Non-Linear Programming: Max. $(a-\alpha)Q -(b-\beta)Q^2$ subject to $Q\geq 0$

Question

I am required to work on the following optimization problem:

Max. $(a-\alpha)Q -(b-\beta)Q^2$ subject to $Q\geq 0$

How do I do so?

My FOC is $(a-\alpha) -2(b-\beta)Q-\lambda(-Q)=0$ and

Complimentary Slackness Condition is

$\lambda \geq0, \lambda=0 \text{ if } -Q <0$

How do I continue from here?

Answer 1

Having the FOC, you can try to solve it. Maybe you could solve your first equation for $Q$ in dependence of $\lambda$ (or the other way round). Then plug this solution into your complementary slackness condition.