Solutions to prime power moduli

I am trying to show:

$f(x)=x^2 - 2x$ has precisely two solutions modulo $p^k$ where $p$ is a 0 odd prime, and $k$ is an positive integer.

I'm thinking I need to use Hensel's lemma, and I have shown it is true for $p^2$, but I'm not sure how to continue.

Homework, so hints are what I'm looking for.

Note that $f(x)=x(x-2) \equiv 0 \pmod {p^k}$.
However, note that $p$ cannot divide both $x$ and $x-2$ since $2$ is even and $p$ is odd.
• Note that $x(x-2) \equiv 0 \pmod {p^k}$ when $x \equiv 0 \pmod {p^{t}}$ and $x \equiv 2 \pmod {p^{k}}$. However, from my answer, $t$ and $k$ cannot both be positive. One has to be $0$. – S.C.B. Feb 27 '16 at 11:42