Let $p$ and $q$ satisfy $pq=9$ and $p+q=6$. What is the value of $p^2 + q^2$?
I notice that this problem involves a product and a sum of two numbers, and I recall that there is a formula that relates them: $(p+q)^2 = p^2 + 2pq + q^2$. If I apply this formula to the given values, I get $(p+q)^2 = 6^2 = 36 = p^2 + 2(9) + q^2$. Simplifying, I obtain $p^2 + q^2 = 36 - 2(9) = 18$. 