Prove that the roots of the equation x^2 + (k + 2)x + 2k = 0 are real for all values of K?

Question

Quadratic Formula.

Answer 1

Well, this factors as (x + 2)(x + k), so it has the roots -2 and -k. These are both real (assuming that k is).

You needn't bother using the quadratic formula to show this.

Answer 2

the discriminant must be positive so

d= (k+2)^2 -8k = k^2+4kx+4 -8k = k^2-4kx+4

this is (k-2)^2 is always positive except for k=2 you have zero

So ,the roots are real.