Prove this function satisfies the time independent schrodinger equation

[keith river]

Prove this function satisfies the time independent Schrödinger equation. When V is constant.
Psi = A e^(i k x) + B e^(-i k x)

Attempt:
Time independent Schrödinger equation : (- (hbar^2) / 2m) * (d^2 Psi / d x^2) + V Psi = E Psi

Second order derivative of Psi : (-k^2 A e^(i k x)) + (- k^2 B e^(-i k x))

Divide all by Psi.

I get (-(hbar^2) / 2m) (-2k^2) + V = E

From here I don't know where to go. Any pointers?

 

[fzero]

k is a parameter in the solution. Are there any values of k for which $$\Psi$$ is a solution?