10. Consider the quadratic congruence $a x^{2}+b x+c \equiv 0(\bmod p)$, where $p$ prime and $a, b$, and $c$ are integers with $p \backslash a$.
a) Let $p=2$. Determine which quadratic congruences $(\bmod 2)$ have solutions.
b) Let $p$ be an odd prime and let $d=b^{2}-4 a c$. Show that the congruence $a x^{2}+b x+c \equiv 0(\bmod p) \quad$ is equivalent to the congruence $y^{2} \equiv d(\bmod p)$, where $y=2 a x+b$. Conclude that if $d \equiv 0(\bmod p)$, then there is exactly one solution $x$ modulo $p$, if $d$ is a quadratic residue of $p$, then there are two incongruent solutions, while if $d$ is a quadratic nonresidue of $p$, then there are no solutions.