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Let u and v be the solutions to 3x^2 + 5x + 7 = x^2 + 4x.  Find u/v + v/u.

 Jun 5, 2022
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Let \(u\) and \(v\) be the solutions to \(3x^2+5x+7=x^2+4x\). Find \(\frac{u}{v}+\frac{v}{u}\).

 

We can apply Vieta's formula to this problem using the fact that \(\frac{u}{v}+\frac{v}{u}=\frac{u^2+v^2}{vu}=\frac{u^2+v^2+2uv-2uv}{uv}=\frac{(u+v)^2-2uv}{uv}\).

 

By moving all the terms to the left side, the equation transforms into \(2x^2+x+7=0\). So then, by Vieta's formula, \(u+v=-\frac{1}{2}\) and \(uv = \frac{7}{2}\).

 

Plugging this in to what we got, we have \(\frac{u}{v}+\frac{v}{u}=\frac{(-\frac{1}{2})^2-2\cdot\frac{7}{2}}{\frac{7}{2}}=-\frac{27}{14}\).

 Jun 6, 2022

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