+842 Let x and y be complex numbers. If $x + y =2$ and $x^3 + y^3 = 5$, then what is $x^2 + y^2$?
+399 \({x}^{3}+{y}^{3}=(x+y)({x}^{2}-xy+{y}^{2})\).
\(5=2({x}^{2}-xy+{y}^{2})\)
\({x}^{2}-xy+{y}^{2}=\frac{5}{2}\)
Because \({x}^{2}-xy+{y}^{2}={(x+y)}^{2}-3xy\)
\({(x+y)}^{2}-3xy=\frac{5}{2}\).
\({2}^{2}-3xy=\frac{5}{2}\)
\(-3xy=-\frac{3}{2}\)
\(xy=\frac{1}{2}\)
\({x}^{2}+{y}^{2}={(x+y)}^{2}-2xy=4-1=3\).
So x2+y2 is 3.
