quality:correction
Examples

An example from high school mathematics says that a single-variable quadratic polynomial has a real root if and only if its discriminant is non-negative:

Here the sentence on the left-hand side involves a quantifier ∃ x ∈ R {\displaystyle \exists x\in \mathbb {R} }, whereas the equivalent sentence on the right does not.

Examples of theories that have been shown decidable using quantifier elimination are Presburger arithmetic, algebraically closed fields, real closed fields, atomless Boolean algebras, term algebras, dense linear orders, abelian groups, random graphs, as well as many of their combinations such as Boolean algebra with Presburger arithmetic, and term algebras with queues.

Quantifier eliminator for the theory of the real numbers as an ordered additive group is Fourier–Motzkin elimination; for the theory of the field of real numbers it is the Tarski–Seidenberg theorem.

Quantifier elimination can also be used to show that "combining" decidable theories leads to new decidable theories (see Feferman–Vaught theorem).