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

Question

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

chihiroasleaf · Answer

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

mathmath333 · Answer

u have to substract both equations

chihiroasleaf · Answer

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

parthkohli · Answer

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

parthkohli · Answer

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

chihiroasleaf · Answer

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

parthkohli · Answer

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

chihiroasleaf · Answer

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