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| 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 (mathematics)|division]], raising to integer powers, and the extraction of roots (square roots, cube roots, etc.).
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| The most well-known example is the solution
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| :<math>
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| x=\frac{-b \pm \sqrt {b^2-4ac\ }}{2a},</math>
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| introduced in secondary school, of the [[quadratic equation]]
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| :<math>ax^2 + bx + c =0\,</math> | |
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| (where ''a'' ≠ 0). | |
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| There exist more complicated algebraic solutions for the general [[cubic equation]]<ref>Nickalls, R. W. D., "A new approach to solving the cubic: Cardano's solution revealed," ''Mathematical Gazette'' 77, November 1993, 354-359.</ref> and [[quartic equation]].<ref>Carpenter, William, "On the solution of the real quartic," ''Mathematics Magazine'' 39, 1966, 28-30.</ref> The [[Abel-Ruffini theorem]]<ref>Jacobson, Nathan (2009), Basic Algebra 1 (2nd ed.), Dover, ISBN 978-0-486-47189-1</ref>{{rp|211}} 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 <math>x^{10} = a</math> can be solved as <math>x=a^{1/10}.</math>
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| Algebraic solutions form a subset of [[closed-form expression]]s, because the latter permit [[transcendental functions]] (non-algebraic functions) such as the exponential function, the logarithmic function, and the trigonometric functions and their inverses.
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| ==See also==
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| *[[sextic equation#Solvable sextics|Solvable sextics]]
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| *[[septic equation#Solvable septics|Solvable septics]]
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| ==References==
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| {{reflist}}
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| {{DEFAULTSORT:Algebraic Solution}}
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| [[Category:Algebra]]
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