+1443 Let $x$ and $y$ be complex numbers. If $x + y =2$ and $x^3 + y^3 = 5$, then what is $x^2 + y^2$?
+1632 Our current goal is to find the value of xy, so we can get x^2 + y^2 = (x + y)^2 - 2xy.
x^3 + y^3 can be factored to (x + y)(x^2 - xy + y^2)
Additionally, x^2 - xy + y^2 = (x + y)^2 - 3xy
So, x^3 + y^3 = (x + y)(x^2 - xy + y^2) = (x + y)[(x + y)^2 - 3xy], now substitute:
2 * (4 - 3xy) = 5 => 4 - 3xy = 2.5 => 1.5 = 3xy => xy = 0.5
Now, to get x^2 + y^2, just do (x + y)^2 - 2xy = 4 - 1 = 3
