Question

Prove that the roots of the equation kx^2+(2k+4)x+8=0 are real for all values of k.

Answer 1

For a quadratic of the form ax^2 + bx + c = 0 to have real roots,
the discriminant b^2 - 4ac ≥ 0

(2k+4)^2 - 4k(8) ≥ 0
4k^2 + 16k + 16 - 32k ≥ 0
k^2 - 4k + 4 ≥ 0
(k-2)^2 ≥ 0
(k - 2) ≥ 0
k-2 ≥ 0 OR -k+2 ≥ 0
k ≥ 2 OR -k ≥ -2 ----> k ≤ 2

Well, k≥ 2 OR k≤2 would refer to all values of k

Answer 2

Recall the discriminant, b^2-4ac
If it is positive, there are two real roots.
So, is (2k+4)^2 - 4*k*8 always positive?

Answer 3

as (k-2) is squared its always positive...so the discriminant is always greater than zero.???