The existence of primitive normal elements of quadratic forms over finite fields

For $q=3^r$ ($r>0$), denote by ${\mathbb{F}}_q$ the finite field of order $q$ and for a positive integer $m\ge 2$, let ${\mathbb{F}}_{q^m}$ be its extension field of degree $m$. We establish a sufficient condition for existence of a primitive normal element $\alpha$ such that $f(\alpha )$ is a primitive element, where $f(x)=a{x}^2+bx+c$, with $a,b,c\in {\mathbb{F}}_{q^m}$ satisfying $b^2\neq ac$in$\Fm$exceptforatmost9exceptionalpairs$(q,m)$.