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Question

A triangle has three sides of the following side lengths: 7, 10, and x^2. What are all of the positive integer values of x such that the triangle exists? Separate your answers using commas and express them in increasing order.

Accepted Answer

For a triangle to exist, the sum of two sides of the triangle must be greater than the third. Therefore, we have three formulas: x^2+7>10 x^2>3, x^2+10>7 x^2>-3, and 7+10>x^2 x^2<17. Thus, we have two quadratics, x^2>3 and x^2<17. Therefore, possible values for x are 2, 3, and 4.