problem: If $(x + y)^2 = 105$ and $x^2 + y^2 = 65$, what is the value of $xy$?
solution: If we expand the left hand side of the first equation, we get $x^2 + 2xy + y^2 = 105$, so $2xy + (x^2 + y^2) = 105$. We are given that $x^2 + y^2 = 65$, so we can substitute for $x^2 + y^2$ to obtain $2xy + 65 = 105$. It follows that $xy = \boxed{20}$.