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TOPIC, AND ITS RELATED TERMS, oUADRATIC EQUATIONS, we, , polynomials,, , It is of the form ax +bx + C = O, a, , have learnt about, , It is a quadratic equation., , In the chapter, and their z e r o e s ., poiynomials, and cubic, quadratic, about quadratic, will study, chapter, we, roots, this, in, finding their, equations, , of, , their applications,, , along with, situations. Quadratic, situations, , ways, , various, , and, , based, , equations, , on, , -20x+ 1) =(*-1)+3), , (B), , -2x +x-2 =x* -x +3x - 3, -X-2 2x - 3, , daily life, several, , in, , arises, , and in different, around the world, , 3x-1 0, , field of, , It is not of the form ax + bx + C = 0, a z 0, , mathematics, , It is not a quadratiC equation., , Quadratic Equation, of the form, , ax, , in x is, A quadratic equation, numbers and, b, and c are real, where, , a,, , Egxx, , -, , a, , +, :, , bx, , + C, , =, , 0,, , not, remember that'term should exist, , 3x+ 7, 50 0 -x +1300x =0, 4x of a Quadratic Equation, , Standard Form, , is, , form p(x) = 0, where p(x), Any equation of the, But, a quadratic equation., quadratic polynomial, is, order, of p(x) in descending, wher we write the terms, the stondard form of the, of their degrees, then we get, 0), is called the, bx + C =0 (a, equatuon Thus., a, , ax, , equations, but, , tg, , they, , Example, , equations, , 1., are, , All the values of variables which sastisfy the given, , quadratic equation, , be, , quadratic, , Any quadratic equation, , be reduced to., , "The, , and x - 2x =(-2) (3 -x), , 306", , (A)x(2x 3)=x1, , can, , have at, , +, , bx, , most two root, , given, , the, , consective, , product of 2, , situation in the, , positive integers is, [CBSE 2020, NCERT], , integers be, Ans. Let, the two consecutive positive, and x + 1., Then, according to the question,, , xx+1), , [NCERT], , (B)(x-2)o+ 1) =(x -1) (x + 3), , ax, , form of a quadratic equation., , can, , Check whether the following, quadratic equations or not:, , roots or zeroes., , Important, , not, , seem, , called its, , are, , Thus, if a is a root of the quadratic equation, +C =0, a 0, then au + ba + C= 0., , do, , 2x- 3), , 11, , to, , Roots ofa Quadratic Equation, , Example 2. Represent, , standard form of o quodratiIc equation., equations,, , ACaution, , While deciding whether a given equation is quadratic or, , 0., , 0.7x-51 0, , Some, , 0., , =, , x, , 306, , x+X = 306, , Ans. (A) We have,, x + X -306, , x(2x + 3) =x*+1, , 2x+3x =x+1, x3x-1 =0, , =, , 0, , required quadratic equation., , which is the, , TOPIC2, METHODS OF SOLVING A QUADRATIC EQUATiON, Solving o quodratiC equation means determining its, , ie., , roots (or solution), , band c., , A, , quodratuc equation con be solved by the following, methods, (1) factorisation method, (2) Compleing the square rmethod, (3) Quadratuc formula, , bx, , Calculate, , + C, , values of, 0 and find the, , =, , of its, , two, , 1e a c, , =, , px, , q andp +q=, , Where, p and, , Step IV:, , standard form, , a, , write, ther, , Sum is equal tO b., , Put the value, , of b,, , obtained, , Write the, , equation in, , linear factors., , b, , factors of, q are, , given equation, , in, , and c. then, factors such that, , the product of a, , sum, , it a s a, , Stepl1:, , Factorisation Method, We use the, following steps, , Step: Write the given equation, , Step II:, , ax, , +, , ac, in, , step, , in, , te, , oroduct of, the form of prod

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Step V: Now, equate each factor, desired roots., , to zero to, , get the, , 2x-14x-120, , Eg., , 0, , x2-7x-60 = 0, , or, , By factorisation method,, , x+7x+ 12 = 0, x+3x + 4x + 12 = 0, , x-12x+ 5x -60 0, , x (x+ 3)+4(x+ 3) = 0, , x - 1 2 ) + 5(x- 12) = 0, , x-12) (x + 5) = 0, , (x+3) (x + 4) =0, x+3 = 0 or x + 4, , X- 12 = 0 or x + 5 = 0, , =0, , X, , X = - 3 orx = -4, , Thus, -3 and -4, , are, , two, , roots of, , the, , X, , Example, , 3., Find the roots of the, quaratic, equation 2 x* + 7x + 52 = 0, by factorisation, method., , [NCERT], , Ans. Given, equation is:, , 5, , But, length cannot be negative., , quadratic, , equation., , = 12 or x = -, , 12, , =, , Hence,, , base of triangle = 12 cm, , and,, , altitude of traingle 12 -7, , 5, , ACaution, Read the word problem twice, before formulating it into, an equation. Be clear about what is asked and how to go, , v2x+7x+5v2 =, , through., , Here, a=v2.b= 7. c= 5v2, , Completing the Square Method, 0,, C =0, a, To solve an equation ax + bx, +, , Then, , v2x5/2 =10, , axC=, , = 5x 2(px), , and, , a +C, , 5+2(p +g) = 7 =b, , 2x +5x4+2x+5v2 =0, , 0, , or, , x=, , v2x+5, , -v2, , Step, , by, , Divide the equation, b, , C, , a, , a, , a, , and, , get, , an, , equation as x+=x+=0, , a, , Step ll: Add square of half of the coefficient of x on, both sides of the equation, obtained in Step, , =0, , orx=, , this, , method, following steps are adopted:, , x+x, , x22x5)-0, =, , by, , Step II: Shift the constant term on RHS to get, , = x+5)-v2|V2x+5)-0, , X+v2, , cm, , -5, , Hence, the required roots are -v2 and, , ACaution, Practicing of these types of problems, helps to increase, the accuracy in such problems., , b-4ac, , Example 4. The altitude of a right triangle is, , 4a, , 7 cm less than its base. If the hypotenuse is 13 cm,, , find the other two sides., , [NCERT, , b2-4ac, 4a, , b, , x* 2a, , Ans. Let, the base of the triangle be 'x cm., , Then, according to question,, altitude oftriangle = (-7) cm, , Given, , 4a2, , hypotenuse = 13 cm, , x=, , or, , Now, by Pythagoras theorem, , Vb-4ac, , 2, , 2a, , (Hypotenuse) =(Perpendicular), , -btvb2-4ac, , +(Base), , 13 (x-7), =, , X=., , 2a, , +*, , 169 x+ 49 14x + x, 2x 14x + 49 169 = O, , Thus, the roots of the quadratic equation ax + bx + C = 0, are given by x =, , -btvb-4ac, 2a

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Find the roots, Example 5., , 4x+ 43x, , (x-7)-(x+4)11, , equation, , (x+4)(x -7), , of completing, , the method, , 0, by, , +3, , ofthe quadratic, , (NCERT], , x-7-x-4, , the square., , 4x+, Ans. Given,equation is, , 43x +3, , by 4,, , sides, On dividing both, , we, , half of square of, , (x+ 4)(x-7), , =0, , get, , -11, , 11, , 3, , x+4x-7x-28, , 30, , x+3x =, Now add, , 30, , -1, , coefficient, , of, , x on, , -3x 28, , 30, , x-3x-28 =-30, both side of equation 1e., , x-3x+ 2 =0, , v3x, , On comparing it with, ax + bx + C = 0, we get, , -3.3, , a, , 1,b=-3, c = 2, , Then,, , D, , b-4ac, =, , :a+2ab +b? = (a +b), Then, , standard equation, , (-3)2-4 x 1x2, 9-8 1>0, , Hence, its roots exist., , Now,, , -btvb-4ac, , X =, , 2a, , -3)+V1, 2x1, X =, , -, , 3, , or x, , =, , -, , 2, , Hence, the roots of the given equation are, and-3, , ACaution, , =, , -Follow oll the steps, , carefully while solving, completing the square method., , equation by, , the, , quadratic, , quadratic equation ax, , say, , and, , a, , -b+ vb-4ac, , bx, , =0, , has two roots,, , formula, , is, , known, , as, , is 8 times the, , quadratic formula., as discriminant., , - 4ac is known, will exist if, b- 4ac>0., =, , Important, formula, This, , 6., , equation, , 1, , x+4, ratic formula., , -, , 4ac)., , And, , larger number. Find the two numbers., [CBSE 2012, NCERT], , Ans. Let, the numbers be, Given that,, , x, , and y, where, , y, , x >, , y., , 8x, , According to the question,, , is also, , Example, , through (b, , numbers is 180. The square of the smaller number, , -b-b4ac, 2a, , where, D b, , roots, , + C, , While calculating the roots by, quadratic formula, check, whether roots exist or not by checking, , Example 7. The difference of squares of two, , and B =, , 2a, , and this, , +, , B. given by, , Hence, the roots are 2 and 1., , ACaution, , Quadratic Formula, A, , 2 or 1, , known, , as, , Find the roots of, the, , ,x, 30, , x-7, , Ans. Given,, equation, , x-y =180, , Sridharacharya formula., , quadratic, , 4, 7 using quadCBSE 2020, NCERT], , is, , X+4 X-7, , x- 8x = 180, from equation (0), , -8x 180 0, x-18x +10x- 180 = 0, x(x-18) + 10(x - 18) = 0, , (x+10)(x- 18), Xt-4,7, , X+ 10, , =, , 0, , 0 or x - 18 = 0, , X= -10 or x = 18

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Then, increased speed =(x + 5) km/h, , If x = 18, then,, , y, , 8x 18, , And, the time taken, , y= v2x4x2x9, , speed =, , to, , cover, , 360 km in increased, , 360h r, x+5, , = 12, , Now, according to the question,, , Ifx= -10, then,, , 360 360 =1, , y= v-8x10.which is not possible, , X+5, , X, , Hence, the numbers are 18 and 12., , 360 x+5X =1, x(X +5), , Example 8. A train travels 360 km at a uniform, , 360 x 5 x, , speed. If the speed had been 5 km/hr more, then it, would have taken 1 hour less for the same journey., Find the speed of the train., , + 5x, , x+5x-1800=0, By factorisation method, , CBSE 2019, NCERT], , x+45x-40x - 1800 =0, , Ans. Let, the speed ofthe train bex km/hr., , xx + 45), , 360, ..Time taken to cover 360 km =20hr, , 40(x + 45), , -, , =, , 0, , xt-40) (x+ 45) =0, X =40 orx = -45, , Time=, , Distance, Speed, , Neglectx= -45, as speed could, , not, , be, , negative, , x = 40, , Hence, the original speed of the train is 40 km/hr., , If the speed is increased by 5 km/hr,, , TOPIC3, NATURE OF ROOTS OF A QUADRATIC EQUATIOON, The nature of roots ofa quadratic equation depends, , Example 9., , upon the value of discriminant D., As, the value of discriminal can be zero, positive or, , Find the nature of roots of the, , quadratic equation 3x, , -, , 4, , V3x, , +4, , =0. Ifreal roots, [NCERT], , exist, then find them., , negative, so three cases arises., , Ans. Given, equation is:, , Quadratic Equation, , ax +bx+ c=0, , 3x2- 43x +4 = 0, , a! 0, , On comparing it with axd + bx+C= 0, we get, , a, , Find discriminant, D =b-4ac, , 3,b =-43 and c= 4, , Then, discriminant,, D b4ac, =, , IfD <0, , IfD 0, , ie.b-4ac<0, , ie.b-4ac =0, Then, roots are, , Then, roots are, , real and equal, , imaginary or does, not exists, , Asx= -bt0, , As, D<0, then D can, , 2a, , not be evaluated, , As D, , -43-4x3x4, , = 48-48 0, 0, then nature of roots are real and, , equal, , Now,, , X=, , -btvD, 2a, , -45)-0, 2x3, , 2/3, IfD>0, , ie.b- 4ac> 0, , 3, 2, , Then, roots are real, , and unequal, , -bt D, , As, x =, , ., , 2a, , Hence, the roots of the quadratic equation are, and real & equal., , 2

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This, , quadratic equation has two, , Here, a = k, b = -2k, c =6, , ACaution, , always compare the, , While finding the nature the, ie. ax*, standard quadratic equation, given equation with the, , of, , roots,, , +bx +C = 0., the quadratic, Find the value of 'k for, equal, 0, if they have two, equation kx (x - 2)+ 6, , Example 10., , So, D = 0, , b-4ac=0, , (-2)-4 xkx6=0, 4k-24k=0, , 4k (k-6)=0, , [NCERT, , real roots., , equal ro0ts, , 4k=24, , is:, Ans. Given, quadratic equation, kx(x 2) +6 0, kx 2kx +6 0, , k=6, , =, , Hence, the value of k is 6., , OBJECTIVE Type Questions, [1 mark, Multiple Choice Questions, 1. The value(s) of k for which the quadratic, , equation 2x +kx+2= 0 has equal roots,is:, (o) 4, , (b)4, , ), , (d), , -4, , [CBSE 2020], , 0, , Ans. (b)4, , (a) +2x+1 (4-x2 +3, , b)-2x (5 -x, (C)k+1)x, , 2x=7, wherek=-1, , (d)-x=(*-1)3, , Explanation:, The equation 2x* + kx +2 =0 has equal roots,, D = (k-4(2)(2) = 0, when, , Explanation: Simplify the given equations., , +2x+1 =(4 -x) +3, , (: D b - 4ac), , x+2x+ 1 16 +x -8x +3, , = 16, k =4, , or, , [CBSE 2020, , Ans. (d) x-x* = (x- 1)3, , 10x= 18, 10x- 18 =0, , 2. The quadratic equation x2 - 4x + k = 0 has, , This equation is not of the form ax + bx+ C, , distinct real roots if, (a) k= 4, , (b) k> 4, , = 0, a : 0., , () k= 16, , (d) k, , Thus, it is not a quadratic equation., , 4, , Ans. (d) k < 4, , -2-5-92x, , Explanation: Equation x - 4x+k = 0 will have, , distinct real roots, if, , D =(-4)-4(1)() > 0, , 50x+ 2x -10 =0, , :D b2-4ac), i.e, , 16-4k>0, , or, , k<4, , 52x- 10=0, , 16> 4k, , This is aso not a quadratic equation as t, also not, , 3. Which of the following is a, equation?, , 2X, , -2x 10x - 2x2-2+, , of the form ax+ bx +, , c=, , 0,a, , quadratic, , (c), , (k+1)x+, , X= 7, where k, , =, , -, , 1, , *0.

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4-8 +5=0, , -1+ 1)x+=7, , 1 # 0, , 0+, , 7, , 2, , 3x, , So, x = 2 is not a root of x, , - 4x + 5, , 0., , (b)Puttingx= 2 in x +3x-12 =0, , 14= 0, , we get, , This is also not a quadratic, equation as it is, also not of the form ax + bx + c, a = 0., =, , (2)+3(2) -12 =0, , 0,, , 4 +6, , x-x =(x-13, , d), , -2 0, , x-=-3(1) 3x{1)? -(1)*, +, , :(a, , -, , b), , a', , =, , -, , 3ab +3ba, , x3-x=x-3x2+3x-1, -x*+ 3x -3x+ 1, , 2x-3x, This represents, , +, , 1, , -, , 12 = 0, , So, x = 2 is not a root of x* + 3x - 12 = 0., , b', , (c) Putting x = 2 in 2x2- 7x+ 6 =0,, we get, , 0, , 2(2-7(2)+ 6= 0, , =0, , 8-14+6 0, , quadratic equation because, itis of the form ax+ bx + c = 0, a 0., , 0 = 0, , a, , 4.Which of the following, , is not, , equation?, , a, , So,x= 2 is a root of the equation 2x-7x + 6 = 0., , (d) Puttingx= 2 in 3x2- 6x -2, , quadratic, , we get, , 3(2)-6(2) -2 = 0, , (a) 2(x 1)? =4x2 2x + 1, , 1212-2 0, , (b) 2x-x2 =x2+5, , -2, , () (2x+V3+=3-Sx, , So, x =, , 2, , [NCERT Exemplar], 5. For what value of k, kx +8x+ 2, , 0 has real, , (b) k»8, , (c) k= 8, , (d) none of these, , A, , real number a is said to be the root of the, , quadratic equation ax + bx + C= 0 ifaa + ba +C = 0., , 7. The positive, (a) 2, , root of y3x+6=3is:, , 4, , (d) 3, , Ans. (d) none of these, Explanation: kx +8x +2 = 0, , root of the equation, , 0., , eConcept Applied, , roots?, , (o) k 8, , 0, , is not a, , 3x-6x-2, , (d) +2x) =x*+3+ 4x3, , (b):1, , Ans. (b) 1, , Here, a = k, b = 8, c = 2, , Explanation: v3x2+6=3, , For real roots,, , Squaring both sides, we get, , b2-4ac20, , 3x+6 9, , 3, , Positive root is 1., , 8-k> 0 k s 8, , 8.Which of the following equations has the, sum of its roots as 3?, , 6. Which of the following equations has 2 as a, , (a) 2x2-3x+ 6 0, , root?, , (o)x-4x, , 5, , =, , (b)x3x, , 0, , ()2x-7x + 6, , (d) 3x2-6x, , 0, , (b) -x*+3x-3 0, , 12 =0, , -2 =0, , (c)2-X+1, 0, 2, , CBSE 2012], , (d) 3x-3x +3 =0, , 7x + 6- 0, , Explanation:, (a), , 3x, , x=1x=+1, , 82-4xkx 20, 64 8k:0, , Ans. (c) 2x, , O0, , Putting x, , =, , 2 in, , x, , -, , we get, , (2-4(2)+ 5 = 0, , 4x, , +, , 5, , =, , 0,, , 9., , Is x- 4x, equation?, (a) Yes, (b) No, , x, , +, , [NCERT Exemplar], 1, , (x, , -2), , a, , quadratic

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So, the equation has no real roots., , (c) Can't say, , (d), , This is, , cubic, , a, , (d) The given equation is:, , equation, , 5x, , Ans. (a) Yes, , 4x, x-4x2 -x+ 1, , Explanation: x*, , xs, , =, , 2x2 13x + 9, , 1, , x +, , -, , -, , -, , a, , 2), , = 5, b = -3, c, , = (-3)-4(5)(1), , 0, , =9, , This is a quadratic equation., , So, the equation has no real roots., , Concept Applied, , quadratic equation ax, , A, , (b)1, , =, , (2) two equal roots if b, , ()1, , - 4ac = 0., - 4ac < 0., , The roots of the quadratic equation, , 12., , has two, 11. Which of the following equations, , 2x-x, , 6, , 0 are:, , distinct real roots?, , =0, , x - 5 =0, , b), , bx +C 0, a :0 has:, , +, , (1) two distinct real roots if b ' - 4ac>0., , (3) no real roots if b, , (a) 2x-3v2x+, , 20, , = -11 <0, , 10.Rootsof -x*x+0 are, , (o)-1, , =1, , D= b - 4ac, , 6x+ 12x, , 8, , -, , (x, , =, , 3x + 1 = 0, , o)-2,, , 6) 22, , -2,-, , (d) 2,, , 3x+ 2 2 0, , (c), , (d) 5xAns. (b) x, , 3x, , +, , +x - 5, , 1, , [NCERT Exemplar], , 0, , 13. The roots of t h e equation x, , 0 , where m is, , =0, , b, Explanation: We know that if D =, +, for a quadrotic equation ax + bx, , - 4ac, C, , =, , >, , 0, , 0, then, , (a), , m,, , m, , +, , a, , 3x -, , constant,, (6), , 3, , m(m + 3), , are:, , -m, m +3, , (d) -m,-(m+ 3), , (c) m, -(m + 3), , [CBSE 2011, , ts roots are real and distinct., , Ans. (b) -m, m + 33, , (a) The given equation is., , Explanation: Given, equation is:, , 2x-32x+ =0, , x, , o 2. b= -32. c =, , 3) m]x - m(m + 3) 0, x - (m + 3) x + mx m (m + 3)= 0, xx - (m + 3)] + m |x- (m+3)] =0, 0, x + m) x (m + 3)] =, x - [m, , D =b2-4ac, , (-32-4(2), , - 3x - m(m + 3) =0, , +, , =, , -, , -, , -, , =1 8, , 18 = 0, , X = -m, (m +3), , So, the equation has real and equal roots., , x+X - 5, , has two, , 0, , =, , D = b- 4ac, = (1) - 4(1)(-5), , = 1, , 20 = 21 >0, , So, the equation has two distinct real roots., (c) The given equotion is:, , x3x+ 2/2 0, , (a)1, , (b), , 0, 2, , (c) 0, 1, , (d), , -1, 0, , +, , (CBSE 2012], , Ans. (a)t1, Explanation:, Given, equation is mx, m, b =2 and, Here, a, =, , O = 1.b = 3, c = 22, b, , equal, , mx, , are:, , O = 1, b = 1, c= -5, , D, , 2x + m = 0,, values of m, roots, then the, , 14. If the quadratic equation, , (b) The given equation is:, , - 4ac, , = (3)- 4(1)(2/2), , = 9 8/2 <0, , So,, , +, , c, , 2x, =, , m, , +, , =0, , m, , D b -4ac, = (2), , 4, , - 4, , 4m, , x mx m