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# Algebra

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Let u and v be the solutions to 3x^2 + 5x + 7 = 0. Find u^2 + v^2.

Feb 14, 2022

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So first we make the coefficient of x^2 = 1.

The new quadratic is: $$x^2 + {5\over3}x + {7\over3} = 0$$

Then since we know by Vieta, the two solutions of the quadratic multiply to the c term, and they add to the b term * -1. The solutions are u and v.

Thus, $$u + v = -{5\over3}$$, and $$uv = {7\over3}$$

Since $$u^2 + v^2 = (u + v)^2 - 2uv$$, we can substitute in our previous values.

Then $$u^2 + v^2 = {25\over9} - {14\over3}$$.

Thus, $$u^2 + v^2 = -{17\over9}$$.

Feb 14, 2022