Past exam of the options II course of the University of Oxford / 2010 / C / C2 / iii / Solution

The Legendre symbol equal to $1$ says that $a$ is a nonzero square modulo $p$. Choose one root $r\ne 0$. Then $-r$ is another root, distinct because $p$ is odd. Any solution $x$ satisfiesin the field ${\mathbb{F}}_p$, so $x=r$ or $x=-r$. Thus there are