Tiffany is solving an equation where both sides are quadratic expressions. She sets each quadratic equation equal to y and graphs the resulting system. If the graph of one quadratic opens upward and the other opens downward, what is the greatest possible number of intersections for these graphs?

At an intersection, the two quadratic expressions have the same y-value, so set them equal and subtract one from the other. Because one parabola opens upward and the other opens downward, their x² coefficients have opposite signs, so the resulting equation is still quadratic and cannot be identically zero. A quadratic can have at most two real solutions, and two intersections are possible. The greatest possible number is two.

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