Algebra

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, \(u^2 + v^2 = -{17\over9}\).