Solution in radicals Facts for Kids

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 $n$ th roots (square roots, cube roots, and other integer roots).

A well-known example is the solution

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

of the quadratic equation

ax^2 + bx + c =0.

There exist more complicated algebraic solutions for cubic equations and quartic equations. The Abel–Ruffini theorem, and, more generally Galois theory, state that some quintic equations, such as

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 $x^{10} = 2$ can be solved as $x=\pm\sqrt[10]2.$ The eight other solutions are nonreal complex numbers, which are also algebraic and have the form $x=\pm r\sqrt[10]2,$ where $r$ is a fifth root of unity, which can be expressed with two nested square roots. See also Solution in radicals § Notes 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.

Solution in radicals facts for kids

See also