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

 Jun 16, 2024
 #1
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x + y  = 2

square both sides

x^2 + 2xy + y^2  = 4

x^2 + y^2 =  4 - 2xy

 

xy = 5  -x^2 - y^2

xy = 5 - (x^2 + y^2)

 

So

 

x^2 + y^2  = 4 -2xy

 

x^2 + y^2  = 4 - 2 [ 5 - (x^2 + y^2) ] 

 

x^2 + y^2  =  4 - 10 + 2(x^2 + y^2)

 

-3(x^2 + y^2)  =  -6

 

x^2 + y^2  = -6 / -3  = 2

 

 

cool cool cool

 Jun 16, 2024

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