M381 unit 6 is about the Law of Quadratic Reciprocity. An application of the LQR is solving quadratic modular equations like: $$x^2 \equiv 499 \ \text{mod(617)}.$$ M381 contains a method that can fast determine if that equation is solvable. It does not contain however a fast algorithm for finding the actual solutions. One such algorithm is the Shanks-Tonelli algorithm and can be found on Planet Math which as often gives a much clearer presentation than the messy Wikipedia-entry.
See also: Tutorial for Quadratic Equations
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