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avatar+683 

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

 Jun 16, 2024
 #1
avatar+1075 
+1

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! :)

 Jun 17, 2024

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