+866 Let x and y be complex numbers. If $x + y =2$ and $x^3 + y^3 = 5$, then what is $x^2 + y^2$?
+9 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
.
