Why does $x^2+y=0$ have 8 zeroes?

Question

Does anyone know why $x^2+y=0$ has 8 zeroes in $(\mathbb{F_9^*})^2$?

Maybe I can show this with the primitive element but I'm not sure if this really works like that.

Answer 1

It's because if $x$ is an non-zero element in a field, then so is $-x^2$.