01T10:39:53","modified_gmt":"2026-07-01T05:09:53","slug":"ncert-class-10-maths-exercise-4-3-solutions-nature-of-roots","status":"publish","type":"post","link":"https:\/\/mymockmate.com\/notes\/ncert-class-10-maths-exercise-4-3-solutions-nature-of-roots\/","title":{"rendered":"NCERT Class 10 Maths Exercise 4.3 Solutions | Nature of Roots"},"content":{"rendered":"<div class=\"mymoc-top mymoc-entity-placement\" id=\"mymoc-2508700579\"><div id=\"mymoc-4059186529\"><a href=\"https:\/\/amzn.to\/43YYZi3\" aria-label=\"smart-phones\"><img src=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/smart-phones.png\" alt=\"\" srcset=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/smart-phones.png 1303w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/smart-phones-300x76.png 300w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/smart-phones-1024x259.png 1024w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/smart-phones-768x194.png 768w\" sizes=\"(max-width: 1303px) 100vw, 1303px\" width=\"1303\" height=\"329\"><\/a><\/div><\/div><div class=\"pdfprnt-buttons pdfprnt-buttons-post pdfprnt-top-bottom-right\"><a href=\"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/posts\/5688?print=print\" class=\"pdfprnt-button pdfprnt-button-print\" target=\"_blank\"><img decoding=\"async\" src=\"https:\/\/mymockmate.com\/notes\/wp-content\/plugins\/pdf-print\/images\/print.png\" alt=\"image_print\" title=\"Print Content\"><span class=\"pdfprnt-button-title pdfprnt-button-print-title\">Print<\/span><\/a> <span class=\"pdfprnt-count-generation\">7<\/span><\/div>\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">Short Introduction<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\">Exercise 4.3 introduces the concept of the <strong>Nature of Roots of Quadratic Equations<\/strong> using the <strong>Discriminant<\/strong>. Instead of solving every quadratic equation completely, students learn to determine whether an equation has <strong>two distinct real roots, two equal real roots, or no real roots<\/strong> simply by calculating the value of the discriminant.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">This exercise also includes application-based questions involving geometry and age problems where quadratic equations help determine practical solutions.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">Quick Information Box<\/h1>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><thead><tr><th>Particular<\/th><th>Details<\/th><\/tr><\/thead><tbody><tr><td>Class<\/td><td>10<\/td><\/tr><tr><td>Subject<\/td><td>Mathematics<\/td><\/tr><tr><td>Chapter<\/td><td>4<\/td><\/tr><tr><td>Exercise<\/td><td>4.3<\/td><\/tr><tr><td>Chapter Name<\/td><td>Quadratic Equations<\/td><\/tr><tr><td>Method Used<\/td><td>Discriminant Method<\/td><\/tr><tr><td>Board<\/td><td>CBSE<\/td><\/tr><tr><td>Difficulty Level<\/td><td>Moderate<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">Learning Objectives<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\">After completing this exercise, students will be able to:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Understand the meaning of discriminant.<\/li>\n\n\n\n<li>Determine the nature of roots without solving completely.<\/li>\n\n\n\n<li>Find repeated (equal) roots.<\/li>\n\n\n\n<li>Solve real-life application problems.<\/li>\n\n\n\n<li>Apply quadratic equations in geometry.<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">Concepts Used (Topics Covered)<\/h1>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Quadratic Equations<\/li>\n\n\n\n<li>Standard Form<\/li>\n\n\n\n<li>Discriminant<\/li>\n\n\n\n<li>Nature of Roots<\/li>\n\n\n\n<li>Equal Roots<\/li>\n\n\n\n<li>Distinct Roots<\/li>\n\n\n\n<li>No Real Roots<\/li>\n\n\n\n<li>Geometry Applications<\/li>\n\n\n\n<li>Age Problems<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">Important Formulas<\/h1>\n\n\n\n<h2 class=\"wp-block-heading\">Standard Form<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><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><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">where<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>a<\/mi><mo mathvariant=\"normal\">&ne;<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">a\\neq0<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h2 class=\"wp-block-heading\">Discriminant<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>D<\/mi><mo>=<\/mo><msup><mi>b<\/mi><mn>2<\/mn><\/msup><mo>&minus;<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">D=b^2-4ac<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h2 class=\"wp-block-heading\">Nature of Roots<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">If<\/h3>\n\n\n\n<div class=\"internal-linking-related-contents\"><a href=\"https:\/\/mymockmate.com\/notes\/ncert-class-10-maths-real-number-exercise-1-2-solutions-mymockmate\/\" class=\"template-2\"><span class=\"cta\">Related Topic to Read more<\/span><span class=\"postTitle\">NCERT Class 10 Maths Real Number Exercise 1.2 Solutions |  MyMockMate<\/span><\/a><\/div><p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>D<\/mi><mo>&gt;<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">D&gt;0<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The equation has<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">&#9989; Two distinct real roots.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h3 class=\"wp-block-heading\">If<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>D<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">D=0<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The equation has<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">&#9989; Two equal real roots.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h3 class=\"wp-block-heading\">If<\/h3>\n\n\n\n<div class=\"internal-linking-related-contents\"><a href=\"https:\/\/mymockmate.com\/notes\/class-10-maths-exercise-2-1-solutions-polynomials-ncert-solutions\/\" class=\"template-2\"><span class=\"cta\">Related Topic to Read more<\/span><span class=\"postTitle\">NCERT Class 10 Maths Exercise 2.1 Solutions | Polynomials<\/span><\/a><\/div><p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>D<\/mi><mo>&lt;<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">D&lt;0<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The equation has<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">&#9989; No real roots.<\/p><div class=\"mymoc-middle mymoc-entity-placement\" id=\"mymoc-1833364552\"><div id=\"mymoc-3827170108\"><a href=\"https:\/\/amzn.to\/4upqwUD\" aria-label=\"laptop-1\"><img src=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/laptop-1.png\" alt=\"\" srcset=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/laptop-1.png 1303w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/laptop-1-300x91.png 300w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/laptop-1-1024x310.png 1024w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/laptop-1-768x233.png 768w\" sizes=\"(max-width: 1303px) 100vw, 1303px\" width=\"1303\" height=\"395\"><\/a><\/div><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h2 class=\"wp-block-heading\">Quadratic Formula<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mfrac><mrow><mo>&minus;<\/mo><mi>b<\/mi><mo>&plusmn;<\/mo><msqrt><mrow><msup><mi>b<\/mi><mn>2<\/mn><\/msup><mo>&minus;<\/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^2-4ac}}{2a}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">Exercise 4.3<\/h1>\n\n\n\n<h2 class=\"wp-block-heading\">Question 1<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Find the nature of the roots of the following quadratic equations. If the real roots exist, find them.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">Question 1(i)<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mn>2<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>&minus;<\/mo><mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>5<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">2x^2-3x+5=0<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Solution<\/h2>\n\n\n\n<figure class=\"wp-block-image size-large\"><img fetchpriority=\"high\" decoding=\"async\" width=\"851\" height=\"1024\" src=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/07\/image-851x1024.png\" alt=\"\" class=\"wp-image-5847\" srcset=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/07\/image-851x1024.png 851w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/07\/image-249x300.png 249w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/07\/image-768x925.png 768w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/07\/image.png 1143w\" sizes=\"(max-width: 851px) 100vw, 851px\"><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">Question 1(ii)<\/h1>\n\n\n\n<div class=\"internal-linking-related-contents\"><a href=\"https:\/\/mymockmate.com\/notes\/class-10-maths-exercise-2-2-solutions-relationship-between-zeroes-and-coefficients\/\" class=\"template-2\"><span class=\"cta\">Related Topic to Read more<\/span><span class=\"postTitle\">Class 10 Maths Exercise 2.2 Solutions | Relationship Between Zeroes and Coefficients<\/span><\/a><\/div><p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mn>3<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>&minus;<\/mo><mn>4<\/mn><msqrt><mn>3<\/mn><\/msqrt><mtext>&thinsp;<\/mtext><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">3x^2-4\\sqrt3\\,x+4=0<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Solution<\/h2>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" width=\"683\" height=\"1024\" src=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/07\/image-2-683x1024.png\" alt=\"\" class=\"wp-image-5853\" srcset=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/07\/image-2-683x1024.png 683w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/07\/image-2-200x300.png 200w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/07\/image-2-768x1152.png 768w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/07\/image-2-600x900.png 600w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/07\/image-2.png 1024w\" sizes=\"(max-width: 683px) 100vw, 683px\"><\/figure>\n\n\n\n<h1 class=\"wp-block-heading\">Question 1(iii)<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mn>2<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>&minus;<\/mo><mn>6<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">2x^2-6x+3=0<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Solution<\/h2>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" width=\"859\" height=\"1024\" src=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/07\/image-1-859x1024.png\" alt=\"\" class=\"wp-image-5849\" srcset=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/07\/image-1-859x1024.png 859w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/07\/image-1-252x300.png 252w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/07\/image-1-768x915.png 768w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/07\/image-1.png 1149w\" sizes=\"(max-width: 859px) 100vw, 859px\"><\/figure>\n\n\n\n<h1 class=\"wp-block-heading\">Key Takeaways from Question 1<\/h1>\n\n\n\n<ul class=\"wp-block-list\">\n<li>If the <strong>discriminant is negative<\/strong>, there are <strong>no real roots<\/strong>.<\/li>\n\n\n\n<li>If the <strong>discriminant is zero<\/strong>, the quadratic equation has <strong>two equal real roots<\/strong>.<\/li>\n\n\n\n<li>If the <strong>discriminant is positive<\/strong>, the equation has <strong>two distinct real roots<\/strong>.<\/li>\n\n\n\n<li>The <strong>quadratic formula<\/strong> is used to calculate the roots whenever real roots exist.<\/li>\n<\/ul>\n\n\n\n<h1 class=\"wp-block-heading\">Question 2<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\">Find the values of <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>k<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">k<\/annotation><\/semantics><\/math>k for each of the following quadratic equations, so that they have two equal roots.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h2 class=\"wp-block-heading\">Concept Used in Question 2<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">A quadratic equation has <strong>two equal roots<\/strong> when its discriminant is zero.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><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><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Discriminant is<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>D<\/mi><mo>=<\/mo><msup><mi>b<\/mi><mn>2<\/mn><\/msup><mo>&minus;<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">D=b^2-4ac<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For equal roots:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>D<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">D=0<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">So,<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msup><mi>b<\/mi><mn>2<\/mn><\/msup><mo>&minus;<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">b^2-4ac=0<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">Question 2(i)<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mn>2<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mi>k<\/mi><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">2x^2+kx+3=0<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Solution<\/h2>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"859\" height=\"1024\" src=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/07\/image-3-859x1024.png\" alt=\"\" class=\"wp-image-5855\" srcset=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/07\/image-3-859x1024.png 859w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/07\/image-3-252x300.png 252w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/07\/image-3-768x915.png 768w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/07\/image-3.png 1149w\" sizes=\"(max-width: 859px) 100vw, 859px\"><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">Question 2(ii)<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>k<\/mi><mi>x<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>&minus;<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>6<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">kx(x-2)+6=0<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Solution<\/h2>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"945\" height=\"1024\" src=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/07\/image-4-945x1024.png\" alt=\"\" class=\"wp-image-5858\" srcset=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/07\/image-4-945x1024.png 945w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/07\/image-4-277x300.png 277w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/07\/image-4-768x832.png 768w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/07\/image-4.png 1205w\" sizes=\"(max-width: 945px) 100vw, 945px\"><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">Quick Recap of Question 2<\/h1>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><thead><tr><th>Question<\/th><th>Equation<\/th><th>Condition<\/th><th>Value of <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>k<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">k<\/annotation><\/semantics><\/math><\/th><\/tr><\/thead><tbody><tr><td>2(i)<\/td><td><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>2<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mi>k<\/mi><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">2x^2+kx+3=0<\/annotation><\/semantics><\/math><\/td><td><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><\/td><td><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo>&plusmn;<\/mo><mn>2<\/mn><msqrt><mn>6<\/mn><\/msqrt><\/mrow><annotation encoding=\"application\/x-tex\">\\pm2\\sqrt6<\/annotation><\/semantics><\/math><\/td><\/tr><tr><td>2(ii)<\/td><td><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>k<\/mi><mi>x<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>&minus;<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>6<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">kx(x-2)+6=0<\/annotation><\/semantics><\/math><\/td><td><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><\/td><td><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>6<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">6<\/annotation><\/semantics><\/math><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">Important Exam Note<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\">In Question 2(ii), many students write:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>k<\/mi><mo>=<\/mo><mn>0<\/mn><mo separator=\"true\">,<\/mo><mn>6<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">k=0,6<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">as the final answer.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">This is incorrect because when<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>k<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">k=0<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">the equation is no longer quadratic. Therefore, only<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mi>k<\/mi><mo>=<\/mo><mn>6<\/mn><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{k=6}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">is accepted.<\/p>\n\n\n\n<h1 class=\"wp-block-heading\">Question 3<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\">Is it possible to design a rectangular mango grove whose length is twice its breadth, and the area is <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>800<\/mn><mtext>&#8197;&#8202;<\/mtext><msup><mi>m<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">800\\;m^2<\/annotation><\/semantics><\/math>? If so, find its length and breadth.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h2 class=\"wp-block-heading\">Solution<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\"><\/h3>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"961\" height=\"1024\" src=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/07\/image-5-961x1024.png\" alt=\"\" class=\"wp-image-5862\" srcset=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/07\/image-5-961x1024.png 961w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/07\/image-5-281x300.png 281w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/07\/image-5-768x819.png 768w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/07\/image-5.png 1215w\" sizes=\"(max-width: 961px) 100vw, 961px\"><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">Question 4<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\">Is the following situation possible? If so, determine their present ages. The sum of the ages of two friends is 20 years. Four years ago, the product of their ages in years was 48.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h2 class=\"wp-block-heading\">Solution<\/h2>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"934\" height=\"1024\" src=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/07\/image-6-934x1024.png\" alt=\"\" class=\"wp-image-5863\" srcset=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/07\/image-6-934x1024.png 934w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/07\/image-6-274x300.png 274w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/07\/image-6-768x842.png 768w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/07\/image-6.png 1198w\" sizes=\"(max-width: 934px) 100vw, 934px\"><\/figure>\n\n\n\n<h1 class=\"wp-block-heading\">Question 5<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Is it possible to design a rectangular park of perimeter 80 m and area 400 m&sup2;? If so, find its length and breadth.<\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">Solution<\/h1>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"834\" height=\"1024\" src=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/07\/image-7-834x1024.png\" alt=\"\" class=\"wp-image-5866\" srcset=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/07\/image-7-834x1024.png 834w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/07\/image-7-244x300.png 244w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/07\/image-7-768x943.png 768w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/07\/image-7.png 1132w\" sizes=\"(max-width: 834px) 100vw, 834px\"><\/figure>\n\n\n\n<h1 class=\"wp-block-heading\">Exercise 4.3 Summary<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\">After completing Exercise 4.3, students should be able to:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Calculate the <strong>discriminant<\/strong> of a quadratic equation.<\/li>\n\n\n\n<li>Determine whether the equation has <strong>real or imaginary roots<\/strong>.<\/li>\n\n\n\n<li>Distinguish between:\n<ul class=\"wp-block-list\">\n<li>Two distinct real roots<\/li>\n\n\n\n<li>Two equal real roots<\/li>\n\n\n\n<li>No real roots<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li>Solve application-based quadratic equation problems involving geometry and ages.<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">Common Mistakes<\/h1>\n\n\n\n<h3 class=\"wp-block-heading\">1. Incorrect Discriminant Formula<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Many students write<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>D<\/mi><mo>=<\/mo><msup><mi>b<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">D=b^2+4ac<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">instead of<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mi>D<\/mi><mo>=<\/mo><msup><mi>b<\/mi><mn>2<\/mn><\/msup><mo>&minus;<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{D=b^2-4ac}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h3 class=\"wp-block-heading\">2. Wrong Interpretation<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Remember:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><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> &rarr; Two distinct real roots<\/li>\n\n\n\n<li><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> &rarr; Two equal real roots<\/li>\n\n\n\n<li><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> &rarr; No real roots<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h3 class=\"wp-block-heading\">3. Sign Errors<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">While calculating<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msup><mi>b<\/mi><mn>2<\/mn><\/msup><mo>&minus;<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">b^2-4ac<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">students often forget that squaring a negative number makes it positive.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Example:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo stretchy=\"false\">(<\/mo><mo>&minus;<\/mo><mn>6<\/mn><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><mo>=<\/mo><mn>36<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">(-6)^2=36<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">not<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo>&minus;<\/mo><mn>36<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">-36<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h3 class=\"wp-block-heading\">4. Not Rejecting Impossible Values<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">In application-based questions,<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Age<\/li>\n\n\n\n<li>Length<\/li>\n\n\n\n<li>Breadth<\/li>\n\n\n\n<li>Number of objects<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">cannot be negative.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Always reject invalid values.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h3 class=\"wp-block-heading\">5. Forgetting Verification<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Always verify the obtained answer using the original conditions.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">Exam Tips<\/h1>\n\n\n\n<h3 class=\"wp-block-heading\">&#10004; Memorise the discriminant formula<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>D<\/mi><mo>=<\/mo><msup><mi>b<\/mi><mn>2<\/mn><\/msup><mo>&minus;<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">D=b^2-4ac<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h3 class=\"wp-block-heading\">&#10004; Learn the three cases<\/h3>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><thead><tr><th>Discriminant<\/th><th>Nature of Roots<\/th><\/tr><\/thead><tbody><tr><td><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><\/td><td>Two distinct real roots<\/td><\/tr><tr><td><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><\/td><td>Two equal real roots<\/td><\/tr><tr><td><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><\/td><td>No real roots<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h3 class=\"wp-block-heading\">&#10004; Solve systematically<\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Identify <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><\/li>\n\n\n\n<li>Calculate <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>D<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">D<\/annotation><\/semantics><\/math><\/li>\n\n\n\n<li>Decide the nature of roots<\/li>\n\n\n\n<li>Find the roots (if required)<\/li>\n<\/ol>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h3 class=\"wp-block-heading\">&#10004; Show every step<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">CBSE awards marks for each mathematical step.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h3 class=\"wp-block-heading\">&#10004; Verify every answer<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Especially in word problems.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">Practice MCQs<\/h1>\n\n\n\n<h3 class=\"wp-block-heading\">Question 1<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">The discriminant of a quadratic equation is<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">A. <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mo>+<\/mo><mi>c<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a+b+c<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">B. <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>b<\/mi><mn>2<\/mn><\/msup><mo>&minus;<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">b^2-4ac<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">C. <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>a<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><msup><mi>b<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">a^2+b^2<\/annotation><\/semantics><\/math>a2+b2<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">D. <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>4<\/mn><mi>a<\/mi><mi>b<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">4ab<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">&#9989; <strong>Answer:<\/strong> B<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h3 class=\"wp-block-heading\">Question 2<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">If<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>D<\/mi><mo>&gt;<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">D&gt;0<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">the equation has<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">A. Equal roots<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">B. No roots<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">C. Two distinct real roots<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">D. Infinite roots<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">&#9989; <strong>Answer:<\/strong> C<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h3 class=\"wp-block-heading\">Question 3<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">If<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>D<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">D=0<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">then the quadratic equation has<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">A. One real root<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">B. Two equal real roots<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">C. No roots<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">D. Three roots<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">&#9989; <strong>Answer:<\/strong> B<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h3 class=\"wp-block-heading\">Question 4<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">If<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>D<\/mi><mo>&lt;<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">D&lt;0<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">the quadratic equation has<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">A. Two distinct roots<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">B. Equal roots<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">C. No real roots<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">D. Infinite roots<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">&#9989; <strong>Answer:<\/strong> C<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h3 class=\"wp-block-heading\">Question 5<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">The perimeter of a rectangle is<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">A. <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>l<\/mi><mi>b<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">lb<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">B. <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>2<\/mn><mo stretchy=\"false\">(<\/mo><mi>l<\/mi><mo>+<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">2(l+b)<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">C. <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>l<\/mi><mo>+<\/mo><mi>b<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">l+b<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">D. <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>2<\/mn><mi>l<\/mi><mi>b<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">2lb<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">&#9989; <strong>Answer:<\/strong> B<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h3 class=\"wp-block-heading\">Question 6<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Area of a rectangle equals<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">A. <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>2<\/mn><mo stretchy=\"false\">(<\/mo><mi>l<\/mi><mo>+<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">2(l+b)<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">B. <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>l<\/mi><mo>&minus;<\/mo><mi>b<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">l-b<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">C. <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>l<\/mi><mo>&times;<\/mo><mi>b<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">l\\times b<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">D. <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>2<\/mn><mi>l<\/mi><mi>b<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">2lb<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">&#9989; <strong>Answer:<\/strong> C<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h3 class=\"wp-block-heading\">Question 7<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">A quadratic equation can have at most<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">A. One root<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">B. Two roots<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">C. Three roots<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">D. Four roots<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">&#9989; <strong>Answer:<\/strong> B<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h3 class=\"wp-block-heading\">Question 8<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">The value of<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo stretchy=\"false\">(<\/mo><mo>&minus;<\/mo><mn>5<\/mn><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">(-5)^2<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">is<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">A. &ndash;25<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">B. 25<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">C. &ndash;10<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">D. 10<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">&#9989; <strong>Answer:<\/strong> B<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h3 class=\"wp-block-heading\">Question 9<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">The roots are equal when<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">A. <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><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">B. <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><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">C. <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><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">D. <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>D<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">D=1<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">&#9989; <strong>Answer:<\/strong> C<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h3 class=\"wp-block-heading\">Question 10<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Exercise 4.3 mainly focuses on<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">A. Graphs<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">B. Probability<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">C. Nature of Roots<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">D. Statistics<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">&#9989; <strong>Answer:<\/strong> C<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">Frequently Asked Questions (FAQ)<\/h1>\n\n\n\n<h3 class=\"wp-block-heading\">Q1. What is the discriminant?<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">The discriminant is the quantity<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>D<\/mi><mo>=<\/mo><msup><mi>b<\/mi><mn>2<\/mn><\/msup><mo>&minus;<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">D=b^2-4ac<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">It helps determine the nature of the roots of a quadratic equation.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h3 class=\"wp-block-heading\">Q2. Why is the discriminant important?<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">It tells us whether the equation has:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Two distinct real roots<\/li>\n\n\n\n<li>Two equal real roots<\/li>\n\n\n\n<li>No real roots<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">without solving the entire equation.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h3 class=\"wp-block-heading\">Q3. Can a quadratic equation have only one real root?<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">A quadratic equation always has two roots. When the discriminant is zero, the two roots are equal (a repeated root).<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h3 class=\"wp-block-heading\">Q4. Why are negative dimensions rejected?<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Length, breadth, age, and the number of objects cannot be negative in real-life situations, so only meaningful positive values are accepted.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h3 class=\"wp-block-heading\">Q5. Which method is used in Exercise 4.3?<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Exercise 4.3 mainly uses the <strong>Discriminant Method<\/strong> to determine the nature of roots and solve application-based problems.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">Final Revision Notes<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\">&#10004; Learn the discriminant formula by heart.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">&#10004; Remember all three cases of the discriminant.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">&#10004; Practice finding <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a<\/annotation><\/semantics><\/math>a, <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>b<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">b<\/annotation><\/semantics><\/math>b, and <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>c<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">c<\/annotation><\/semantics><\/math>c correctly.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">&#10004; Verify every solution in word problems.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">&#10004; Show complete calculations to score full marks in CBSE examinations.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<p class=\"wp-block-paragraph\">&#128640; <strong>Prepare Smarter with MyMockMate!<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Looking for complete NCERT solutions, chapter-wise notes, important questions, mock tests, MCQs, previous year papers, and CBSE exam preparation resources?<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Visit <strong><a href=\"http:\/\/www.mymockmate.com\">www.mymockmate.com<\/a><\/strong> and explore:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>&#128216; Complete NCERT Class 10 Maths Solutions<\/li>\n\n\n\n<li>&#128221; Chapter-wise Revision Notes<\/li>\n\n\n\n<li>&#127919; Topic-wise Practice Questions<\/li>\n\n\n\n<li>&#128202; CBSE Mock Tests<\/li>\n\n\n\n<li>&#10067; Assertion &amp; Reason Questions<\/li>\n\n\n\n<li>&#128218; Previous Year Question Papers<\/li>\n\n\n\n<li>&#127942; Exam-Oriented Study Material<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Practice regularly, understand the concepts thoroughly, and you&rsquo;ll be well-prepared to score excellent marks in Quadratic Equations and your CBSE Class 10 Mathematics examination.<\/strong><\/p>\n<div class=\"pdfprnt-buttons pdfprnt-buttons-post pdfprnt-top-bottom-right\"><a href=\"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/posts\/5688?print=print\" class=\"pdfprnt-button pdfprnt-button-print\" target=\"_blank\"><img decoding=\"async\" src=\"https:\/\/mymockmate.com\/notes\/wp-content\/plugins\/pdf-print\/images\/print.png\" alt=\"image_print\" title=\"Print Content\"><span class=\"pdfprnt-button-title pdfprnt-button-print-title\">Print<\/span><\/a> <span class=\"pdfprnt-count-generation\">7<\/span><\/div><div class=\"mymoc-bottom mymoc-entity-placement\" id=\"mymoc-1929389174\"><div id=\"mymoc-2676638580\"><a href=\"https:\/\/amzn.to\/4xnw9oY\" aria-label=\"head-phones\"><img src=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/head-phones.png\" alt=\"\" srcset=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/head-phones.png 1301w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/head-phones-300x80.png 300w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/head-phones-1024x274.png 1024w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/head-phones-768x205.png 768w\" sizes=\"(max-width: 1301px) 100vw, 1301px\" width=\"1301\" height=\"348\"><\/a><\/div><\/div>","protected":false},"excerpt":{"rendered":"<p> 7 Short Introduction Exercise 4.3 introduces the concept of the Nature of Roots of Quadratic Equations using the Discriminant. Instead of solving every quadratic&#8230;<\/p>\n","protected":false},"author":1,"featured_media":5695,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"postBodyCss":"","postBodyMargin":[],"postBodyPadding":[],"postBodyBackground":{"backgroundType":"classic","gradient":""},"footnotes":""},"categories":[5,10],"tags":[3610,3607,44,3609,3606,47,3608],"class_list":["post-5688","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-class-10","category-maths-class-10","tag-cbse-maths-chapter-4","tag-chapter-4-nature-of-roots","tag-class-10-maths","tag-discriminant-questions","tag-ncert-class-10-maths-exercise-4-3-solutions","tag-ncert-solutions","tag-quadratic-equations-class-10"],"featured_image_src":"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/NCERT-Class-10-Maths-Exercise-4.3-Solutions.png","author_info":{"display_name":"Team Mymockmate","author_link":"https:\/\/mymockmate.com\/notes\/author\/bsm_adm\/"},"_links":{"self":[{"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/posts\/5688","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/comments?post=5688"}],"version-history":[{"count":18,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/posts\/5688\/revisions"}],"predecessor-version":[{"id":5868,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/posts\/5688\/revisions\/5868"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/media\/5695"}],"wp:attachment":[{"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/media?parent=5688"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/categories?post=5688"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/tags?post=5688"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}