imc

In a quadratic equation of the form $x^2+bx+c=0$, the product of the roots is $c$ (Vieta's Formulas). Using this property, we have that $ab=2$ and $q=(a+\frac{1}{b})\cdot (b+\frac{1}{a})=\frac{ab+1}{b}\cdot \frac{ab+1}{a}=\frac{(ab+1{)}^2}{ab}=\frac{(2+1{)}^2}{2}=\boxed{\frac{9}{2}}$. * Notice the fact that we never actually found the roots.