+613 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 First off, note that (x+y)3=x3+3x2y+3xy2+y3.
We already know most of the terms, so we have 8=5+3xy(x+y), which is the same as 8=5+6xy. Now, we have xy=1/2.
You may be wondering why in the world would we want xy, but it will make sense later.
Now, when we see x^2+y^2, (x+y)^2 instantly comes to mind.
We know that (x+y)2=x2+2xy+y2. This means that x2+y2=(x+y)2−2xy! We already know all these terms! We can easily plug in 2 and 1/2 to solve our problem!
x2+y2=4−1=3.
Our final answer is just 3.
Thanks! :)
