Suppose that the equation a x^2+b x+c=0 has roots and , both of which are different from 1 3 , then an equation whose roots are 1 3 -1 and 1 3 -1 is

Step-by-step solution

a x^2+b x+c=0...(i) Let and are the roots of Eq. (i). + =- b a and = c a Now, the quadratic equation whose roots are aligned & 1 3 -1 , 1 3 -1 & Sum of roots = 1 3 -1 + 1 3 -1 & = 3 -1+3 -1 (3 -1)(3 -1) = 3( + )-2 9 -3( + )+1 & = 3 (- b a )-2 9 ( c a )-3 (- b a )+1 & = -3 b-2 a 9 c+3 b+a =- (3 b+2 a) a+3 b+9 c & aligned Product of roots = 1 (3 -1)(3 -1) aligned & = 1 9 -3( + )+1 & = 1 9 ( c a )-3 (- b a )+1 & = a a+3 b+9 c aligned Now, the required equation x^2- - ( 3 b+2 a a+3 b+9 c ) x + a a+3 b+9 c =0 [ . for an