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# Find the sums of the squares of the solutions of the equation 3z^2 - 5z + 8 = 0.

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Find the sums of the squares of the solutions of the equation 3z^2 - 5z + 8 = 0.

Jul 3, 2022

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The roots of 3z^2 - 5z + 8 = 0 are $$\frac{5 \pm i \sqrt{71}}{6}$$

Then $$(\frac{5 + i \sqrt{71}}{6})^2 + (\frac{5 - i \sqrt{71}}{6})^2 = -\frac{53}{36}$$

Jul 3, 2022
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Let the roots be x and y. Note that $$x^2 + y^2 = ( x+ y)^2 - 2xy$$

By Vieta's, $$x + y = -{b \over a} = {5 \over 3}$$ and $$xy = {c \over a} = {8 \over 3}$$

So, $$x^2 + y^2 = {25 \over 9} - {16 \over 3} = {25 \over 9} - {48 \over 9} = \color{brown}\boxed{-{ 23 \over 9}}$$

Jul 4, 2022