Suppose that $f(x)$ is a quadratic monic polynomial in $x$ with integer coefficients; for instance, $$ f(x) = x^2 + 3 x + 7.$$
Using quadratic reciprocity, it is easy to find necessary and sufficient congruence conditions on a prime $p$ such that $f(x)$ has either (1) no root or (2) two unequal roots modulo $p$. (There can be a multiple root for only finitely many $p$, which we ignore for the moment.)
Now suppose $p$ is such that (2) holds, and let $u_1, u_2$ denote the roots. Can we give necessary and sufficient conditions on $p$ which distinguish between the following cases?
(A) The Legendre symbols $(u_1/p)$ and $(u_2/p)$ are both $1$. (B) They are both $-1$. (C) One of them is $1$ and the other is $-1$.