Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 4.8 AIME Find the answer

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.

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

If we use the system of equations to solve for $y$ , we get $y^{2}+y-29=0$ (since $x^{2}=y+7$ ). 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 $2 \cdot(-1+7 \cdot 2)=26$ .

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