{"id":13346,"date":"2020-12-10T10:20:23","date_gmt":"2020-12-10T04:50:23","guid":{"rendered":"https:\/\/cbselibrary.com\/?p=13346"},"modified":"2020-12-10T11:15:06","modified_gmt":"2020-12-10T05:45:06","slug":"quadratic-equations","status":"publish","type":"post","link":"https:\/\/cbselibrary.com\/quadratic-equations\/","title":{"rendered":"Quadratic Equations"},"content":{"rendered":"<h2>Quadratic Equations<\/h2>\n<p><iframe loading=\"lazy\" title=\"Quadratic Equations - Basics | Don&#039;t Memorise\" width=\"1200\" height=\"675\" src=\"https:\/\/www.youtube.com\/embed\/UZTvYYoOrmI?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/p>\n<p>An equation in which the highest power of the unknown quantity is two is called <strong>quadratic equation<\/strong>.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-128865 aligncenter\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Quadratic-Equations-1.jpg\" alt=\"Quadratic Equations 1\" width=\"701\" height=\"458\" srcset=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Quadratic-Equations-1.jpg 701w, https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Quadratic-Equations-1-300x196.jpg 300w\" sizes=\"(max-width: 701px) 100vw, 701px\" \/><\/p>\n<h3>Types of quadratic equation<\/h3>\n<p>Quadratic equations are of two types:<\/p>\n<table border=\"2\">\n<tbody>\n<tr>\n<td style=\"text-align: center;\"><strong>Purely quadratic<\/strong><\/td>\n<td style=\"text-align: center;\"><strong>Adfected quadratic<\/strong><\/td>\n<\/tr>\n<tr>\n<td>ax<sup>2<\/sup> + c = 0 where a, c \u2208 C and b = 0, a \u2260 0<\/td>\n<td>ax<sup>2<\/sup> + bx + c = 0 where a, b, c \u2208 C and a \u2260 0, b \u2260 0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><strong>Roots of a quadratic equation:<\/strong><br \/>\nThe values of variable x which satisfy the quadratic equation is called roots of quadratic equation.<\/p>\n<h3>Solution of quadratic equation<\/h3>\n<p><strong>(1) Factorization method<\/strong><br \/>\nLet ax<sup>2<\/sup> + bx + c = a(x \u2013 \u03b1)(x \u2013 \u03b2) = 0.<br \/>\nThen x = \u03b1 and x = \u03b2 will satisfy the given equation.<br \/>\nHence, factorize the equation and equating each factor to zero gives roots of the equation.<br \/>\n<strong>Example:<\/strong> 3x<sup>2<\/sup> \u2013 2x + 1 = 0 \u21d2(x \u2013 1)(3x + 1) = 0; x = 1, -1\/3<\/p>\n<p><strong>(2) Sri Dharacharya method\u00a0<\/strong><br \/>\nBy completing the perfect square as<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-128866\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Quadratic-Equations-2.jpg\" alt=\"Quadratic Equations 2\" width=\"597\" height=\"297\" srcset=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Quadratic-Equations-2.jpg 597w, https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Quadratic-Equations-2-300x149.jpg 300w\" sizes=\"(max-width: 597px) 100vw, 597px\" \/><br \/>\nEvery quadratic equation has two and only two roots.<\/p>\n<p>https:\/\/www.youtube.com\/watch?v=Nf1Fs8ln8tw<\/p>\n<h3>Nature of roots<\/h3>\n<p>In a quadratic equation ax<sup>2<\/sup> + bx + c = 0, let us suppose that are real and a \u2260 0. The following is true about the nature of its roots.<\/p>\n<ol>\n<li>The equation has real and distinct roots if and only if D \u2261 b<sup>2<\/sup> \u2013 4ac &gt; 0.<\/li>\n<li>The equation has real and coincident (equal) roots if and only if D \u2261 b<sup>2<\/sup> \u2013 4ac =\u00a00.<\/li>\n<li>The equation has complex roots of the form \u03b1 \u00b1 i\u03b2, \u03b1 \u2260 0\u00a0if and only if D \u2261 b<sup>2<\/sup> \u2013 4ac &lt;\u00a00.<\/li>\n<li>The equation has rational roots if and only if a, b, c \u2208 Q\u00a0(the set of rational numbers) and D \u2261 b<sup>2<\/sup> \u2013 4ac is a perfect square (of a rational number).<\/li>\n<li>The equation has (unequal) irrational (surd form) roots if and only if D \u2261 b<sup>2<\/sup> \u2013 4ac &gt; 0 and not a perfect square even if a, b and c are rational. In this case if p + \u221aq, p,q rational is an irrational root, then p &#8211;\u00a0\u221aq\u00a0is also a root (a, b, c being rational).<\/li>\n<li>\u03b1 +\u00a0i\u03b2 (\u03b2 \u2260 0\u00a0and \u03b1, \u03b2 \u2208 R) is a root if and only if its conjugate \u03b1 &#8211;\u00a0i\u03b2\u00a0is a root, that is complex roots occur in pairs in a quadratic equation. In case the equation is satisfied by more than two complex numbers, then it reduces to an identity.<br \/>\n0.x<sup>2<\/sup> + 0.x + 0 = 0, i.e., a = 0 = b = c.<\/li>\n<\/ol>\n<h3>Relations between roots and coefficients<\/h3>\n<p>(1) Relation between roots and coefficients of quadratic equation: If \u03b1 and \u03b2 are the roots of quadratic equation , (a \u2260 0) then<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-128868\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Quadratic-Equations-3.jpg\" alt=\"Quadratic Equations 3\" width=\"463\" height=\"125\" srcset=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Quadratic-Equations-3.jpg 463w, https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Quadratic-Equations-3-300x81.jpg 300w\" sizes=\"(max-width: 463px) 100vw, 463px\" \/><br \/>\n(2) Formation of an equation with given roots: A quadratic equation whose roots are \u03b1 and \u03b2 is given by (x \u2013 \u03b1)(x \u2013 \u03b2) = 0.<br \/>\n\u2234 x<sup>2<\/sup> \u2013 (\u03b1+i\u03b2)x + \u03b1\u03b2 = 0<br \/>\ni.e. x<sup>2<\/sup> \u2013 (sum of roots)x + (product of roots) = 0<br \/>\n\u2234 x<sup>2<\/sup> \u2013 Sx + P = 0<br \/>\n(3) Symmetric function of the roots : A function of \u03b1 and \u03b2\u00a0is said to be a symmetric function, if it remains unchanged when \u03b1 and \u03b2\u00a0are interchanged.<br \/>\nFor example, \u03b1<sup>2<\/sup> + \u03b2<sup>2<\/sup> + 2\u03b1\u03b2\u00a0is a symmetric function of \u03b1 and \u03b2 whereas \u03b1<sup>2<\/sup> + \u03b2<sup>2<\/sup> + 2\u03b1\u03b2\u00a0is not a symmetric function of \u03b1 and \u03b2.<br \/>\nIn order to find the value of a symmetric function of \u03b1 and \u03b2, express the given function in terms of \u03b1 +\u00a0\u03b2 and \u03b1\u03b2. The following results may be useful.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-128869\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Quadratic-Equations-4.jpg\" alt=\"Quadratic Equations 4\" width=\"482\" height=\"283\" srcset=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Quadratic-Equations-4.jpg 482w, https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Quadratic-Equations-4-300x176.jpg 300w\" sizes=\"(max-width: 482px) 100vw, 482px\" \/><\/p>\n<h3>Properties of quadratic equation<\/h3>\n<ol>\n<li>If f(a) and f(b) are of opposite signs then at least one or in general odd number of roots of the equation lie between a and b.<\/li>\n<li>If then f(a) = f(b) there exists a point c between a and b such that f(c) = 0, a&lt;c&lt;b.<\/li>\n<li>If \u03b1 is a root of the equation f(x) = 0 then the polynomial f(x) is exactly divisible by (x &#8211; \u03b1), then (x &#8211; \u03b1)\u00a0is factor of f(x).<\/li>\n<li>If the roots of the quadratic equations a<sub>1<\/sub>x<sup>2<\/sup> + b<sub>1<\/sub>x + c<sub>1<\/sub> = 0 and a<sub>2<\/sub>x<sup>2<\/sup> + b<sub>2<\/sub>x + c<sub>2<\/sub> = 0\u00a0are in the same ratio [i.e. \u03b1<sub>1<\/sub>\/\u03b2<sub>1<\/sub> = \u03b1<sub>2<\/sub>\/\u03b2<sub>2<\/sub>] then b<sub>1<\/sub><sup>2<\/sup>\/b<sub>2<\/sub><sup>2<\/sup> = a<sub>1<\/sub>c<sub>1<\/sub>\/a<sub>2<\/sub>c<sub>2<\/sub>.<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>Quadratic Equations An equation in which the highest power of the unknown quantity is two is called quadratic equation. Types of quadratic equation Quadratic equations are of two types: Purely quadratic Adfected quadratic ax2 + c = 0 where a, c \u2208 C and b = 0, a \u2260 0 ax2 + bx + c &#8230; <a title=\"Quadratic Equations\" class=\"read-more\" href=\"https:\/\/cbselibrary.com\/quadratic-equations\/\" aria-label=\"More on Quadratic Equations\">Read more<\/a><\/p>\n","protected":false},"author":5,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[5],"tags":[4859,4861,64,4860,4858,4857],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v17.8 (Yoast SEO v18.4.1) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Quadratic Equations - CBSE Library<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/cbselibrary.com\/quadratic-equations\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Quadratic Equations\" \/>\n<meta property=\"og:description\" content=\"Quadratic Equations An equation in which the highest power of the unknown quantity is two is called quadratic equation. 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