+2729 Let $x$ and $y$ be complex numbers. If $x + y = 2$ and $x^3 + y^3 = 5$, then what is $x^2 + y^2$?
+1952 We can use the information we already know to solve this problem!
First off, let's note that (x+y)3=x3+3x2y+3xy2+y3, which we can simplify to 8=5+6xy. Solving for xy, we get xy=1/2.
This may seem useless, but it will come into play later.
Now, we know that (x+y)2=x2+2xy+y2, which contains x^2+y^2. Isolating this, we get x2+y2=(x+y)2−2xy.
We already know all the terms! We can plug in 1/2 and 2 which we know from above! We get x2+y2=4−1=3.
So 3 is our answer!
Thanks! :)
