# What are the fundamental principles/algorithms on the process of equation solving?

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?

• Can you clarify what you are looking for? Symbolic solvers, or numerical? Single variable, or multiple? Only polynomials, or involving some specific transcendentals? Involving differentials? Do you know how many solutions there are, or does the algorithm need to guess? Are any of the solutions complex? – Wandering Logic Jan 26 '15 at 2:54
• You might be looking for computer algebra systems (CAS). – Yuval Filmus Jan 26 '15 at 4:04
• Some more things to look for are real closed fields (a.k.a. Tarski algebra) and Galois theory. If you're feeling brave, look into algebraic geometry and its techniques (e.g. Gröbner basis). – Pseudonym Jan 26 '15 at 6:46
• one basic strategy is solving group of eqns in terms of $n$ variables, then $n-1$, ... and so on, and then backsubstitution. some of the basic strategies come from algebra, encoded as algorithms. – vzn Jan 26 '15 at 16:07