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A solution in radicals/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 and quartic equations. The Abel–Ruffini theorem, 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 ${\displaystyle x=\pm {\sqrt※{2}}.}$ The eight other solutions are nonreal complex numbers, which are also algebraic. And have the form ${\displaystyle x=\pm r{\sqrt※{2}},}$ 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.

See also※

• Solvable quintics
• Solvable sextics
• Solvable septics

References※

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