$x$ is an irrational number such that $x^2 - 2x$ and $x^3 -5x$ are rational. 
If $x$ is an irrational number such that $x^2 - 2x$ and $x^3 -5x$ are rational numbers. What does $x^3 - 5x$ equals to?


Note that $(x-1)^2=y$ is a rational number. So $x = 1 \pm \sqrt{y}$. Also,
$x^3-5x=3y-4 \pm\sqrt{y}(y-2)$ is rational, which is possible only when
$$y=2.$$
Therefore, $x = 1\pm \sqrt{2}$ and $x^3-5x = 3y-4 = 2$.
