TITLE: Solving a quadratic congruence: $4x^2 \equiv 2 \ (\text{mod} \ 7)$
QUESTION [5 upvotes]: How does one go about solving the following quadratic congruence?
$4x^2 \equiv 2 \ (\text{mod} \ 7)$

REPLY [3 votes]: You can take the square roots of both sides: $$4x^2 \equiv 2 \pmod 7$$ $$2x \equiv 3, 4 \pmod 7$$ Then halve both sides: $$x \equiv 5, 2 \pmod 7.$$
Example: $4 \times 12^2 = 576 = 82 \times 7 + 2$.