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, $u^2 + v^2 = -{17\over9}$.