jmc

We have two equations and two variables, so it's possible to solve for $a$ and $b$ directly and then calculate $a^2$ and $b^2$ separately to get our answer. However, doing so involves a fair amount of computation with complex numbers and square roots, so we look for an alternative approach. We square the second equation to get $(a+b{)}^2=a^2+2ab+b^2=25$, which is close to what we want but has the extra $2ab$ term. Since we know that $ab=7$, we can substitute to get $$a^2+2(7)+b^2=25⟹a^2+b^2=\boxed{11}.$$