Let $p$ and $q$ satisfy $pq=9$ and $p+q=6$. What is the value of $p^2 + q^2$?
We have two equations and two variables, so it's possible to solve for $p$ and $q$ directly and then calculate $p^2$ and $q^2$ 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 $$(p+q)^2 = p^2 + 2pq + q^2 = 36,$$which is close to what we want but has the extra $2pq$ term. Since we know that $pq=9$, we can substitute to get $$p^2 + 2(9) +q^2 = 36 \implies p^2+q^2 = \boxed{18}.$$Note that our task was made easier by only solving for what the problem asked rather than by trying to solve for $p$ and $q$ individually.