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Let x and y be complex numbers.  If $x + y =2$ and $x^3 + y^3 = 5$, then what is $x^2 + y^2$?

 Feb 23, 2024
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\({x}^{3}+{y}^{3}=(x+y)({x}^{2}-xy+{y}^{2})\)

\(5=2({x}^{2}-xy+{y}^{2})\)

\({x}^{2}-xy+{y}^{2}=\frac{5}{2}\)

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.

 Feb 23, 2024

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