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?

Intersections occur where the two quadratic expressions have the same y-value. Setting them equal gives a quadratic equation; because one parabola opens upward and the other downward, their leading coefficients have opposite signs, so the equation cannot reduce to an identity. A quadratic equation can have at most two solutions, and two intersections are possible. Therefore, the greatest possible number is two, choice C.

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