jmc

The discriminant of the quadratic polynomial $a{x}^2+bx+c$ is given by $b^2-4ac$. Substituting, the answer is ${\left(3+\frac{1}{3}\right)}^2-4\cdot 3\cdot \frac{1}{3}=3^2+2+\frac{1}{3^2}-4=3^2-2+\frac{1}{3^2}={\left(3-\frac{1}{3}\right)}^2=\boxed{\frac{64}{9}}$.