Let $p$ be an odd prime, and let $a,b,c \in \mathbb{Z}$ satisfy $p \nmid a$. Define the discriminant $\Delta := b^2 - 4ac$. Then the number of residue classes $\bar{x} \in \mathbb{Z}/p\mathbb{Z}$ satisfying
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\begin{align*}
a x^2 + bx + c \equiv 0 \pmod p
\end{align*}