Solve the equation $x^2+x+9\equiv 0\pmod {63}$

Question

Solve the equation $x^2+x+9\equiv 0\pmod {63}$

Quadratic equation $x^2+x+9=0$ can't be factorized (with integer roots).

Also, $63$ is not a prime, and I have checked the method of completing the square.

What method to use for this congruence relation?

Answer 1

$63 = 9 \times 7$. Do it mod $9$ and mod $7$ (these are small enough that you can look at each case individually), then put the results together with the Chinese Remainder Theorem.