I have seen a lot of solvers that are capable of, for example, getting an equation such as x ^ 2 + x = 12 and finding x = [3, -4]. I know some of them are implemented by hardcoding special cases. For example, when finding an equation on the quadratic form, they use a hardcoded bhaskara formula.
What I'm looking for is something different, I want to learn more about the nature of solving equations. That is, I want simple algorithms or systems that generalize the process of solving an equation, in the same principle that lambda calculus encapsules the process of computation.
What are the names I'm looking for?