Let $n, m, k \in \mathbb{N}$ and $p \in \mathbb{P}$ be an odd prime with gcd$(n, p) =1$. How would one approach proving:
$$ \#\{x \in \mathbb{F}_p\,\vert\ nx^2 + mx + k \equiv 0 \pmod{p}\} = 1 + \frac{n^2 - 4mk}{p} $$
How many solutions are there for when gcd$(n, p) \neq 1$?
I know about quadratic reciprocity, but am stuck on how to apply it in a useful way here.