Problem

Factor and solve \(x^{2}-3x-10=0\). Justify the factor pair using its product and sum, apply the zero-product property, and check both solutions in the original equation.

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What this problem is really about

For a monic trinomial, a valid factor pair must reproduce both the constant product and the signed middle-term sum. We’ll verify those conditions, write the product equal to zero, solve every factor equation with the signs reversed correctly, and check both results in the original equation.

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x² − 3x − 10 = (x − 5)(x + 2), so the zeros are x = 5 and x = −2. No product-and-sum evidence or check is supplied.

x² − 3x − 10 = (x − 5)(x + 2), so the zeros are x = −5 and x = 2. Those signs do not solve the factor equations.

x² − 3x − 10 = (x − 10)(x + 1), so the zeros are x = 10 and x = −1. Expanding gives x² − 9x − 10, not the original equation.

Because (−5)(2) = −10 and −5 + 2 = −3, x² − 3x − 10 = (x − 5)(x + 2). The zeros are x = 5 and x = −2; expanding confirms the original equation.

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Factor and solve \(x^{2}-3x-10=0\). Justify the factor pair using its product and sum, apply the zero-product property, and check both solutions in the original equation.

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