Equation solving

In mathematics, to solve an equation is to find its solutions, which are the values (numbers, functions, sets, etc.)

that fulfill the condition stated by the equation, consisting generally of two expressions related by an equals sign.

A solution is an assignment of values to the unknown variables that makes the equality in the equation true.

If the solution set is empty, then there are no values of the unknowns that satisfy simultaneously all equations and inequalities.

For representing them, a parametrization is often useful, which consists of expressing the solutions in terms of some of the unknowns or auxiliary variables.

Such infinite solution sets can naturally be interpreted as geometric shapes such as lines, curves (see picture), planes, and more generally algebraic varieties or manifolds.

For several classes of equations, algorithms have been found for solving them, some of which have been implemented and incorporated in computer algebra systems, but often require no more sophisticated technology than pencil and paper.

It may be the case, though, that the number of possibilities to be considered, although finite, is so huge that an exhaustive search is not practically feasible; this is, in fact, a requirement for strong encryption methods.

Equations involving linear or simple rational functions of a single real-valued unknown, say x, such as can be solved using the methods of elementary algebra.

Polynomial equations of degree up to four can be solved exactly using algebraic methods, of which the quadratic formula is the simplest example.

Polynomial equations with a degree of five or higher require in general numerical methods (see below) or special functions such as Bring radicals, although some specific cases may be solvable algebraically, for example (by using the rational root theorem), and (by using the substitution x = z1⁄3, which simplifies this to a quadratic equation in z).

Often, root-finding algorithms like the Newton–Raphson method can be used to find a numerical solution to an equation, which, for some applications, can be entirely sufficient to solve some problem.

Equations involving matrices and vectors of real numbers can often be solved by using methods from linear algebra.

There is a vast body of methods for solving various kinds of differential equations, both numerically and analytically.