Solution in radicals

A solution in radicals or algebraic solution is a closed-form expression, and more specifically a closed-form algebraic expression, that is the solution of a polynomial equation, and relies only on addition, subtraction, multiplication, division, raising to integer powers, and the extraction of nth roots (square roots, cube roots, and other integer roots).

A well-known example is the solution

${\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac\ }}}{2a}}}$

of the quadratic equation

${\displaystyle ax^{2}+bx+c=0.}$

There exist more complicated algebraic solutions for cubic equations^[1] and quartic equations.^[2] The Abel–Ruffini theorem,^[3]: 211 and, more generally Galois theory, state that some quintic equations, such as

${\displaystyle x^{5}-x+1=0,}$

do not have any algebraic solution. The same is true for every higher degree. However, for any degree there are some polynomial equations that have algebraic solutions; for example, the equation ${\displaystyle x^{10}=2}$ can be solved as ${\displaystyle x=\pm {\sqrt[{10}]{2}}.}$ The eight other solutions are nonreal complex numbers, which are also algebraic and have the form ${\displaystyle x=\pm r{\sqrt[{10}]{2}},}$ where r is a fifth root of unity, which can be expressed with two nested square roots. See also Quintic function § Other solvable quintics for various other examples in degree 5.

Évariste Galois introduced a criterion allowing one to decide which equations are solvable in radicals. See Radical extension for the precise formulation of his result.

Algebraic solutions form a subset of closed-form expressions, because the latter permit transcendental functions (non-algebraic functions) such as the exponential function, the logarithmic function, and the trigonometric functions and their inverses.