The line with equation $y=5 x+a$ passes through the point $\left(a, a^{2}\right)$. If $a \neq 0$, what is the value of $a$ ?
Since $\left(a, a^{2}\right)$ lies on the line with equation $y=5 x+a$, then $a^{2}=5 a+a$ or $a^{2}=6 a$.

Since $a \neq 0$, then $a=6$.