Question

If alpha and beta are the zeros of the quadratic polynomial f(x) = kx^2+4x+4 such that a^2 + beta^2 =24, find the values of k

Answer 1

F(x). = kx2 + 4x. +4

Let alpha =A and beta =B

We know that

A + B = -b/a

A + B = -4/k

And

AB = c/a

AB = 4/k

Now it is given that

A2 + B2 =. 24

(A+B)2 - 2AB = 24

(-4/k)2 - 2*4/k = 24

16/k2 - 8/k = 24

16 - 8k/k2 = 24

16 - 8k = 24k2

24k2 +8k -16 =0

8(3k2 +k -2) =0

3k2 + k -2 =0

3k2 + 3k -2k -2 =0

3k (k +1 ) -2(k +1)=0

(3k -2)(k+1) =0


k = 2/3 and k = -1