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

 May 30, 2024
 #1
avatar+1332 
+1

First off, note that \((x+y)^3 = x^{3}+3x^{2}y+3xy^{2}+y^{3}\)

 

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 \((x+y)^2 = x^2+2xy+y^2\). This means that \(x^2+y^2 = (x+y)^2 - 2xy\)! We already know all these terms! We can easily plug in 2 and 1/2 to solve our problem!

 

\(x^2 + y^2 = 4 - 1 = 3\)

 

Our final answer is just 3. 

 

Thanks! :)

 May 30, 2024

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