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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, \(u^2 + v^2 = -{17\over9}\).

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 Feb 14, 2022

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