quartic formula

The four roots $r_{1}, r_{2}, r_{3}, r_{4}$ of a quartic polynomial equation $x^{4} + a x^{3} + b x^{2} + c x + d = 0$ are given by

$r_{1}$	$=$	$\frac{- a}{4} - \frac{1}{2} \sqrt{\frac{a^{2}}{4} - \frac{2 b}{3} + \frac{2^{\frac{1}{3}} (b^{2} - 3 a c + 12 d)}{3 {(2 b^{3} - 9 a b c + 27 c^{2} + 27 a^{2} d - 72 b d + \sqrt{- 4 {(b^{2} - 3 a c + 12 d)}^{3} + {(2 b^{3} - 9 a b c + 27 c^{2} + 27 a^{2} d - 72 b d)}^{2}})}^{\frac{1}{3}}} + {(\frac{2 b^{3} - 9 a b c + 27 c^{2} + 27 a^{2} d - 72 b d + \sqrt{- 4 {(b^{2} - 3 a c + 12 d)}^{3} + {(2 b^{3} - 9 a b c + 27 c^{2} + 27 a^{2} d - 72 b d)}^{2}}}{54})}^{\frac{1}{3}}} - \frac{1}{2} \sqrt{\frac{a^{2}}{2} - \frac{4 b}{3} - \frac{2^{\frac{1}{3}} (b^{2} - 3 a c + 12 d)}{3 {(2 b^{3} - 9 a b c + 27 c^{2} + 27 a^{2} d - 72 b d + \sqrt{- 4 {(b^{2} - 3 a c + 12 d)}^{3} + {(2 b^{3} - 9 a b c + 27 c^{2} + 27 a^{2} d - 72 b d)}^{2}})}^{\frac{1}{3}}} - {(\frac{2 b^{3} - 9 a b c + 27 c^{2} + 27 a^{2} d - 72 b d + \sqrt{- 4 {(b^{2} - 3 a c + 12 d)}^{3} + {(2 b^{3} - 9 a b c + 27 c^{2} + 27 a^{2} d - 72 b d)}^{2}}}{54})}^{\frac{1}{3}} - \frac{- a^{3} + 4 a b - 8 c}{4 \sqrt{\frac{a^{2}}{4} - \frac{2 b}{3} + \frac{2^{\frac{1}{3}} (b^{2} - 3 a c + 12 d)}{3 {(2 b^{3} - 9 a b c + 27 c^{2} + 27 a^{2} d - 72 b d + \sqrt{- 4 {(b^{2} - 3 a c + 12 d)}^{3} + {(2 b^{3} - 9 a b c + 27 c^{2} + 27 a^{2} d - 72 b d)}^{2}})}^{\frac{1}{3}}} + {(\frac{2 b^{3} - 9 a b c + 27 c^{2} + 27 a^{2} d - 72 b d + \sqrt{- 4 {(b^{2} - 3 a c + 12 d)}^{3} + {(2 b^{3} - 9 a b c + 27 c^{2} + 27 a^{2} d - 72 b d)}^{2}}}{54})}^{\frac{1}{3}}}}}$
$r_{2}$	$=$	$\frac{- a}{4} - \frac{1}{2} \sqrt{\frac{a^{2}}{4} - \frac{2 b}{3} + \frac{2^{\frac{1}{3}} (b^{2} - 3 a c + 12 d)}{3 {(2 b^{3} - 9 a b c + 27 c^{2} + 27 a^{2} d - 72 b d + \sqrt{- 4 {(b^{2} - 3 a c + 12 d)}^{3} + {(2 b^{3} - 9 a b c + 27 c^{2} + 27 a^{2} d - 72 b d)}^{2}})}^{\frac{1}{3}}} + {(\frac{2 b^{3} - 9 a b c + 27 c^{2} + 27 a^{2} d - 72 b d + \sqrt{- 4 {(b^{2} - 3 a c + 12 d)}^{3} + {(2 b^{3} - 9 a b c + 27 c^{2} + 27 a^{2} d - 72 b d)}^{2}}}{54})}^{\frac{1}{3}}} + \frac{1}{2} \sqrt{\frac{a^{2}}{2} - \frac{4 b}{3} - \frac{2^{\frac{1}{3}} (b^{2} - 3 a c + 12 d)}{3 {(2 b^{3} - 9 a b c + 27 c^{2} + 27 a^{2} d - 72 b d + \sqrt{- 4 {(b^{2} - 3 a c + 12 d)}^{3} + {(2 b^{3} - 9 a b c + 27 c^{2} + 27 a^{2} d - 72 b d)}^{2}})}^{\frac{1}{3}}} - {(\frac{2 b^{3} - 9 a b c + 27 c^{2} + 27 a^{2} d - 72 b d + \sqrt{- 4 {(b^{2} - 3 a c + 12 d)}^{3} + {(2 b^{3} - 9 a b c + 27 c^{2} + 27 a^{2} d - 72 b d)}^{2}}}{54})}^{\frac{1}{3}} - \frac{- a^{3} + 4 a b - 8 c}{4 \sqrt{\frac{a^{2}}{4} - \frac{2 b}{3} + \frac{2^{\frac{1}{3}} (b^{2} - 3 a c + 12 d)}{3 {(2 b^{3} - 9 a b c + 27 c^{2} + 27 a^{2} d - 72 b d + \sqrt{- 4 {(b^{2} - 3 a c + 12 d)}^{3} + {(2 b^{3} - 9 a b c + 27 c^{2} + 27 a^{2} d - 72 b d)}^{2}})}^{\frac{1}{3}}} + {(\frac{2 b^{3} - 9 a b c + 27 c^{2} + 27 a^{2} d - 72 b d + \sqrt{- 4 {(b^{2} - 3 a c + 12 d)}^{3} + {(2 b^{3} - 9 a b c + 27 c^{2} + 27 a^{2} d - 72 b d)}^{2}}}{54})}^{\frac{1}{3}}}}}$
$r_{3}$	$=$	$\frac{- a}{4} + \frac{1}{2} \sqrt{\frac{a^{2}}{4} - \frac{2 b}{3} + \frac{2^{\frac{1}{3}} (b^{2} - 3 a c + 12 d)}{3 {(2 b^{3} - 9 a b c + 27 c^{2} + 27 a^{2} d - 72 b d + \sqrt{- 4 {(b^{2} - 3 a c + 12 d)}^{3} + {(2 b^{3} - 9 a b c + 27 c^{2} + 27 a^{2} d - 72 b d)}^{2}})}^{\frac{1}{3}}} + {(\frac{2 b^{3} - 9 a b c + 27 c^{2} + 27 a^{2} d - 72 b d + \sqrt{- 4 {(b^{2} - 3 a c + 12 d)}^{3} + {(2 b^{3} - 9 a b c + 27 c^{2} + 27 a^{2} d - 72 b d)}^{2}}}{54})}^{\frac{1}{3}}} - \frac{1}{2} \sqrt{\frac{a^{2}}{2} - \frac{4 b}{3} - \frac{2^{\frac{1}{3}} (b^{2} - 3 a c + 12 d)}{3 {(2 b^{3} - 9 a b c + 27 c^{2} + 27 a^{2} d - 72 b d + \sqrt{- 4 {(b^{2} - 3 a c + 12 d)}^{3} + {(2 b^{3} - 9 a b c + 27 c^{2} + 27 a^{2} d - 72 b d)}^{2}})}^{\frac{1}{3}}} - {(\frac{2 b^{3} - 9 a b c + 27 c^{2} + 27 a^{2} d - 72 b d + \sqrt{- 4 {(b^{2} - 3 a c + 12 d)}^{3} + {(2 b^{3} - 9 a b c + 27 c^{2} + 27 a^{2} d - 72 b d)}^{2}}}{54})}^{\frac{1}{3}} + \frac{- a^{3} + 4 a b - 8 c}{4 \sqrt{\frac{a^{2}}{4} - \frac{2 b}{3} + \frac{2^{\frac{1}{3}} (b^{2} - 3 a c + 12 d)}{3 {(2 b^{3} - 9 a b c + 27 c^{2} + 27 a^{2} d - 72 b d + \sqrt{- 4 {(b^{2} - 3 a c + 12 d)}^{3} + {(2 b^{3} - 9 a b c + 27 c^{2} + 27 a^{2} d - 72 b d)}^{2}})}^{\frac{1}{3}}} + {(\frac{2 b^{3} - 9 a b c + 27 c^{2} + 27 a^{2} d - 72 b d + \sqrt{- 4 {(b^{2} - 3 a c + 12 d)}^{3} + {(2 b^{3} - 9 a b c + 27 c^{2} + 27 a^{2} d - 72 b d)}^{2}}}{54})}^{\frac{1}{3}}}}}$
$r_{4}$	$=$	$\frac{- a}{4} + \frac{1}{2} \sqrt{\frac{a^{2}}{4} - \frac{2 b}{3} + \frac{2^{\frac{1}{3}} (b^{2} - 3 a c + 12 d)}{3 {(2 b^{3} - 9 a b c + 27 c^{2} + 27 a^{2} d - 72 b d + \sqrt{- 4 {(b^{2} - 3 a c + 12 d)}^{3} + {(2 b^{3} - 9 a b c + 27 c^{2} + 27 a^{2} d - 72 b d)}^{2}})}^{\frac{1}{3}}} + {(\frac{2 b^{3} - 9 a b c + 27 c^{2} + 27 a^{2} d - 72 b d + \sqrt{- 4 {(b^{2} - 3 a c + 12 d)}^{3} + {(2 b^{3} - 9 a b c + 27 c^{2} + 27 a^{2} d - 72 b d)}^{2}}}{54})}^{\frac{1}{3}}} + \frac{1}{2} \sqrt{\frac{a^{2}}{2} - \frac{4 b}{3} - \frac{2^{\frac{1}{3}} (b^{2} - 3 a c + 12 d)}{3 {(2 b^{3} - 9 a b c + 27 c^{2} + 27 a^{2} d - 72 b d + \sqrt{- 4 {(b^{2} - 3 a c + 12 d)}^{3} + {(2 b^{3} - 9 a b c + 27 c^{2} + 27 a^{2} d - 72 b d)}^{2}})}^{\frac{1}{3}}} - {(\frac{2 b^{3} - 9 a b c + 27 c^{2} + 27 a^{2} d - 72 b d + \sqrt{- 4 {(b^{2} - 3 a c + 12 d)}^{3} + {(2 b^{3} - 9 a b c + 27 c^{2} + 27 a^{2} d - 72 b d)}^{2}}}{54})}^{\frac{1}{3}} + \frac{- a^{3} + 4 a b - 8 c}{4 \sqrt{\frac{a^{2}}{4} - \frac{2 b}{3} + \frac{2^{\frac{1}{3}} (b^{2} - 3 a c + 12 d)}{3 {(2 b^{3} - 9 a b c + 27 c^{2} + 27 a^{2} d - 72 b d + \sqrt{- 4 {(b^{2} - 3 a c + 12 d)}^{3} + {(2 b^{3} - 9 a b c + 27 c^{2} + 27 a^{2} d - 72 b d)}^{2}})}^{\frac{1}{3}}} + {(\frac{2 b^{3} - 9 a b c + 27 c^{2} + 27 a^{2} d - 72 b d + \sqrt{- 4 {(b^{2} - 3 a c + 12 d)}^{3} + {(2 b^{3} - 9 a b c + 27 c^{2} + 27 a^{2} d - 72 b d)}^{2}}}{54})}^{\frac{1}{3}}}}}$

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