Which integers $a, b, c$ satisfy the equation $a \sqrt 2 − b = c \sqrt5$?

Question

This is a non-calculator question:

Which integers $a, b, c$ satisfy the equation $a \sqrt 2 − b = c \sqrt5$?

I've tried solving it through trial and error and the only solution I seem to be getting is $0,0,0$.

Answer 1

Let us assume there are $a,b,c \in \mathbb{Z},ac \neq 0$ such that $a \sqrt 2 − b = c \sqrt5$

$$b=a\sqrt{2} -c\sqrt{5}$$ $$b^2=2a^2+5c^2-2ac\sqrt{10}$$ $$\sqrt{10}=\frac{2a^2+5c^2-b^2}{2ac}$$ $$\implies \sqrt{10} \space \text{is rational (contradiction)}$$

EDIT: As suggested by A.P., we will need to rule out the cases when $a=0$ and when $c=0$ for a complete solution.

When $a=0$,

$$\sqrt{5}=-\frac b{c} \space \Rightarrow\Leftarrow$$

When $c=0$,

$$\sqrt{2}=\frac b{a} \space \Rightarrow\Leftarrow$$

Answer 2

Hint: square both sides of the equation and remember that $\sqrt2$ and $\sqrt5$ are irrational.