We know that each congruence has two solutions, and that if we can find one solution $\displaystyle x\equiv s$ then the second solution is $\displaystyle x\equiv-s$. But I don't know any algorithms I can use to solve each of the first solutions, except trial and error, which of course is far too inefficient.list all solutions of...the ten congruences $\displaystyle x^2\equiv a\mod 11^2$ where $\displaystyle a=1,3,4,5,9$.

This is an exercise in the chapter for quadratic residues, so presumably that has something to do with it.

Any ideas would be appreciated.