[Math] Let $p$ be an odd prime s.t $\gcd(a, p) = 1$. Show that $x^2 -a \equiv 0 \mod p$ has either $0$ or $2$ solutions modulo $p$

Question

Let $p$ be an odd prime and $a$ be an integer with $\gcd(a, p) = 1$. Show that $x^2 – a \equiv 0 \mod p$ has either $0$ or $2$ solutions modulo $p$

I am clueless with this one. Hints please.

Answer 1

If $x^2-a=0 \mod p$ has some solution $b$, it means that $b^2=a$, and hence you original question becomes

$$x^2-b^2 \equiv 0 \mod p$$

Can you prove now that this equation has exactly two solutions?