Find real and distinct solutions

Question

Find real and distinct solutions to the equations $$r^2+s^2=u^2+v^2$$ and $$r^3+s^3=u^3+v^3$$ I don't know how to solve this problem please help

Answer 1

Here's one example (but there are lots of others) . . .

Let $r,s$ be the roots of the quadratic equation $$x^2 - x - 1 = 0$$ and let $u,v$ be the roots of the quadratic equation $$x^2 - ax + b = 0$$ where \begin{align*} a&=\frac{\sqrt{33}-1}{2}\\[4pt] b&=\frac{11-\sqrt{33}}{4}\\[4pt] \end{align*} Then $r,s,u,v$ satisfy $$r^2 + s^2 = 3 = u^2 + v^2$$ $$r^3 + s^3 = 4 = u^3 + v^3$$