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quartic formula
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(Theorem)
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The four roots $r_1, r_2, r_3, r_4$ of a quartic polynomial equation $x^4 + ax^3 + bx^2 + cx + d = 0$ are given by \begin{eqnarray*} r_1 & = & {\frac{-a}{4} - \frac{1}{2}{\sqrt{\frac{a^2}{4} - \frac{2b}{3} + \frac{2^{\frac{1}{3}}\left( b^2 - 3ac + 12d \right) } {3{\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd + {\sqrt{-4{\left( b^2 - 3ac + 12d \right) }^3 + {\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd \right) }^2}} \right) }^{\frac{1}{3}}} + \left(\frac{{ 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd + {\sqrt{-4 {\left( b^2 - 3ac + 12d \right) }^3 + {\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd \right) }^2}} }} {54}\right)^\frac{1}{3}}} -
\frac{1}{2}{\sqrt{\frac{a^2}{2} - \frac{4b}{3} - \frac{2^{\frac{1}{3}}\left( b^2 - 3ac + 12d \right) } {3{\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd + {\sqrt{-4{\left( b^2 - 3ac + 12d \right) }^3 + {\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd \right) }^2}} \right) }^{\frac{1}{3}}} - \left(\frac{{ 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd + {\sqrt{-4 {\left( b^2 - 3ac + 12d \right) }^3 + {\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd \right) }^2}} }} {54}\right)^\frac{1}{3} - \frac{-a^3 + 4ab - 8c} {4{\sqrt{\frac{a^2}{4} - \frac{2b}{3} + \frac{2^{\frac{1}{3}} \left( b^2 - 3ac + 12d \right) }{3 {\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd + {\sqrt{-4 {\left( b^2 - 3ac + 12d \right) }^3 + {\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd \right) }^2}} \right) }^ {\frac{1}{3}}} + \left( \frac{{ 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd + {\sqrt{-4 {\left( b^2 - 3ac + 12d \right) }^3 + {\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd \right) }^2}} } }{54}\right)^\frac{1}{3}}}}}}} \\ r_2 & = & {\frac{-a}{4} -
\frac{1}{2}{\sqrt{\frac{a^2}{4} - \frac{2b}{3} + \frac{2^{\frac{1}{3}}\left( b^2 - 3ac + 12d \right) } {3{\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd + {\sqrt{-4{\left( b^2 - 3ac + 12d \right) }^3 + {\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd \right) }^2}} \right) }^{\frac{1}{3}}} + \left( \frac{{ 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd + {\sqrt{-4 {\left( b^2 - 3ac + 12d \right) }^3 + {\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd \right) }^2}} }} {54}\right)^\frac{1}{3}}} + \frac{1}{2}{\sqrt{\frac{a^2}{2} - \frac{4b}{3} - \frac{2^{\frac{1}{3}}\left( b^2 - 3ac + 12d \right) } {3{\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd + {\sqrt{-4{\left( b^2 - 3ac + 12d \right) }^3 + {\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd \right) }^2}} \right) }^{\frac{1}{3}}} - \left( \frac{{ 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd + {\sqrt{-4 {\left( b^2 - 3ac + 12d \right) }^3 + {\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd \right) }^2}} }} {54}\right)^\frac{1}{3} - \frac{-a^3 + 4ab - 8c} {4{\sqrt{\frac{a^2}{4} - \frac{2b}{3} +
\frac{2^{\frac{1}{3}} \left( b^2 - 3ac + 12d \right) }{3 {\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd + {\sqrt{-4 {\left( b^2 - 3ac + 12d \right) }^3 + {\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd \right) }^2}} \right) }^ {\frac{1}{3}}} + \left( \frac{{ 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd + {\sqrt{-4 {\left( b^2 - 3ac + 12d \right) }^3 + {\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd \right) }^2}} } }{54}\right)^\frac{1}{3}}}}}}} \\ r_3 & = & {\frac{-a}{4} + \frac{1}{2}{\sqrt{\frac{a^2}{4} - \frac{2b}{3} + \frac{2^{\frac{1}{3}}\left( b^2 - 3ac + 12d \right) } {3{\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd + {\sqrt{-4{\left( b^2 - 3ac + 12d \right) }^3 + {\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd \right) }^2}} \right) }^{\frac{1}{3}}} + \left( \frac{{ 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd + {\sqrt{-4 {\left( b^2 - 3ac + 12d \right) }^3 + {\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd \right) }^2}} }} {54}\right)^\frac{1}{3}}} - \frac{1}{2}{\sqrt{\frac{a^2}{2} - \frac{4b}{3} - \frac{2^{\frac{1}{3}}\left( b^2 -
3ac + 12d \right) } {3{\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd + {\sqrt{-4{\left( b^2 - 3ac + 12d \right) }^3 + {\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd \right) }^2}} \right) }^{\frac{1}{3}}} - \left( \frac{{ 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd + {\sqrt{-4 {\left( b^2 - 3ac + 12d \right) }^3 + {\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd \right) }^2}} }} {54}\right)^\frac{1}{3} + \frac{-a^3 + 4ab - 8c} {4{\sqrt{\frac{a^2}{4} - \frac{2b}{3} + \frac{2^{\frac{1}{3}} \left( b^2 - 3ac + 12d \right) }{3 {\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd + {\sqrt{-4 {\left( b^2 - 3ac + 12d \right) }^3 + {\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd \right) }^2}} \right) }^ {\frac{1}{3}}} + \left( \frac{{ 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd + {\sqrt{-4 {\left( b^2 - 3ac + 12d \right) }^3 + {\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd \right) }^2}} } }{54}\right)^\frac{1}{3}}}}}}} \\ r_4 & = & {\frac{-a}{4} + \frac{1}{2}{\sqrt{\frac{a^2}{4} - \frac{2b}{3} + \frac{2^{\frac{1}{3}}\left( b^2 - 3ac + 12d \right) }
{3{\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd + {\sqrt{-4{\left( b^2 - 3ac + 12d \right) }^3 + {\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd \right) }^2}} \right) }^{\frac{1}{3}}} + \left( \frac{{ 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd + {\sqrt{-4 {\left( b^2 - 3ac + 12d \right) }^3 + {\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd \right) }^2}} }} {54}\right)^\frac{1}{3}}} + \frac{1}{2}{\sqrt{\frac{a^2}{2} - \frac{4b}{3} - \frac{2^{\frac{1}{3}}\left( b^2 - 3ac + 12d \right) } {3{\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd + {\sqrt{-4{\left( b^2 - 3ac + 12d \right) }^3 + {\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd \right) }^2}} \right) }^{\frac{1}{3}}} - \left( \frac{{ 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd + {\sqrt{-4 {\left( b^2 - 3ac + 12d \right) }^3 + {\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd \right) }^2}} }} {54} \right)^\frac{1}{3} + \frac{-a^3 + 4ab - 8c} {4{\sqrt{\frac{a^2}{4} - \frac{2b}{3} + \frac{2^{\frac{1}{3}} \left( b^2 - 3ac + 12d \right) }{3 {\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd + {\sqrt{-4
{\left( b^2 - 3ac + 12d \right) }^3 + {\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd \right) }^2}} \right) }^ {\frac{1}{3}}} + \left( \frac{{ 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd + {\sqrt{-4 {\left( b^2 - 3ac + 12d \right) }^3 + {\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd \right) }^2}} } }{54}\right)^\frac{1}{3}}}}}}} \end{eqnarray*} The formulas for the roots are much too unwieldy to be used for solving quartic equations by radicals, even with the help of a computer. A practical algorithm for solving quartic equations by radicals is
given in the concluding paragraph of the Galois-theoretic derivation of the quartic formula.
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"quartic formula" is owned by djao.
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Cross-references: Galois-theoretic derivation of the quartic formula, algorithm, computer, even, radicals, formulas, equation, polynomial, roots
There are 8 references to this entry.
This is version 3 of quartic formula, born on 2002-01-22, modified 2005-07-07.
Object id is 1525, canonical name is QuarticFormula.
Accessed 129836 times total.
Classification:
| AMS MSC: | 12D10 (Field theory and polynomials :: Real and complex fields :: Polynomials: location of zeros ) |
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Pending Errata and Addenda
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