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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 26, 2024
 #1
avatar+806 
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We can use the information we already know to solve this problem!

 

First off, let's note that \((x+y)^3 = x^{3}+3x^{2}y+3xy^{2}+y^{3}\), 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 \((x+y)^2 = x^2+2xy+y^2\), which contains x^2+y^2. Isolating this, we get \(x^2+y^2 = (x+y)^2 - 2xy\)

 

We already know all the terms! We can plug in 1/2 and 2 which we know from above! We get \(x^2 + y^2 = 4 - 1 = 3\)

 

So 3 is our answer!

 

Thanks! :)

 May 30, 2024

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