Solve system of congruences which involves a quadratic term

Question

I am studying for an admission exam and I came to this system of congruences

$$x^2 \equiv 2 \text{ mod } 7 \hspace{1cm} x \equiv 1 \text{ mod } 5 $$

I know how to solve linear systems, but I don't know what to do with a quadratic one.

Could anyone explain me ho to solve it?

Thanks

Answer 1

You can try to sub in first $3$ rests by $7$, so $1, 2, 3$. We get $$x^2\equiv 1, 4, 2\pmod7 $$ From this, $$x\equiv 3\,\lor x\equiv7-3=4 \pmod7$$ Now it's linear.