## anonymous one year ago If x and y are real numbers, with $x^3 - 3xy^2 = 44$ and $y^3 - 3x^2y = 8$ What is the value of $x^2 + y^2$

1. anonymous

If we add both equations, we'll get $x^3 - 3xy^2 + 3x^2y - y^3 = 52$ While $(x-y)^3 = x^3 - 3x^2y + 3xy ^2 - y^3$

2. mathmath333

u have to substract both equations

3. anonymous

Umm What I mean above is If we subtract, we get $x^3 - 3xy^2 + 3x^2y - y^3 = 36$

4. ParthKohli

$(x+iy)^3 = x^3 + 3x^2y i - 3xy^2 - iy^3 = (x^3 - 3x y^2) + (3x^2 y - y^3) i$$(x-iy)^3 = (x^3 - 3x y^2) + (y^3 - 3x ^2 y) i$$x^2 + y^2 = (x+iy)(x-iy)$

5. ParthKohli

$\Rightarrow (x+iy)^3 = 44-8i, (x-iy)^3 = 44 + 8i$$\Rightarrow (x^2+y^2)^3 = (44+8i)(44-8i) = 2000$$\Rightarrow x^2 + y^2 = \sqrt[3]{2000}$

6. anonymous

Thank you @ParthKohli :) How did you get the idea?

7. ParthKohli

I don't know either. Maybe the $$x^2 + y^2$$ triggered me and I started thinking along the lines of complex numbers.

8. anonymous

I didn't think to use conplex numbers :-|