Solution:

Suppose $x \neq 0$. Then $\frac{x \sqrt{2}}{x} = \sqrt{2}$ is the quotient of two nonzero rationals, and so is rational. However, it is well-known that $\sqrt{2}$ is irrational. Therefore any solution $x$ must satisfy $x = 0$. We can see that $0$ is rational and $0 \sqrt{2} = 0$ is rational, so the answer is indeed $x = 0$.