Algebraic solution

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

The most well-known example is the solution

introduced in secondary school, of the quadratic equation

(where a ≠ 0).

There exist more complicated algebraic solutions for the general cubic equation and quartic equation. The Abel-Ruffini theorem states that the general quintic equation lacks an algebraic solution, and this directly implies that the general polynomial equation of degree n, for n ≥ 5, cannot be solved algebraically. However, under certain conditions algebraic solutions can be obtained; for example, the equation ${\displaystyle x^{10}=a}$ can be solved as ${\displaystyle x=a^{1/10}.}$

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