File:Quartic Formula.svg

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All solutions of the equation $x^4+ax^3+bx^2+cx+d=0$
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<math>\begin{align}
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^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2} } }\right)}^{\frac{1}{3} } }+\left(\frac{ {2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-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^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2} } }\right)}^{\frac{1}{3} } }-\left(\frac{ {2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-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^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2} } }\right)}^{\frac{1}{3} } }+\left(\frac{ {2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-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^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2} } }\right)}^{\frac{1}{3} } }+\left(\frac{ {2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-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^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2} } }\right)}^{\frac{1}{3} } }-\left(\frac{ {2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-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^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2} } }\right)}^{\frac{1}{3} } }+\left(\frac{ {2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-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^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2} } }\right)}^{\frac{1}{3} } }+\left(\frac{ {2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-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^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2} } }\right)}^{\frac{1}{3} } }-\left(\frac{ {2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-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^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2} } }\right)}^{\frac{1}{3} } }+\left(\frac{ {2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-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^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2} } }\right)}^{\frac{1}{3} } }+\left(\frac{ {2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-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^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2} } }\right)}^{\frac{1}{3} } }-\left(\frac{ {2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-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^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2} } }\right)}^{\frac{1}{3} } }+\left(\frac{ {2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2} } } } }{54}\right)^{\frac{1}{3} } } } } } }
\end{align}</math>
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Summary

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Description
English: All 4 roots of a quartic equation (x^4+ax^3+bx^2+cx+d=0).
Source https://planetmath.org/quarticformula
Author David Jao

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This work is ineligible for copyright and therefore in the public domain because it consists entirely of information that is common property and contains no original authorship.

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current00:10, 17 May 2013Thumbnail for version as of 00:10, 17 May 201314,406 × 1,443 (326 KB)Linket (talk | contribs){{subst:Upload marker added by en.wp UW}} {{Information |Description = {{en|All 4 roots of a quartic equation (x^4+ax^3+bx^2+cx+d=0).}} |Source = http://planetmath.org/quarticformula |Author = David Jao }} Category:Mathematical equations

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