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- Feb 14, 2012
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Two of the roots of the equation \(\displaystyle 2x^3-8x^2+9x+p=0\) are also roots of the equation \(\displaystyle 2x^3+8x^2-7x+q=0\). Evaluate \(\displaystyle 19q+99p\).
Use Vieta's relations. The sum of the roots of the first equation is $4$, and the sum of the roots of the second equation is $-4$. So if the roots of the first equation are $\alpha,\ \beta$ and $\gamma$, then the roots of the second equation are $\alpha,\ \beta$ and $\gamma-8$. Vieta's relations tell us thatTwo of the roots of the equation \(\displaystyle 2x^3-8x^2+9x+p=0\) are also roots of the equation \(\displaystyle 2x^3+8x^2-7x+q=0\). Evaluate \(\displaystyle 19q+99p\).