Menaechmus's cube duplication

To construct a line of length $\sqrt[3]{2}$.

Deaw the parabola with equation $y = x^{2}$.
Draw the parabola with equation $y^{2} = 2x$.
These curves meet at $P$.

The $x$-coordinate of $P$ is $\sqrt[3]{2}$.

Of course the construction does not satisfy the compass and straight edge rules.