quartic formula

The four roots $r_1,r_2,r_3,r_4$ of a quartic polynomial equation $x^4+a{x}^3+b{x}^2+cx+d=0$ are given by

$r_1$ $=$ ${\displaystyle \frac{-a}{4}}-{\displaystyle \frac{1}{2}}\sqrt{{\displaystyle \frac{a^2}{4}}-{\displaystyle \frac{2b}{3}}+{\displaystyle \frac{2^{\frac{1}{3}}\left(b^2-3ac+12d\right)}{3{\left(2b^3-9abc+27c^2+27a^{2}d-72bd+\sqrt{-4{\left(b^2-3ac+12d\right)}^3+{\left(2b^3-9abc+27c^2+27a^{2}d-72bd\right)}^2}\right)}^{\frac{1}{3}}}}+{\left({\displaystyle \frac{2b^3-9abc+27c^2+27a^{2}d-72bd+\sqrt{-4{\left(b^2-3ac+12d\right)}^3+{\left(2b^3-9abc+27c^2+27a^{2}d-72bd\right)}^2}}{54}}\right)}^{\frac{1}{3}}}-{\displaystyle \frac{1}{2}}\sqrt{{\displaystyle \frac{a^2}{2}}-{\displaystyle \frac{4b}{3}}-{\displaystyle \frac{2^{\frac{1}{3}}\left(b^2-3ac+12d\right)}{3{\left(2b^3-9abc+27c^2+27a^{2}d-72bd+\sqrt{-4{\left(b^2-3ac+12d\right)}^3+{\left(2b^3-9abc+27c^2+27a^{2}d-72bd\right)}^2}\right)}^{\frac{1}{3}}}}-{\left({\displaystyle \frac{2b^3-9abc+27c^2+27a^{2}d-72bd+\sqrt{-4{\left(b^2-3ac+12d\right)}^3+{\left(2b^3-9abc+27c^2+27a^{2}d-72bd\right)}^2}}{54}}\right)}^{\frac{1}{3}}-{\displaystyle \frac{-a^3+4ab-8c}{4\sqrt{\frac{a^2}{4}-\frac{2b}{3}+\frac{2^{\frac{1}{3}}\left(b^2-3ac+12d\right)}{3{\left(2b^3-9abc+27c^2+27a^{2}d-72bd+\sqrt{-4{\left(b^2-3ac+12d\right)}^3+{\left(2b^3-9abc+27c^2+27a^{2}d-72bd\right)}^2}\right)}^{\frac{1}{3}}}+{\left(\frac{2b^3-9abc+27c^2+27a^{2}d-72bd+\sqrt{-4{\left(b^2-3ac+12d\right)}^3+{\left(2b^3-9abc+27c^2+27a^{2}d-72bd\right)}^2}}{54}\right)}^{\frac{1}{3}}}}}}$

$r_2$ $=$ ${\displaystyle \frac{-a}{4}}-{\displaystyle \frac{1}{2}}\sqrt{{\displaystyle \frac{a^2}{4}}-{\displaystyle \frac{2b}{3}}+{\displaystyle \frac{2^{\frac{1}{3}}\left(b^2-3ac+12d\right)}{3{\left(2b^3-9abc+27c^2+27a^{2}d-72bd+\sqrt{-4{\left(b^2-3ac+12d\right)}^3+{\left(2b^3-9abc+27c^2+27a^{2}d-72bd\right)}^2}\right)}^{\frac{1}{3}}}}+{\left({\displaystyle \frac{2b^3-9abc+27c^2+27a^{2}d-72bd+\sqrt{-4{\left(b^2-3ac+12d\right)}^3+{\left(2b^3-9abc+27c^2+27a^{2}d-72bd\right)}^2}}{54}}\right)}^{\frac{1}{3}}}+{\displaystyle \frac{1}{2}}\sqrt{{\displaystyle \frac{a^2}{2}}-{\displaystyle \frac{4b}{3}}-{\displaystyle \frac{2^{\frac{1}{3}}\left(b^2-3ac+12d\right)}{3{\left(2b^3-9abc+27c^2+27a^{2}d-72bd+\sqrt{-4{\left(b^2-3ac+12d\right)}^3+{\left(2b^3-9abc+27c^2+27a^{2}d-72bd\right)}^2}\right)}^{\frac{1}{3}}}}-{\left({\displaystyle \frac{2b^3-9abc+27c^2+27a^{2}d-72bd+\sqrt{-4{\left(b^2-3ac+12d\right)}^3+{\left(2b^3-9abc+27c^2+27a^{2}d-72bd\right)}^2}}{54}}\right)}^{\frac{1}{3}}-{\displaystyle \frac{-a^3+4ab-8c}{4\sqrt{\frac{a^2}{4}-\frac{2b}{3}+\frac{2^{\frac{1}{3}}\left(b^2-3ac+12d\right)}{3{\left(2b^3-9abc+27c^2+27a^{2}d-72bd+\sqrt{-4{\left(b^2-3ac+12d\right)}^3+{\left(2b^3-9abc+27c^2+27a^{2}d-72bd\right)}^2}\right)}^{\frac{1}{3}}}+{\left(\frac{2b^3-9abc+27c^2+27a^{2}d-72bd+\sqrt{-4{\left(b^2-3ac+12d\right)}^3+{\left(2b^3-9abc+27c^2+27a^{2}d-72bd\right)}^2}}{54}\right)}^{\frac{1}{3}}}}}}$

$r_3$ $=$ ${\displaystyle \frac{-a}{4}}+{\displaystyle \frac{1}{2}}\sqrt{{\displaystyle \frac{a^2}{4}}-{\displaystyle \frac{2b}{3}}+{\displaystyle \frac{2^{\frac{1}{3}}\left(b^2-3ac+12d\right)}{3{\left(2b^3-9abc+27c^2+27a^{2}d-72bd+\sqrt{-4{\left(b^2-3ac+12d\right)}^3+{\left(2b^3-9abc+27c^2+27a^{2}d-72bd\right)}^2}\right)}^{\frac{1}{3}}}}+{\left({\displaystyle \frac{2b^3-9abc+27c^2+27a^{2}d-72bd+\sqrt{-4{\left(b^2-3ac+12d\right)}^3+{\left(2b^3-9abc+27c^2+27a^{2}d-72bd\right)}^2}}{54}}\right)}^{\frac{1}{3}}}-{\displaystyle \frac{1}{2}}\sqrt{{\displaystyle \frac{a^2}{2}}-{\displaystyle \frac{4b}{3}}-{\displaystyle \frac{2^{\frac{1}{3}}\left(b^2-3ac+12d\right)}{3{\left(2b^3-9abc+27c^2+27a^{2}d-72bd+\sqrt{-4{\left(b^2-3ac+12d\right)}^3+{\left(2b^3-9abc+27c^2+27a^{2}d-72bd\right)}^2}\right)}^{\frac{1}{3}}}}-{\left({\displaystyle \frac{2b^3-9abc+27c^2+27a^{2}d-72bd+\sqrt{-4{\left(b^2-3ac+12d\right)}^3+{\left(2b^3-9abc+27c^2+27a^{2}d-72bd\right)}^2}}{54}}\right)}^{\frac{1}{3}}+{\displaystyle \frac{-a^3+4ab-8c}{4\sqrt{\frac{a^2}{4}-\frac{2b}{3}+\frac{2^{\frac{1}{3}}\left(b^2-3ac+12d\right)}{3{\left(2b^3-9abc+27c^2+27a^{2}d-72bd+\sqrt{-4{\left(b^2-3ac+12d\right)}^3+{\left(2b^3-9abc+27c^2+27a^{2}d-72bd\right)}^2}\right)}^{\frac{1}{3}}}+{\left(\frac{2b^3-9abc+27c^2+27a^{2}d-72bd+\sqrt{-4{\left(b^2-3ac+12d\right)}^3+{\left(2b^3-9abc+27c^2+27a^{2}d-72bd\right)}^2}}{54}\right)}^{\frac{1}{3}}}}}}$

$r_4$ $=$ ${\displaystyle \frac{-a}{4}}+{\displaystyle \frac{1}{2}}\sqrt{{\displaystyle \frac{a^2}{4}}-{\displaystyle \frac{2b}{3}}+{\displaystyle \frac{2^{\frac{1}{3}}\left(b^2-3ac+12d\right)}{3{\left(2b^3-9abc+27c^2+27a^{2}d-72bd+\sqrt{-4{\left(b^2-3ac+12d\right)}^3+{\left(2b^3-9abc+27c^2+27a^{2}d-72bd\right)}^2}\right)}^{\frac{1}{3}}}}+{\left({\displaystyle \frac{2b^3-9abc+27c^2+27a^{2}d-72bd+\sqrt{-4{\left(b^2-3ac+12d\right)}^3+{\left(2b^3-9abc+27c^2+27a^{2}d-72bd\right)}^2}}{54}}\right)}^{\frac{1}{3}}}+{\displaystyle \frac{1}{2}}\sqrt{{\displaystyle \frac{a^2}{2}}-{\displaystyle \frac{4b}{3}}-{\displaystyle \frac{2^{\frac{1}{3}}\left(b^2-3ac+12d\right)}{3{\left(2b^3-9abc+27c^2+27a^{2}d-72bd+\sqrt{-4{\left(b^2-3ac+12d\right)}^3+{\left(2b^3-9abc+27c^2+27a^{2}d-72bd\right)}^2}\right)}^{\frac{1}{3}}}}-{\left({\displaystyle \frac{2b^3-9abc+27c^2+27a^{2}d-72bd+\sqrt{-4{\left(b^2-3ac+12d\right)}^3+{\left(2b^3-9abc+27c^2+27a^{2}d-72bd\right)}^2}}{54}}\right)}^{\frac{1}{3}}+{\displaystyle \frac{-a^3+4ab-8c}{4\sqrt{\frac{a^2}{4}-\frac{2b}{3}+\frac{2^{\frac{1}{3}}\left(b^2-3ac+12d\right)}{3{\left(2b^3-9abc+27c^2+27a^{2}d-72bd+\sqrt{-4{\left(b^2-3ac+12d\right)}^3+{\left(2b^3-9abc+27c^2+27a^{2}d-72bd\right)}^2}\right)}^{\frac{1}{3}}}+{\left(\frac{2b^3-9abc+27c^2+27a^{2}d-72bd+\sqrt{-4{\left(b^2-3ac+12d\right)}^3+{\left(2b^3-9abc+27c^2+27a^{2}d-72bd\right)}^2}}{54}\right)}^{\frac{1}{3}}}}}}$

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.

Title	quartic formula
Canonical name	QuarticFormula
Date of creation	2013-03-22 12:12:29
Last modified on	2013-03-22 12:12:29
Owner	djao (24)
Last modified by	djao (24)
Numerical id	7
Author	djao (24)
Entry type	Theorem
Classification	msc 12D10
Synonym	biquadratic formula
Synonym	quartic equation
Synonym	biquadratic equation
Related topic	GaloisTheoreticDerivationOfTheCubicFormula
Related topic	CubicFormula
Related topic	CardanosDerivationOfTheCubicFormula
Related topic	FundamentalTheoremOfGaloisTheory