I am an undergraduate and have been working on some research that I plan to publish. Part of the proof for one of my theorems relies on proving that an equation of the form $ax^2 + bx + c \equiv 0 \pmod{p^y}$ ($p$ an odd prime and $y > 1$ if that ends up mattering) is not "trivial" (by which I mean not every $x$ is a solution) for $x < p$.
My current understanding is that a polynomial of degree $d$ over a finite field either is identically true in the field or has not more than $d$ roots in the field (please correct me if this is wrong). Without getting into what $a,b,c$ are in my proofs (it really doesn't matter for this question), I have proven that $a$ is relatively prime to $p$ (and thus relatively prime to the modulus).
Shouldn't this be sufficient to prove that this polynomial isn't "trivial"? And one step further, the data I've looked at suggests that this polynomial should always have either 0 or two roots - is this sufficient to prove that?