+820 Let $x$ and $y$ be complex numbers. If $x + y = 2$ and $x^3 + y^3 = 5$, then what is $x^2 + y^2$?
+1804 We can note something really important to solve this problem.
First off, let's note that
\((x+y)^3 = x^{3}+3x^{2}y+3xy^{2}+y^{3}\)
Plugging in all the information we already have from the problem, we get that
\(8=5+6xy\\ xy=1/2\)
The reason we need the value of xy will come into play later.
Now, let's also note that
\((x+y)^2 = x^2+2xy+y^2\)
Isolating x^2+y^2, we get that
\(x^2+y^2 = (x+y)^2 - 2xy\)
We already know all the terms of the equation we needed to find x^2 + y^2.
Plugging in 1/2 and 2, we get
\(x^2 + y^2 = 4 - 1 = 3\)
So our answer is 3.
Thanks! :)
