  We prove that for all $q>211$, there always exists a primitive root $g$ in
the finite field $\mathbb{F}_{q}$ such that $Q(g)$ is also a primitive root,
where $Q(x)= ax^2 + bx + c$ is a quadratic polynomial with $a, b, c\in
\mathbb{F}_{q}$ such that $b^{2} - 4ac \neq 0$.
