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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 l

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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

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, $$x_1^2+x_2^2=\left(\frac{-3}{2}\right)^2-2\left(\frac{-7}{4}\right)=\frac{9}{4}+\frac{14}{4}=\boxed{\frac{23}{4}}$$.