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The quadratic 8x^2+12x-14 has two real roots. What is the sum of the squares of these roots? Express your answer as a common fraction in lowest terms.

 Dec 18, 2020
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Let \(x_1\) and \(x_2\) be the roots of the equation \(8x^2+12x-14 \). We want to find \(x_1^2+x_2^2\). Note that \(x_1^2+x_2^2=(x_1+x_2)^2-2x_1x_2\). We know that \(x_1+x_2\), the sum of the roots, is equal to \(\frac{-b}{a}\), which for this equation is \(\frac{-12}{8}=\frac{-3}{2}\). Likewise, we know that \(x_1x_2\), the product of the roots, is equal to \(\frac{c}{a}\), which for this equation is \(\frac{-14}{8}=\frac{-7}{4}\). Thus, \(x_1^2+x_2^2=\left(\frac{-3}{2}\right)^2-2\left(\frac{-7}{4}\right)=\frac{9}{4}+\frac{14}{4}=\boxed{\frac{23}{4}}\).

 Mar 9, 2021

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