  Let $\mathbb{F}_q$ be the finite field of $q$ elements, and let $k\mid q-1$
be a positive integer. Let $f(x)=ax^2+bx+c$ be a quadratic polynomial in
$\mathbb{F}_q[x]$ with $b^2-4ac\ne0$. In this paper, we show that if
$q>\max\{e^{e^3},(2k)^6\}$, then there is a primitive element $g$ of
$\mathbb{F}_q$ such that $f(g)\in\mathbb{F}_q^{\times k}=\{x^k:
x\in\mathbb{F}_q\setminus\{0\}\}$. Moreover, we shall confirm a conjecture
posed by Sun.
