Find the value of $(x^{2}+y^{2})$

Question

If $x^{3}-3xy^{2}=10$, $y^{3}-3x^{2}y=30$

Find $$(x^{2}+y^{2})$$

I tried but I got nothing, any help please?

Answer 1

$$(x+iy)^3 = (x^3-3xy^2) + i(3x^2y-y^3) $$ gives $(x+iy)^3 = 10-30 i $. If we multiply by the conjugate (i.e. take norms): $$ (x^2+y^2)^3 = 10^2+30^2 = 1000 $$ we immediately get $x^2+y^2=\color{red}{10}$.