An algebraic solution is an explicit formula for one or more roots of an algebraic equation that can be expressed using a finite combination of the basic arithmetic operations and root extractions. In other words, the values of the unknown(s) are written in terms of the equation's coefficients by applying addition, subtraction, multiplication, division and taking nth roots (square roots, cube roots, etc.). Such formulas are often described as being “solvable by radicals.”

Classic examples

The best-known closed-form result is the formula for the general quadratic equation. The solutions of

for the quadratic

(with a ≠ 0) are expressed in terms of the coefficients a, b and c by arithmetic and a square root.

There also exist algebraic formulas for the general cubic and the general quartic, though these expressions are considerably more involved than the quadratic formula. These higher-degree formulas still use only the permitted operations and radicals.

Limits: why not all polynomials have algebraic solutions

The Abel–Ruffini theorem establishes that for a general polynomial of degree five or higher there is no solution expressible solely by the listed operations and a finite number of root extractions. That is, a generic quintic cannot be solved by radicals. Modern understanding of when a polynomial admits an algebraic solution comes from Galois theory, which gives criteria (the structure of the polynomial's Galois group) determining whether its roots are obtainable by radicals.

Special cases

Even though the general polynomial of degree ≥ 5 is not solvable by radicals, many particular higher-degree equations are. For example, the simple equation

has the algebraic solution

More generally, whether a particular polynomial admits an algebraic formula depends on its symmetry properties rather than only on its degree.

Operations used in algebraic solutions