Quadratic congruences are much more complex than linear congruences. If you’re interested in the details, see these notes on quadratic congruences.
When I started looking into this, I thought my number theory class years ago covered the details but that I’d forgotten them. Looking back, I see that my instructor and my textbook neglected some cases. Then I looked at a few other references and saw that they also left out some cases. I put together some notes by pulling together information from several incomplete but complementary references.
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