{"id":22067,"date":"2024-01-10T15:58:25","date_gmt":"2024-01-10T10:28:25","guid":{"rendered":"https:\/\/apboardsolutions.in\/?p=22067"},"modified":"2024-01-11T12:25:29","modified_gmt":"2024-01-11T06:55:29","slug":"inter-2nd-year-maths-2a-quadratic-expressions-formulas","status":"publish","type":"post","link":"https:\/\/apboardsolutions.in\/inter-2nd-year-maths-2a-quadratic-expressions-formulas\/","title":{"rendered":"Inter 2nd Year Maths 2A Quadratic Expressions Formulas"},"content":{"rendered":"<p>Use these <a href=\"https:\/\/apboardsolutions.in\/inter-2nd-year-maths-2a-formulas\/\">Inter 2nd Year Maths 2A Formulas<\/a> PDF Chapter 3 Quadratic Expressions to solve questions creatively.<\/p>\n<h2>Intermediate 2nd Year Maths 2A Quadratic Expressions Formulas<\/h2>\n<p>\u2192 If a, b, c are real or complex numbers and a \u2260 0, then the expression ax<sup>2<\/sup> + bx + c is called a quadratic expression in the variable x.<br \/>\nEg: 4x<sup>2<\/sup> &#8211; 2x + 3<\/p>\n<p>\u2192 If a, b, c are real or complex numbers and a \u2260 0, then ax<sup>2<\/sup> + bx + c = 0 is called a quadratic equation in x.<br \/>\nEg: 2x<sup>2<\/sup> &#8211; 5x + 6 = 0<\/p>\n<p><img loading=\"lazy\" src=\"https:\/\/apboardsolutions.com\/wp-content\/uploads\/2022\/07\/AP-Board-Solutions.png\" alt=\"Inter 2nd Year Maths 2A Quadratic Expressions Formulas\" width=\"196\" height=\"17\" \/><\/p>\n<p>\u2192 A complex number \u03b1 is said to be a root or solution of the quadratic equation ax<sup>2<\/sup> + bx + c = 0 if a\u03b1<sup>2<\/sup> + b\u03b1 + c = 0<\/p>\n<p>\u2192 The roots of ax<sup>2<\/sup> + bx + c = 0 are \\(\\frac{-b \\pm \\sqrt{b^{2}-4 a c}}{2 a}\\)<\/p>\n<p>\u2192 If \u03b1, \u03b2 are roots of ax<sup>2<\/sup> + bx + c = 0, then \u03b1 + \u03b2 = \\(\\frac{-b}{a}\\) and \u03b1\u03b2 = \\(\\frac{C}{a}\\)<\/p>\n<p>\u2192 The equation of whose roots are \u03b1, \u03b2 is x<sup>2<\/sup> &#8211; (\u03b1 + \u03b2)x + \u03b1\u03b2 = 0<\/p>\n<p>\u2192 Nature of the roots: \u2206 = b<sup>2<\/sup> &#8211; 4ac is called the discriminant of the quadratic equation ax<sup>2<\/sup> + bx + c = 0. Let \u03b1, \u03b2 be the roots of the quadratic equation ax<sup>2<\/sup> + bx + c = 0<br \/>\nCase 1: If a, b, c are real numbers, then<\/p>\n<ul>\n<li>\u2206 = 0 \u21d4 \u03b1 = \u03b2 = \\(\\frac{-b}{2 a}\\) (a repeated root or double root)<\/li>\n<li>\u2206 &gt; 0 \u21d4 \u03b1 and \u03b2 are real and distinct.<\/li>\n<li>\u2206 &lt; 0, \u21d4 \u03b1 and \u03b2 are non- real complex numbers conjugate to each other.<\/li>\n<\/ul>\n<p>Case 2: If a, b, c are rational numbers, then<\/p>\n<ul>\n<li>\u2206 = 0 \u21d4 \u03b1 and \u03b2 are rational and equal (ei) \u03b1 = \\(\\frac{-b}{2 a}\\), a double root or a repeated root.<\/li>\n<li>\u2206 &gt; 0 and is a square of a rational number \u21d4 \u03b1 and \u03b2 are rational and distinct.<\/li>\n<li>\u2206 &gt; 0 but not a square of a rational number \u21d4 \u03b1 and \u03b2 are conjugate surds.<\/li>\n<li>\u2206 &lt; 0, \u21d4 \u03b1 and \u03b2 are non- real \u21d4 \u03b1 and \u03b2 are non-real con conjugate complex numbers.<\/li>\n<\/ul>\n<p>\u2192 Let a, b and c are rational numbers, \u03b1 and \u03b2 be the roots of the equations ax<sup>2<\/sup> + bx + c = 0. Then<\/p>\n<ul>\n<li>\u03b1, \u03b2 are equal rational numbers if \u2206 = 0.<\/li>\n<li>\u03b1, \u03b2 are distinct rational numbers if \u2206 is the square of a non zero rational numbers.<\/li>\n<li>\u03b1, \u03b2 are conjugate surds if \u2206 &gt; 0 and \u2206 is not the square of a nonzero square of a rational number.<\/li>\n<\/ul>\n<p>\u2192 If a<sub>1<\/sub>x<sup>2<\/sup> + b<sub>1<\/sub>x + c<sub>1<\/sub> = 0, a<sub>2<\/sub>x<sup>2<\/sup> + b<sub>2<\/sub>x + c<sub>2<\/sub> = 0 have two same roots, then \\(\\frac{a_{1}}{a_{2}}=\\frac{b_{1}}{b_{2}}=\\frac{c_{1}}{c_{2}}\\)<\/p>\n<p>\u2192 If \u03b1, \u03b2 are roots of ax<sup>2<\/sup> + bx + c = 0,<\/p>\n<ul>\n<li>the equation whose roots are \\(\\frac{1}{\\alpha}, \\frac{1}{\\beta}\\) is f \\(\\left(\\frac{1}{x}\\right)\\) = 0. If c \u2260 0 (ie) \u03b1\u03b2 \u2260 0<\/li>\n<li>the equation whose roots are \u03b1 + k, \u03b2 + k is f(x &#8211; k) = 0<\/li>\n<li>the equation whose roots are k\u03b1 and k\u03b2 is f\\(\\left(\\frac{x}{k}\\right)\\) = 0<\/li>\n<li>the equation whose roots are equal but opposite in sign is f(-x) = 0<br \/>\n(ie) the equation whose roots are &#8211; \u03b1, &#8211; \u03b2 is f(-x) = 0.<\/li>\n<\/ul>\n<p>\u2192 If the roots of ax<sup>2<\/sup> + bx + c = 0 are complex roots then for x \u2208 R, ax<sup>2<\/sup> + bx + c and &#8216;a&#8217; have the same sign.<\/p>\n<p>\u2192 If \u03b1 and \u03b2 (\u03b1 &lt; \u03b2) are the roots of ax<sup>2<\/sup> + bx + c = 0 then<\/p>\n<ul>\n<li>ax<sup>2<\/sup> + bx + c and &#8216;a&#8217; are of opposite sign when \u03b1 &lt; x &lt; \u03b2<\/li>\n<li>ax<sup>2<\/sup> + bx + c and &#8216;a&#8217; are of the same sign if x &lt; \u03b1 or x &gt; \u03b2.<\/li>\n<\/ul>\n<p>\u2192 Let f(x) = ax<sup>2<\/sup> + bx + c be a quadratic function<\/p>\n<ul>\n<li>If a &gt; 0 then f(x) has minimum value at x = \\(\\frac{-b}{2 a}\\) and the minimum value is given by \\(\\frac{4 a c-b^{2}}{4 a}\\)<\/li>\n<li>If a &lt; 0 then f(x) has maximum value at x = \\(\\frac{-b}{2 a}\\) and the maximum value is given by \\(\\frac{4 a c-b^{2}}{4 a}\\)<\/li>\n<\/ul>\n<p><img loading=\"lazy\" src=\"https:\/\/apboardsolutions.com\/wp-content\/uploads\/2022\/07\/AP-Board-Solutions.png\" alt=\"Inter 2nd Year Maths 2A Quadratic Expressions Formulas\" width=\"196\" height=\"17\" \/><\/p>\n<p>\u2192 A necessary and sufficient condition for the quadratic equation 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 to have a common root is (c<sub>1<\/sub>a<sub>2<\/sub> &#8211; c<sub>2<\/sub>a<sub>1<\/sub>)<sup>2<\/sup> = (a<sub>1<\/sub>b<sub>2<\/sub> &#8211; a<sub>2<\/sub>b<sub>1<\/sub>) (b<sub>1<\/sub>c<sub>2<\/sub> &#8211; b<sub>2<\/sub>c<sub>1<\/sub>).<\/p>\n<p>\u2192 If a<sub>1<\/sub>b<sub>2<\/sub> &#8211; a<sub>2<\/sub>b<sub>1<\/sub> = 0 then common root of a<sub>1<\/sub>x<sup>2<\/sup> + b<sub>1<\/sub>x + c<sub>1<\/sub> = 0, a<sub>2<\/sub>x<sup>2<\/sup> + b<sub>2<\/sub>x + c<sub>2<\/sub> = 0 is \\(\\frac{c_{1} a_{2}-c_{2} a_{1}}{a_{1} b_{2}-a_{2} b_{1}}\\).<\/p>\n<p>\u2192 The standard form of a quadratic ax<sup>2<\/sup> + bx + c = 0 where a, b, c \u2208 R and a \u2260 0<\/p>\n<p>\u2192 The roots of ax<sup>2<\/sup> + bx + c = 0 are \\(\\frac{-b \\pm \\sqrt{b^{2}-4 a c}}{2 a}\\)<\/p>\n<p>\u2192 For the equation ax<sup>2<\/sup> + bx + c = 0, sum of the roots = \\(-\\frac{b}{a}\\), product of the roots = \\(\\frac{c}{a}\\).<\/p>\n<p>\u2192 If the roots of a quadratic are known, the equation is x<sup>2<\/sup> &#8211; (sum of the roots)x +(product of the roots)= 0<\/p>\n<p>\u2192 \u201cIrrational roots\u201d of a quadratic equation with \u201crational coefficients\u201d occur in conjugate pairs. If p + \u221aq is a root of ax<sup>2<\/sup> + bx + c = 0, then p &#8211; \u221aq is also a root of the equation.<\/p>\n<p>\u2192 \u201cImaginary\u201d or \u201cComplex Roots\u201d of a quadratic equation with \u201creal coefficients\u201d occur in conjugate pairs. If p + iq is a root of ax<sup>2<\/sup> + bx + c = 0. Then p &#8211; iq is also a root of the equation.<\/p>\n<p>\u2192 Nature of the roots of ax<sup>2<\/sup> + bx + c = 0<\/p>\n<table border=\"2\">\n<tbody>\n<tr>\n<td>Nature of the Roots<\/td>\n<td>Condition<\/td>\n<\/tr>\n<tr>\n<td>Imagine<\/td>\n<td>b<sup>2<\/sup> \u2013 4ac &lt; 0<\/td>\n<\/tr>\n<tr>\n<td>Equal<\/td>\n<td>b<sup>2<\/sup> \u2013 4ac = 0<\/td>\n<\/tr>\n<tr>\n<td>Real<\/td>\n<td>b<sup>2<\/sup> \u2013 4ac \u2265 0<\/td>\n<\/tr>\n<tr>\n<td>Real and different<\/td>\n<td>b<sup>2<\/sup> \u2013 4ac &gt; 0<\/td>\n<\/tr>\n<tr>\n<td>Rational<\/td>\n<td>b<sup>2<\/sup> \u2013 4ac is a perfect square a, b, c being rational<\/td>\n<\/tr>\n<tr>\n<td>Equal in magnitude and opposite in sign<\/td>\n<td>b = 0<\/td>\n<\/tr>\n<tr>\n<td>Reciprocal to each other<\/td>\n<td>c = a<\/td>\n<\/tr>\n<tr>\n<td>Both positive<\/td>\n<td>b has a sign opposite to that of a and c<\/td>\n<\/tr>\n<tr>\n<td>Both negative<\/td>\n<td>a, b, c all have same sign<\/td>\n<\/tr>\n<tr>\n<td>Opposite sign<\/td>\n<td>a, c are of opposite sign<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u2192 Two equations a<sub>1<\/sub>x<sup>2<\/sup> + b<sub>1<\/sub>x + c<sub>1<\/sub> = 0, a<sub>2<\/sub>x<sup>2<\/sup> + b<sub>2<\/sub>x + c<sub>2<\/sub> = 0 have exactly the same roots if \\(\\frac{a_{1}}{a_{2}}=\\frac{b_{1}}{b_{2}}=\\frac{c_{1}}{c_{2}}\\)<\/p>\n<p>\u2192 The equations a<sub>1<\/sub>x<sup>2<\/sup> + b<sub>1<\/sub>x + c<sub>1<\/sub> = 0, a2x<sup>2<\/sup> + b<sub>2<\/sub>x + c<sub>2<\/sub> = 0 have a common root, if (c<sub>1<\/sub>a<sub>2<\/sub> &#8211; c<sub>2<\/sub>a<sub>1<\/sub>)<sup>2<\/sup> = (a<sub>1<\/sub>b<sub>2<\/sub> &#8211; a<sub>2<\/sub>b<sub>1<\/sub>)(b<sub>1<\/sub>c<sub>2<\/sub> &#8211; b<sub>2<\/sub>c<sub>1<\/sub>) and the common root is \\(\\frac{c_{1} a_{2}-c_{2} a_{1}}{a_{1} b_{2}-a_{2} b_{1}}\\) if a<sub>1<\/sub>b<sub>2<\/sub> \u2260 a<sub>2<\/sub>b<sub>1<\/sub><\/p>\n<p>\u2192 If f(x) = 0 is a quadratic equation, then the equation whose roots are<\/p>\n<ul>\n<li>The reciprocals of the roots of f(x) = 0 is f\\(\\left(\\frac{1}{x}\\right)\\) = 0<\/li>\n<li>The roots of f(x) = 0, each \u2018increased\u2019 by k is f(x &#8211; k) = 0<\/li>\n<li>The roots of f(x) = 0, each \u2018diminished\u2019 by k is f(x + k) = 0<\/li>\n<li>The roots of f(x) = 0 with sign changed is f(-x) = 0<\/li>\n<li>The roots of f(x) = 0 each multiplied by k(\u22600) is f\\(\\left(\\frac{x}{k}\\right)\\) = 0<\/li>\n<\/ul>\n<p><img loading=\"lazy\" src=\"https:\/\/apboardsolutions.com\/wp-content\/uploads\/2022\/07\/AP-Board-Solutions.png\" alt=\"Inter 2nd Year Maths 2A Quadratic Expressions Formulas\" width=\"196\" height=\"17\" \/><\/p>\n<p>\u2192 Sign of the expression ax<sup>2<\/sup> + bx + c = 0<\/p>\n<ul>\n<li>The sign of the expression ax<sup>2<\/sup> + bx + c is same as that of \u2018a\u2019 for all values of x if b<sup>2<\/sup> &#8211; 4ac \u2264 0 i.e. if the roots of ax<sup>2<\/sup> + bx + c = 0 are imaginary or equal.<\/li>\n<li>If the roots of the equation ax<sup>2<\/sup> + bx + c = 0 are real and different i.e b<sup>2<\/sup> &#8211; 4ac &gt; 0, the sign of the expression is same as that of \u2018a\u2019 if x does not lie between the two roots of the equation and opposite to that of \u2018a\u2019 if x lies between the roots of the equation.<\/li>\n<\/ul>\n<p>\u2192 The expression ax<sup>2<\/sup> + bx + c is positive for all real values of x if b<sup>2<\/sup> &#8211; 4ac &lt; 0 and a &gt; 0.<\/p>\n<p>\u2192 The expression ax<sup>2<\/sup> + bx + c has a maximum value when \u2018a\u2019 is negative and x = &#8211;\\(\\frac{\\mathrm{b}}{2 \\mathrm{a}}\\). Maximum value of the expression = \\(\\frac{4 a c-b^{2}}{4 a}\\)<\/p>\n<p>\u2192 The expression ax<sup>2<\/sup> + bx + c has a maximum value when \u2018a\u2019 is positive and x = &#8211;\\(\\frac{\\mathrm{b}}{2 \\mathrm{a}}\\). Minimum value of the expression = \\(\\frac{4 a c-b^{2}}{4 a}\\)<\/p>\n<p>Theorem 1:<br \/>\nIf the roots of ax<sup>2<\/sup> + bx + c = 0 are imaginary, then for x \u2208 R , ax<sup>2<\/sup> + bx + c and a have the same sign.<br \/>\nProof:<br \/>\nThe root are imaginary<br \/>\nb<sup>2<\/sup> &#8211; 4ac &lt; 0 4ac &#8211; b<sup>2<\/sup> &gt; 0<br \/>\n\\(\\frac{a x^{2}+b x+c}{a}=x^{2}+\\frac{b}{a} x+\\frac{c}{a}=\\left(x+\\frac{b}{2 a}\\right)^{2}+\\frac{c}{a}-\\frac{b^{2}}{4 a^{2}}=\\left(x+\\frac{b}{2 a}\\right)^{2}+\\frac{4 a c-b^{2}}{4 a^{2}}\\)<br \/>\n\u2234 For x \u2208 R, ax<sup>2<\/sup> + bx + c = 0 and a have the same sign.<\/p>\n<p>Theorem 2.<br \/>\nIf the roots of ax<sup>2<\/sup> + bx + c = 0 are real and equal to \u03b1 = \\(\\frac{-b}{2 a}\\), then \u03b1 \u2260 x \u2208 R ax<sup>2<\/sup> + bx + c and a will have same sign.<br \/>\nProof:<br \/>\nThe roots of ax<sup>2<\/sup> + bx + c = 0 are real and equal<br \/>\n\u21d2 b<sup>2<\/sup> = 4ac \u21d2 4ac &#8211; b<sup>2<\/sup> = 0<br \/>\n\\(\\frac{a x^{2}+b x+c}{a}\\) = x + \\(\\frac{b}{a}\\)x + \\(\\frac{c}{a}\\)<br \/>\n= \\(\\left(x+\\frac{b}{2 a}\\right)^{2}+\\frac{c}{a}-\\frac{b^{2}}{4 a^{2}}\\)<br \/>\n= \\(\\left(x+\\frac{b}{2 a}\\right)^{2}+\\frac{4 a c-b^{2}}{4 a^{2}}\\)<br \/>\n= \\(\\left(x+\\frac{b}{2 a}\\right)^{2}\\) &gt; 0 for x \u2260 \\(\\frac{-b}{2 a}\\) = \u03b1<br \/>\nFor \u03b1 \u2260 x \u2208 R, ax<sup>2<\/sup> + bx + c and a have the same sign.<\/p>\n<p>Theorem 3.<br \/>\nLet be the real roots of ax<sup>2<\/sup> + bx + c = 0 and \u03b1 &lt; \u03b2. Then<br \/>\n(i) x \u2208 R, \u03b1 &lt; x&lt; \u03b2 ax<sup>2<\/sup> + bx + c and a have the opposite signs<br \/>\n(ii) x \u2208 R, x &lt; \u03b1 or x&gt; \u03b2 ax<sup>2<\/sup> + bx + c and a have the same sign.<br \/>\nProof:<br \/>\n\u03b1, \u03b2 are the roots of ax<sup>2<\/sup> + bx + c = 0<br \/>\nTherefore, ax<sup>2<\/sup> + bx + c = a(x &#8211; \u03b1)(x &#8211; \u03b2)<br \/>\n\\(\\frac{a x^{2}+b x+c}{a}\\) = (x &#8211; \u03b1)(x &#8211; \u03b2)<\/p>\n<p>(i) Suppose x \u2208 R, \u03b1 &lt; x &lt; \u03b2<br \/>\n\u21d2 x &lt; \u03b1 &lt; \u03b2 then x &#8211; \u03b1 &lt; 0, x &#8211; \u03b2 &lt; 0 \u21d2 (x &#8211; \u03b1)(x &#8211; \u03b2) &gt; 0 \u21d2 \\(\\frac{a x^{2}+b x+c}{a}\\) &gt; 0<br \/>\n\u21d2 ax<sup>2<\/sup> + bx + c, a have a same sign<\/p>\n<p>(ii) Suppose x \u2208 R, x &gt; \u03b2, x &gt; \u03b2 &gt; \u03b1 then x &#8211; \u03b1 &gt; 0, x &#8211; \u03b2 &gt; 0<br \/>\n\u21d2 (x &#8211; \u03b1)(x &#8211; \u03b2) &gt; 0 \u21d2 \\(\\frac{a x^{2}+b x+c}{a}\\) &gt; 0<br \/>\n\u21d2 ax<sup>2<\/sup> + bx + c, a have same sign<br \/>\n\u2234 x \u2208 R, x &lt; \u03b1 or x &gt; \u03b2 \u21d2 ax<sup>2<\/sup> + bx + c and a have the same sign.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Use these Inter 2nd Year Maths 2A Formulas PDF Chapter 3 Quadratic Expressions to solve questions creatively. Intermediate 2nd Year Maths 2A Quadratic Expressions Formulas \u2192 If a, b, c are real or complex numbers and a \u2260 0, then the expression ax2 + bx + c is called a quadratic expression in the variable &#8230; <a title=\"Inter 2nd Year Maths 2A Quadratic Expressions Formulas\" class=\"read-more\" href=\"https:\/\/apboardsolutions.in\/inter-2nd-year-maths-2a-quadratic-expressions-formulas\/\" aria-label=\"More on Inter 2nd Year Maths 2A Quadratic Expressions Formulas\">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":[15],"tags":[],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/apboardsolutions.in\/wp-json\/wp\/v2\/posts\/22067"}],"collection":[{"href":"https:\/\/apboardsolutions.in\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/apboardsolutions.in\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/apboardsolutions.in\/wp-json\/wp\/v2\/users\/5"}],"replies":[{"embeddable":true,"href":"https:\/\/apboardsolutions.in\/wp-json\/wp\/v2\/comments?post=22067"}],"version-history":[{"count":1,"href":"https:\/\/apboardsolutions.in\/wp-json\/wp\/v2\/posts\/22067\/revisions"}],"predecessor-version":[{"id":22068,"href":"https:\/\/apboardsolutions.in\/wp-json\/wp\/v2\/posts\/22067\/revisions\/22068"}],"wp:attachment":[{"href":"https:\/\/apboardsolutions.in\/wp-json\/wp\/v2\/media?parent=22067"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/apboardsolutions.in\/wp-json\/wp\/v2\/categories?post=22067"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/apboardsolutions.in\/wp-json\/wp\/v2\/tags?post=22067"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}