Let and be the roots of the equation ax^2 + bx + c = 0 , where a, c 0 . If the quadratic equation whose roots are 1 a +b and 1 a +b is x^2 - 8x + 2 = 0 , then the value of b is :

Step-by-step solution

Since is a root of ax^2 + bx + c = 0 , it satisfies a ^2 + b + c = 0 . Factoring out from the first two terms gives (a + b) = -c , which implies a + b = - c . Therefore, the first new root simplifies to: 1 a +b = 1 - c = - c Similarly, the second new root is: 1 a +b = - c The sum of these new roots is given by the coefficient of x in x^2 - 8x + 2 = 0 : S = - c - c = - + c Since + = - b a , we have: S = - - b a c = b ac We are given that S = 8 , so b ac = 8 . The product of the new roots is given by the constant ter