HMMT-Nov 2009 guts, Problem 15

The curves $x^{2}+y^{2}=36$ and $y=x^{2}-7$ intersect at four points. Find the sum of the squares of the $x$-coordinates of these points.

Solution: If we use the system of equations to solve for $y$, we get $y^{2}+y-29=0\left(\right.$ since $\left.x^{2}=y+7\right)$. The sum of the roots of this equation is -1 . Combine this with $x^{2}=y+7$ to see that the sum of the square of the possible values of $x$ is $2 \cdot(-1+7 \cdot 2)=26$. $\fbox{26}$.

Answer: 26