We looked at linear congruences and now we want to look at congruences of the form

$$ \begin{align*} ax^2 + bx + c \equiv 0 \pmod{p} \end{align*} $$

If we complete the square then we get

$$ \begin{align*} a(x + b/2)^2 \equiv \frac{b^2 - 4ac}{4a} \pmod{p} \end{align*} $$

References