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

 Apr 2, 2024
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To find the value of

x^2+y^2

, we can use the identities:

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

(x3+y3)=(x+y)(x2+y2−xy)

From the given equations, we know that

x+y=2

and

x^3+y^3=5

.

Substituting

x+y=2

into the first identity, we get:

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

4=x^2+y^2+2xy

We can rearrange this to find an expression for

xy

:

xy=2−(x^2+y^2)/2​

Substituting

x+y=2

and

xy=2−(x^2+y^2)/2

into the second identity, we get:

5=2(2−((x^2+y^2)/2)​−((x^2+y^2​)/20)

5=4−2(x^2+y^2)

Solving for

x^2+y^2

gives us:

x^2+y^2=(4-5)/-2=1/2

So,

x^2+y^2=1/2

 

 

edit 1 : im not 100% sure if i'm correct

.

 Apr 2, 2024
edited by coolerthanu  Apr 2, 2024

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