{"id":33927,"date":"2024-07-26T07:00:48","date_gmt":"2024-07-26T07:00:48","guid":{"rendered":"https:\/\/mp.moonpreneur.com\/math-corner\/?p=33927"},"modified":"2024-07-26T12:33:41","modified_gmt":"2024-07-26T12:33:41","slug":"quadratic-formula","status":"publish","type":"post","link":"https:\/\/mp.moonpreneur.com\/math-corner\/quadratic-formula\/","title":{"rendered":"Quadratic Formula Practice Problems"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-post\" data-elementor-id=\"33927\" class=\"elementor elementor-33927\" data-elementor-post-type=\"post\">\n\t\t\t\t\t\t<div class=\"elementor-inner\">\n\t\t\t\t<div class=\"elementor-section-wrap\">\n\t\t\t\t\t\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-7f48241 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"7f48241\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-fb260c3\" data-id=\"fb260c3\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-d1b8fde elementor-widget elementor-widget-text-editor\" data-id=\"d1b8fde\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<h2><b>What is the Quadratic formula?<\/b><\/h2>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-6214cfa elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"6214cfa\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-833438c\" data-id=\"833438c\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-a105685 elementor-widget elementor-widget-text-editor\" data-id=\"a105685\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<p><span style=\"font-weight: 400;\">In mathematics, the quadratic formula is a highly effective technique for solving quadratic equations and finding their roots.\u00a0<\/span><\/p><p><span style=\"font-weight: 400;\">The formula for calculating the roots of a quadratic equation\u00a0<\/span><\/p><p><a href=\"https:\/\/mp.moonpreneur.com\/math-corner\/wp-content\/uploads\/2024\/07\/quadratic-equation-1.webp\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mp.moonpreneur.com\/math-corner\/wp-content\/uploads\/2024\/07\/quadratic-equation-1.webp\" alt=\"Quadratic Equation 1\" width=\"1200\" height=\"628\" \/><\/a><\/p><p><span style=\"font-weight: 400;\">ax2 + bx + c = 0 is given by x = [-b \u00b1 \u221a(b2 \u2013 4ac)]\/2a<\/span><\/p><p><span style=\"font-weight: 400;\">Here, D=b2\u22124acD = b^2 &#8211; 4acD=b2\u22124ac is known as the discriminant of the quadratic equation. The value of the discriminant determines the nature of the roots of the quadratic equation<\/span><span style=\"font-weight: 400;\">.<\/span><\/p><ul><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">If D &lt; 0, the roots are not real, i.e., imaginary.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">If D = 0, the roots are real and equal.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">If D &gt; 0, the roots are real and unequal.<\/span><\/li><\/ul><p><span style=\"font-weight: 400;\"><a href=\"https:\/\/mp.moonpreneur.com\/math-corner\/wp-content\/uploads\/2024\/07\/quadratic-equation.webp\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mp.moonpreneur.com\/math-corner\/wp-content\/uploads\/2024\/07\/quadratic-equation.webp\" alt=\"Quadratic Equation\" width=\"1200\" height=\"628\" \/><\/a>We have compiled Quadratic formula questions to assist students in Classes 10 and 11 with solving quadratic equations, particularly those with imaginary roots.\u00a0<\/span><\/p><p><span style=\"font-weight: 400;\">While the factorization technique is often straightforward for finding the roots of quadratic equations, it may not always be applicable. In such instances, the quadratic formula becomes an essential tool, enabling us to find the roots of quadratic equations effortlessly.<\/span><\/p><p><strong>Recommended reading: <\/strong><a href=\"https:\/\/mp.moonpreneur.com\/math-corner\/color-by-number-books-for-kids\/\">The Best Color by Number Books for Kids<\/a><\/p><p><span style=\"font-weight: 400;\">However, the quadratic formula is used to find the roots of a quadratic equation when other methods are not sufficient, particularly for finding imaginary roots. Let\u2019s delve into a detailed discussion about the quadratic formula and its derivation here.<\/span><\/p>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-21272f0 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"21272f0\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-c583179\" data-id=\"c583179\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-8720f2c elementor-widget elementor-widget-text-editor\" data-id=\"8720f2c\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<h3 style=\"text-align: center;\"><span style=\"color: #000080;\"><b>Problems with Solution:<\/b><\/span><\/h3>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-567c3df elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"567c3df\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-6069cb8\" data-id=\"6069cb8\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-9ba4caf elementor-widget elementor-widget-toggle\" data-id=\"9ba4caf\" data-element_type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-1631\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-1631\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">\u200b 1. Problem: <\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-1631\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-1631\"><p>m2 \u2212 31 \u2212 2m = \u22126 \u2212 3m2 \u2212 4m<\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-1632\" class=\"elementor-tab-title\" data-tab=\"2\" role=\"button\" aria-controls=\"elementor-tab-content-1632\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Solution:<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-1632\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"2\" role=\"region\" aria-labelledby=\"elementor-tab-title-1632\"><p><span style=\"font-weight: 400;\">Given,<\/span><\/p><p><span style=\"font-weight: 400;\">m2 \u2212 31 \u2212 2m = \u22126 \u2212 3m2 \u2212 4m<\/span><\/p><p><span style=\"font-weight: 400;\">m2 \u2013 31 \u2013 2m + 6 + 3m2 + 4m = 0<\/span><\/p><p><span style=\"font-weight: 400;\">4m2 + 2m \u2013 25 = 0<\/span><\/p><p><span style=\"font-weight: 400;\">Comparing with the standard form of quadratic equation ax2 + bx + c = 0, we get;<\/span><\/p><p><span style=\"font-weight: 400;\">a = 4, b = 2 and c = -25<\/span><\/p><p><span style=\"font-weight: 400;\">Now, b2 \u2013 4ac = (2)2 \u2013 4(4)(-25)<\/span><\/p><p><span style=\"font-weight: 400;\">= 4 + 400<\/span><\/p><p><span style=\"font-weight: 400;\">= 404 &gt; 0<\/span><\/p><p><span style=\"font-weight: 400;\">That means the given equation has two real and unequal roots.<\/span><\/p><p><span style=\"font-weight: 400;\">Using the quadratic formula,<\/span><\/p><p><span style=\"font-weight: 400;\">x = [-b \u00b1 \u221a(b2 \u2013 4ac)]\/2a<\/span><\/p><p><span style=\"font-weight: 400;\">m = [-2 \u00b1 \u221a404]\/ 2(4)<\/span><\/p><p><span style=\"font-weight: 400;\">= [-2 \u00b1 \u221a(4 \u00d7 101)]\/8<\/span><\/p><p><span style=\"font-weight: 400;\">= (-2 \u00b1 2\u221a101)\/8<\/span><\/p><p><span style=\"font-weight: 400;\">Therefore, m = (-2 + 2\u221a101)\/8 and m = (-2 \u2013 2\u221a101)\/8.<\/span><\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-cbe6a29 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"cbe6a29\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-3296a16\" data-id=\"3296a16\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-2adce35 elementor-widget elementor-widget-toggle\" data-id=\"2adce35\" data-element_type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-4491\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-4491\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">2. Problem:<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-4491\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-4491\"><p>Write the roots of the quadratic equation 3&#215;2 +3x = -3 using the quadratic formula.<\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-4492\" class=\"elementor-tab-title\" data-tab=\"2\" role=\"button\" aria-controls=\"elementor-tab-content-4492\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Solution<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-4492\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"2\" role=\"region\" aria-labelledby=\"elementor-tab-title-4492\"><p><b>Solution:<\/b><span style=\"font-weight: 400;\">Given,<\/span><\/p><p><span style=\"font-weight: 400;\">3&#215;2 +3x = -3<\/span><\/p><p><span style=\"font-weight: 400;\">3&#215;2 +3x + 3 = 0<\/span><\/p><p><span style=\"font-weight: 400;\">3(x2 + x + 1) = 0<\/span><\/p><p><span style=\"font-weight: 400;\">x2 + x + 1 = 0<\/span><\/p><p><span style=\"font-weight: 400;\">Here, a = 1, b = 1 and c = 1.<\/span><\/p><p><span style=\"font-weight: 400;\">Using the quadratic formula,<\/span><\/p><p><span style=\"font-weight: 400;\">x = [-b \u00b1 \u221a(b2 \u2013 4ac)]\/2a<\/span><\/p><p><span style=\"font-weight: 400;\">= [-1 \u00b1 \u221a{12 \u2013 4(1)(1)}]\/ 2(1)<\/span><\/p><p><span style=\"font-weight: 400;\">= [-1 \u00b1 \u221a(1 \u2013 4)]\/ 2<\/span><\/p><p><span style=\"font-weight: 400;\">= (-1 \u00b1 \u221a-3)\/2<\/span><\/p><p><span style=\"font-weight: 400;\">= (-1 \u00b1 i\u221a3)\/2<\/span><\/p><p><span style=\"font-weight: 400;\">x = (-1 + i\u221a3)\/2, x = (-1 \u2013 i\u221a3)\/2<\/span><\/p><p><span style=\"font-weight: 400;\">Therefore, the roots of 3&#215;2 + 3x = -3 are (-1 + i\u221a3)\/2 and (-1 \u2013 i\u221a3)\/2<\/span><\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-202ce3e elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"202ce3e\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-f28fc3d\" data-id=\"f28fc3d\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-a73154a elementor-widget elementor-widget-toggle\" data-id=\"a73154a\" data-element_type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-1751\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-1751\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">  3.Problem: <\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-1751\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-1751\"><p>Using the quadratic formula, find the roots of the quadratic equation 2&#215;2 \u2013 7x + 6 = 0.<\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-1752\" class=\"elementor-tab-title\" data-tab=\"2\" role=\"button\" aria-controls=\"elementor-tab-content-1752\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Solution<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-1752\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"2\" role=\"region\" aria-labelledby=\"elementor-tab-title-1752\"><p><span style=\"font-weight: 400;\">Given,<\/span><\/p><p><span style=\"font-weight: 400;\">2&#215;2 \u2013 7x + 6 = 0<\/span><\/p><p><span style=\"font-weight: 400;\">When compared with the standard form, we have;<\/span><\/p><p><span style=\"font-weight: 400;\">a = 2, b = -7 and c = 6<\/span><\/p><p><span style=\"font-weight: 400;\">D = b2 \u2013 4ac<\/span><\/p><p><span style=\"font-weight: 400;\">= (-7)2 \u2013 4(2)(6)<\/span><\/p><p><span style=\"font-weight: 400;\">= 49 \u2013 48<\/span><\/p><p><span style=\"font-weight: 400;\">= 1 &gt; 0<\/span><\/p><p><span style=\"font-weight: 400;\">That means the roots are real and unequal.<\/span><\/p><p><span style=\"font-weight: 400;\">Using the quadratic formula,<\/span><\/p><p><span style=\"font-weight: 400;\">x = [-b \u00b1 \u221a(b2 \u2013 4ac)]\/2a<\/span><\/p><p><span style=\"font-weight: 400;\">= [-(-7) \u00b1 \u221a1]\/ 2(2)<\/span><\/p><p><span style=\"font-weight: 400;\">= (7 \u00b1 1)\/4<\/span><\/p><p><span style=\"font-weight: 400;\">x = (7 + 1)\/4, x = (7 \u2013 1)\/4<\/span><\/p><p><span style=\"font-weight: 400;\">x = 8\/4, x = 6\/4<\/span><\/p><p><span style=\"font-weight: 400;\">x = 2, x = 3\/2<\/span><\/p><p><span style=\"font-weight: 400;\">Therefore, 2 and 3\/2 are the roots of the given equation.<\/span><\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-ce711bc elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"ce711bc\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-9fe008f\" data-id=\"9fe008f\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-8112d08 elementor-widget elementor-widget-toggle\" data-id=\"8112d08\" data-element_type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-1351\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-1351\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">4. Problem:<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-1351\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-1351\"><p>Find whether the equation 5&#215;2 \u2013 2x \u2013 10 = 0 has real roots. If real roots exist, find them<b>.<\/b><\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-1352\" class=\"elementor-tab-title\" data-tab=\"2\" role=\"button\" aria-controls=\"elementor-tab-content-1352\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Solution:<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-1352\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"2\" role=\"region\" aria-labelledby=\"elementor-tab-title-1352\"><p><span style=\"font-weight: 400;\">Given,<\/span><\/p><p><span style=\"font-weight: 400;\">5&#215;2 \u2013 2x \u2013 10 = 0<\/span><\/p><p><span style=\"font-weight: 400;\">Here, a = 5, b = -2, and c = -10<\/span><\/p><p><span style=\"font-weight: 400;\">Let us calculate the discriminant value.<\/span><\/p><p><span style=\"font-weight: 400;\">D = b2 \u2013 4ac<\/span><\/p><p><span style=\"font-weight: 400;\">= (-2)2 \u2013 4(5)(-10)<\/span><\/p><p><span style=\"font-weight: 400;\">= 4 + 200<\/span><\/p><p><span style=\"font-weight: 400;\">= 204 &gt; 0<\/span><\/p><p><span style=\"font-weight: 400;\">That means the given quadratic equation has two real and unequal roots.<\/span><\/p><p><span style=\"font-weight: 400;\">Now, using the quadratic formula, we have;<\/span><\/p><p><span style=\"font-weight: 400;\">x = [-b \u00b1 \u221a(b2 \u2013 4ac)]\/2a<\/span><\/p><p><span style=\"font-weight: 400;\">= [-(-2) \u00b1 \u221a204]\/ 2(5)<\/span><\/p><p><span style=\"font-weight: 400;\">= (2 \u00b1 \u221a(4 \u00d7 51))\/10<\/span><\/p><p><span style=\"font-weight: 400;\">x = (2 + 2\u221a51)\/10, x = (2 \u2013 2\u221a51)\/10<\/span><\/p><p><span style=\"font-weight: 400;\">x = 2(1 + \u221a51)\/10, x = 2(1 \u2013 \u221a51)\/10<\/span><\/p><p><span style=\"font-weight: 400;\">x = (1 + \u221a51)\/5 , x = (1 \u2013 \u221a51)\/5<\/span><\/p><p><span style=\"font-weight: 400;\">Therefore, the real roots of the given quadratic equation are (1 + \u221a51)\/5 and (1 \u2013 \u221a51)\/5.<\/span><\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-ad6db83 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"ad6db83\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-aea0e6a\" data-id=\"aea0e6a\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-7312db9 elementor-widget elementor-widget-toggle\" data-id=\"7312db9\" data-element_type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-1201\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-1201\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">5. Problem<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-1201\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-1201\"><p>Calculate the roots of the quadratic equation x \u2013 1\/x = 3, x \u2260 0, using the quadratic formula.<\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-1202\" class=\"elementor-tab-title\" data-tab=\"2\" role=\"button\" aria-controls=\"elementor-tab-content-1202\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Solution:<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-1202\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"2\" role=\"region\" aria-labelledby=\"elementor-tab-title-1202\"><p><span style=\"font-weight: 400;\">Given the quadratic equation is:<\/span><\/p><p><span style=\"font-weight: 400;\">x \u2013 1\/x = 3<\/span><\/p><p><span style=\"font-weight: 400;\">(x2 \u2013 1)\/x = 3<\/span><\/p><p><span style=\"font-weight: 400;\">x2 \u2013 1 = 3x<\/span><\/p><p><span style=\"font-weight: 400;\">x2 \u2013 3x \u2013 1 = 0<\/span><\/p><p><span style=\"font-weight: 400;\">Here, a = 1, b = -3 and c = -1<\/span><\/p><p><span style=\"font-weight: 400;\">D = b2 \u2013 4ac<\/span><\/p><p><span style=\"font-weight: 400;\">= (-3)2 \u2013 4(1)(-1)<\/span><\/p><p><span style=\"font-weight: 400;\">= 9 + 4<\/span><\/p><p><span style=\"font-weight: 400;\">= 13 &gt; 0<\/span><\/p><p><span style=\"font-weight: 400;\">Thus, the roots of the given quadratic equation are real and unequal.<\/span><\/p><p><span style=\"font-weight: 400;\">Using the quadratic formula,<\/span><\/p><p><span style=\"font-weight: 400;\">x = [-b \u00b1 \u221a(b2 \u2013 4ac)]\/2a<\/span><\/p><p><span style=\"font-weight: 400;\">= [-(-3) \u00b1 \u221a13]\/ 2(1)<\/span><\/p><p><span style=\"font-weight: 400;\">= (3 \u00b1 \u221a13)\/2<\/span><\/p><p><span style=\"font-weight: 400;\">x = (3 + \u221a13)\/2, x = (3 \u2013 \u221a13)\/2<\/span><\/p><p><span style=\"font-weight: 400;\">Hence, (3 + \u221a13)\/2 and (3 \u2013 \u221a13)\/2 are the roots of the given quadratic equation.<\/span><\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-83f2a4a elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"83f2a4a\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-a1155f2\" data-id=\"a1155f2\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-e31620a elementor-widget elementor-widget-toggle\" data-id=\"e31620a\" data-element_type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-2381\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-2381\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">6. Problem:<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-2381\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-2381\"><p>What are the roots of the quadratic equation 2&#215;2 \u2013 \u221a5x + 1 = 0?<\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-2382\" class=\"elementor-tab-title\" data-tab=\"2\" role=\"button\" aria-controls=\"elementor-tab-content-2382\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Solution:<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-2382\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"2\" role=\"region\" aria-labelledby=\"elementor-tab-title-2382\"><p><span style=\"font-weight: 400;\">Given,<\/span><\/p><p><span style=\"font-weight: 400;\">2&#215;2 \u2013 \u221a5x + 1 = 0<\/span><\/p><p><span style=\"font-weight: 400;\">Here, a = 2, b = -\u221a5 and c = 1<\/span><\/p><p><span style=\"font-weight: 400;\">D = b2 \u2013 4ac<\/span><\/p><p><span style=\"font-weight: 400;\">= (-\u221a5)2 \u2013 4(2)(1)<\/span><\/p><p><span style=\"font-weight: 400;\">= 5 \u2013 8<\/span><\/p><p><span style=\"font-weight: 400;\">= -3 &lt; 0<\/span><\/p><p><span style=\"font-weight: 400;\">That means the roots are imaginary.<\/span><\/p><p><span style=\"font-weight: 400;\">Using the quadratic formula, we have;<\/span><\/p><p><span style=\"font-weight: 400;\">x = [-b \u00b1 \u221a(b2 \u2013 4ac)]\/2a<\/span><\/p><p><span style=\"font-weight: 400;\">= [-(-\u221a5) \u00b1 \u221a(-3)]\/ 2(2)<\/span><\/p><p><span style=\"font-weight: 400;\">= (\u221a5 \u00b1 i\u221a3)\/4<\/span><\/p><p><span style=\"font-weight: 400;\">x = (\u221a5 + i\u221a3)\/4, (\u221a5 \u2013 i\u221a3)\/4<\/span><\/p><p><span style=\"font-weight: 400;\">Hence, (\u221a5 + i\u221a3)\/4 and (\u221a5 \u2013 i\u221a3)\/4 are the roots of the quadratic equation 2&#215;2 \u2013 \u221a5x + 1 = 0.<\/span><\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-8625a51 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"8625a51\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-9b5c324\" data-id=\"9b5c324\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-fc4bf0f elementor-widget elementor-widget-toggle\" data-id=\"fc4bf0f\" data-element_type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-2641\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-2641\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">7. Problem:<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-2641\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-2641\"><p>\u00a0Find the roots of the quadratic equation 4&#215;2 + 4\u221a3x + 3 = 0<\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-2642\" class=\"elementor-tab-title\" data-tab=\"2\" role=\"button\" aria-controls=\"elementor-tab-content-2642\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Solution: <\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-2642\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"2\" role=\"region\" aria-labelledby=\"elementor-tab-title-2642\"><p><span style=\"font-weight: 400;\">Given,<\/span><\/p><p><span style=\"font-weight: 400;\">4&#215;2 + 4\u221a3x + 3 = 0<\/span><\/p><p><span style=\"font-weight: 400;\">Here, a = 4, b = 4\u221a3 and c = 3<\/span><\/p><p><span style=\"font-weight: 400;\">b2 \u2013 4ac = (4\u221a3)2 \u2013 4(4)(3)<\/span><\/p><p><span style=\"font-weight: 400;\">= 48 \u2013 48<\/span><\/p><p><span style=\"font-weight: 400;\">= 0<\/span><\/p><p><span style=\"font-weight: 400;\">That means the roots of the given quadratic equation are real and equal.<\/span><\/p><p><span style=\"font-weight: 400;\">By using the quadratic formula, we get,<\/span><\/p><p><span style=\"font-weight: 400;\">x = [-b \u00b1 \u221a(b2 \u2013 4ac)]\/2a<\/span><\/p><p><span style=\"font-weight: 400;\">x = (-4\u221a3 \u00b1 \u221a0)\/8<\/span><\/p><p><span style=\"font-weight: 400;\">x = (-4\u221a3)\/8<\/span><\/p><p><span style=\"font-weight: 400;\">x = -\u221a3\/2<\/span><\/p><p><span style=\"font-weight: 400;\">Therefore, -\u221a3\/2 and -\u221a3\/2 are the roots of the given equation.<\/span><\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-de6b75f elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"de6b75f\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-48c2a35\" data-id=\"48c2a35\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-11ffd59 elementor-widget elementor-widget-toggle\" data-id=\"11ffd59\" data-element_type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-1881\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-1881\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">8. Problem<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-1881\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-1881\"><p>What will be the roots of the quadratic equation 10&#215;2 \u2212 9x + 6 = 0?<\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-1882\" class=\"elementor-tab-title\" data-tab=\"2\" role=\"button\" aria-controls=\"elementor-tab-content-1882\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Solution:<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-1882\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"2\" role=\"region\" aria-labelledby=\"elementor-tab-title-1882\"><p><span style=\"font-weight: 400;\">Given the quadratic equation is:<\/span><\/p><p><span style=\"font-weight: 400;\">10&#215;2 \u2212 9x + 6 = 0<\/span><\/p><p><span style=\"font-weight: 400;\">Here, a = 10, b = -9 and c = 6.<\/span><\/p><p><span style=\"font-weight: 400;\">b2 \u2013 4ac = (-9)2 \u2013 4(10)(6)<\/span><\/p><p><span style=\"font-weight: 400;\">= 81 \u2013 240<\/span><\/p><p><span style=\"font-weight: 400;\">= -159<\/span><\/p><p><span style=\"font-weight: 400;\">Using the quadratic formula, we have;<\/span><\/p><p><span style=\"font-weight: 400;\">x = [-(-9) \u00b1 \u221a(b2 \u2013 4ac)]\/2a<\/span><\/p><p><span style=\"font-weight: 400;\">= [9 \u00b1 \u221a(-159)]\/ 2(10)<\/span><\/p><p><span style=\"font-weight: 400;\">= (9 \u00b1 i\u221a159)\/20<\/span><\/p><p><span style=\"font-weight: 400;\">x = (9 + i\u221a159)\/20, (9 \u2013 i\u221a159)\/20<\/span><\/p><p><span style=\"font-weight: 400;\">Hence, (9 + i\u221a159)\/20 and (9 \u2013 i\u221a159)\/20 are the roots of the quadratic equation 10&#215;2 \u2212 9x + 6 = 0.<\/span><\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-00e1ac1 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"00e1ac1\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-1ba5dcf\" data-id=\"1ba5dcf\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-17820a2 elementor-widget elementor-widget-toggle\" data-id=\"17820a2\" data-element_type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-2461\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-2461\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">9. Problem:<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-2461\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-2461\"><p>Solve the following equation using the quadratic formula:4&#215;2 \u2013 1 = -8x<\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-2462\" class=\"elementor-tab-title\" data-tab=\"2\" role=\"button\" aria-controls=\"elementor-tab-content-2462\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Solution:<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-2462\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"2\" role=\"region\" aria-labelledby=\"elementor-tab-title-2462\"><p><span style=\"font-weight: 400;\">Given,<\/span><\/p><p><span style=\"font-weight: 400;\">4&#215;2 \u2013 1 = -8x<\/span><\/p><p><span style=\"font-weight: 400;\">4&#215;2 + 8x \u2013 1 = 0<\/span><\/p><p><span style=\"font-weight: 400;\">Here, a = 4, b = 8 and c = -1<\/span><\/p><p><span style=\"font-weight: 400;\">Now, b2 \u2013 4ac = (8)2 \u2013 4(4)(-1)<\/span><\/p><p><span style=\"font-weight: 400;\">= 64 + 16<\/span><\/p><p><span style=\"font-weight: 400;\">= 80 &gt; 0<\/span><\/p><p><span style=\"font-weight: 400;\">That means the given quadratic equation contains two real and unequal roots.<\/span><\/p><p><span style=\"font-weight: 400;\">Now, by using the quadratic formula, we have;<\/span><\/p><p><span style=\"font-weight: 400;\">x = [-b \u00b1 \u221a(b2 \u2013 4ac)]\/2a<\/span><\/p><p><span style=\"font-weight: 400;\">= [-8 \u00b1 \u221a80]\/ 2(4)<\/span><\/p><p><span style=\"font-weight: 400;\">= (-8 \u00b1 \u221a(16 \u00d7 5))\/8<\/span><\/p><p><span style=\"font-weight: 400;\">x = (-8 + 4\u221a5)\/8, x = (-8 \u2013 4\u221a5)\/8<\/span><\/p><p><span style=\"font-weight: 400;\">x = 4(-2 + \u221a5)\/8, x = 4(-2 \u2013 \u221a5)\/8<\/span><\/p><p><span style=\"font-weight: 400;\">x = (-2 + \u221a5)\/2, x = (-2 \u2013 \u221a5)\/2<\/span><\/p><p><span style=\"font-weight: 400;\">Therefore, the real roots of the given quadratic equation are (-2 + \u221a5)\/2 and x = (-2 \u2013 \u221a5)\/2.<\/span><\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-f2f52cc elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"f2f52cc\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-60e9f0e\" data-id=\"60e9f0e\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-529f590 elementor-widget elementor-widget-toggle\" data-id=\"529f590\" data-element_type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-8661\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-8661\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">10.Problem:<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-8661\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-8661\"><p>What is the nature of the roots of 5&#215;2 + 3x + 1 = 0? Find them.<\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-8662\" class=\"elementor-tab-title\" data-tab=\"2\" role=\"button\" aria-controls=\"elementor-tab-content-8662\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Solution:<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-8662\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"2\" role=\"region\" aria-labelledby=\"elementor-tab-title-8662\"><p><span style=\"font-weight: 400;\">Given,<\/span><\/p><p><span style=\"font-weight: 400;\">5&#215;2 + 3x + 1 = 0<\/span><\/p><p><span style=\"font-weight: 400;\">Here, a = 5, b = 3 and c = 1.<\/span><\/p><p><span style=\"font-weight: 400;\">Let us calculate the discriminant to determine the nature of the roots of 5&#215;2 + 3x + 1 = 0.<\/span><\/p><p><span style=\"font-weight: 400;\">D = b2 \u2013 4ac<\/span><\/p><p><span style=\"font-weight: 400;\">= (3)2 \u2013 4(5)(1)<\/span><\/p><p><span style=\"font-weight: 400;\">= 9 \u2013 20<\/span><\/p><p><span style=\"font-weight: 400;\">= -11 &lt; 0<\/span><\/p><p><span style=\"font-weight: 400;\">Thus, the given quadratic equation contains two imaginary roots.<\/span><\/p><p><span style=\"font-weight: 400;\">Using the quadratic formula, we have;<\/span><\/p><p><span style=\"font-weight: 400;\">x = [-b \u00b1 \u221a(b2 \u2013 4ac)]\/2a<\/span><\/p><p><span style=\"font-weight: 400;\">= [-3 \u00b1 \u221a(-11)]\/ 2(5)<\/span><\/p><p><span style=\"font-weight: 400;\">= (-3 \u00b1 i\u221a11)\/10<\/span><\/p><p><span style=\"font-weight: 400;\">x = (-3 + i\u221a11)\/10, (-3 \u2013 i\u221a11)\/10<\/span><\/p><p><span style=\"font-weight: 400;\">Hence, (-3 + i\u221a11)\/10 and (-3 \u2013 i\u221a11)\/10 are the roots of the given quadratic equation.<\/span><\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-fa331b2 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"fa331b2\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-2243571\" data-id=\"2243571\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-5e5cd0d elementor-widget elementor-widget-toggle\" data-id=\"5e5cd0d\" data-element_type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-9891\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-9891\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">11. Problem:<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-9891\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-9891\"><p>Solve the quadratic equation using the quadratic formula. 2\ud835\udc65\u00b2\u00a0 \u2212\ud835\udc65=1<\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-9892\" class=\"elementor-tab-title\" data-tab=\"2\" role=\"button\" aria-controls=\"elementor-tab-content-9892\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Solution:<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-9892\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"2\" role=\"region\" aria-labelledby=\"elementor-tab-title-9892\"><p>2 x\u00b2 \u2013<i>x<\/i>\u20131=0<\/p><p>\ud835\udc4e= 2<\/p><p>\ud835\udc4f= \u20131<\/p><p>\ud835\udc50= \u20131<\/p><p>\u00a0\ud835\udc65=\u00a0 \u2013\ud835\udc4f\u00b1 \u221a\ud835\udc4f\u00b2 \u20134\ud835\udc4e\ud835\udc50\/2\ud835\udc4e\u00a0 \u00a0\u00a0\u00a0<\/p><p>\u00a0= \u2013 (\u20131) \u00b1\u221a(\u20131)\u00b2 \u20134(2) (\u20131)\/2(2)<\/p><p>=1\u00b1\u221a1+8\/4<\/p><p>=1\u00b1\u221a9\/4<\/p><p>=1\u00b13\/4<\/p><p>X1=1+3\/4<\/p><p>=4\/4<\/p><p>=1<\/p><p>X2=1-3\/4<\/p><p>= \u20132\/4<\/p><p>= \u20131\/2<\/p><p>Therefore, the answers are \ud835\udc651=1\u00a0 \ud835\udc652=\u00a0 \u20131\/2<\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-d0585dd elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"d0585dd\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-c5ef2cd\" data-id=\"c5ef2cd\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-24f0812 elementor-widget elementor-widget-video\" data-id=\"24f0812\" data-element_type=\"widget\" data-settings=\"{&quot;youtube_url&quot;:&quot;https:\\\/\\\/youtu.be\\\/5Va05uRRxR4?si=DpTemhKhSrAAWLr0&quot;,&quot;video_type&quot;:&quot;youtube&quot;,&quot;controls&quot;:&quot;yes&quot;}\" data-widget_type=\"video.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-wrapper elementor-open-inline\">\n\t\t\t<div class=\"elementor-video\"><\/div>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-ecb7381 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"ecb7381\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-cb5c8dd\" data-id=\"cb5c8dd\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-964af76 elementor-widget elementor-widget-text-editor\" data-id=\"964af76\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<h3 style=\"text-align: center;\"><span style=\"color: #008000;\"><b>Conclusion<\/b><\/span><\/h3>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-3241f11 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"3241f11\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-83b9a49\" data-id=\"83b9a49\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-851fca6 elementor-widget elementor-widget-text-editor\" data-id=\"851fca6\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<p><span style=\"font-weight: 400;\">Practicing these problems will boost your confidence in using the quadratic formula to solve any quadratic equation. Remember, you have to carefully follow each step and double-check your calculation. With regular practice, you\u2019ll find solving quadratic equations using the quadratic formula to be straightforward and efficient. <\/span><\/p><p><span style=\"font-weight: 400;\">Want to excite your child about math and sharpen their math skills? Moonpreneur&#8217;s online math curriculum is unique as it helps children understand math skills through hands-on lessons, assists them in building real-life applications, and excites them to learn math.\u00a0<\/span><\/p><p><span style=\"font-weight: 400;\">You can opt for our <\/span><a href=\"https:\/\/moonpreneur.com\/innovator-program\/advanced-math\/\"><span style=\"font-weight: 400;\">Advanced Math<\/span><\/a><span style=\"font-weight: 400;\"> or Vedic Math+Mental Math courses. Our <\/span><a href=\"https:\/\/mp.moonpreneur.com\/math-quiz-for-kids\/\"><span style=\"font-weight: 400;\">Math Quiz<\/span><\/a><span style=\"font-weight: 400;\"> for grades 3rd, 4th, 5th, and 6th helps in further exciting and engaging in mathematics with hands-on lessons.<\/span><\/p><p><strong>Recommended reading: <\/strong><a href=\"https:\/\/mp.moonpreneur.com\/math-corner\/the-international-mathematical-olympiad\/\">The International Mathematical Olympiad<\/a><\/p>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>What is the Quadratic formula? In mathematics, the quadratic formula is a highly effective technique for solving quadratic equations and finding their roots.\u00a0 The formula for calculating the roots of a quadratic equation\u00a0 ax2 + bx + c = 0 is given by x = [-b \u00b1 \u221a(b2 \u2013 4ac)]\/2a Here, D=b2\u22124acD = b^2 &#8211; [&hellip;]<\/p>\n","protected":false},"author":116,"featured_media":33930,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false},"categories":[979],"tags":[],"acf":[],"_links":{"self":[{"href":"https:\/\/mp.moonpreneur.com\/math-corner\/wp-json\/wp\/v2\/posts\/33927"}],"collection":[{"href":"https:\/\/mp.moonpreneur.com\/math-corner\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mp.moonpreneur.com\/math-corner\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mp.moonpreneur.com\/math-corner\/wp-json\/wp\/v2\/users\/116"}],"replies":[{"embeddable":true,"href":"https:\/\/mp.moonpreneur.com\/math-corner\/wp-json\/wp\/v2\/comments?post=33927"}],"version-history":[{"count":11,"href":"https:\/\/mp.moonpreneur.com\/math-corner\/wp-json\/wp\/v2\/posts\/33927\/revisions"}],"predecessor-version":[{"id":33942,"href":"https:\/\/mp.moonpreneur.com\/math-corner\/wp-json\/wp\/v2\/posts\/33927\/revisions\/33942"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/mp.moonpreneur.com\/math-corner\/wp-json\/wp\/v2\/media\/33930"}],"wp:attachment":[{"href":"https:\/\/mp.moonpreneur.com\/math-corner\/wp-json\/wp\/v2\/media?parent=33927"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mp.moonpreneur.com\/math-corner\/wp-json\/wp\/v2\/categories?post=33927"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mp.moonpreneur.com\/math-corner\/wp-json\/wp\/v2\/tags?post=33927"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}