quartic formula


The four roots r1,r2,r3,r4 of a quartic polynomial equation x4+ax3+bx2+cx+d=0 are given by

r1 = -a4-12a24-2b3+213(b2-3ac+12d)3(2b3-9abc+27c2+27a2d-72bd+-4(b2-3ac+12d)3+(2b3-9abc+27c2+27a2d-72bd)2)13+(2b3-9abc+27c2+27a2d-72bd+-4(b2-3ac+12d)3+(2b3-9abc+27c2+27a2d-72bd)254)13-12a22-4b3-213(b2-3ac+12d)3(2b3-9abc+27c2+27a2d-72bd+-4(b2-3ac+12d)3+(2b3-9abc+27c2+27a2d-72bd)2)13-(2b3-9abc+27c2+27a2d-72bd+-4(b2-3ac+12d)3+(2b3-9abc+27c2+27a2d-72bd)254)13--a3+4ab-8c4a24-2b3+213(b2-3ac+12d)3(2b3-9abc+27c2+27a2d-72bd+-4(b2-3ac+12d)3+(2b3-9abc+27c2+27a2d-72bd)2)13+(2b3-9abc+27c2+27a2d-72bd+-4(b2-3ac+12d)3+(2b3-9abc+27c2+27a2d-72bd)254)13
r2 = -a4-12a24-2b3+213(b2-3ac+12d)3(2b3-9abc+27c2+27a2d-72bd+-4(b2-3ac+12d)3+(2b3-9abc+27c2+27a2d-72bd)2)13+(2b3-9abc+27c2+27a2d-72bd+-4(b2-3ac+12d)3+(2b3-9abc+27c2+27a2d-72bd)254)13+12a22-4b3-213(b2-3ac+12d)3(2b3-9abc+27c2+27a2d-72bd+-4(b2-3ac+12d)3+(2b3-9abc+27c2+27a2d-72bd)2)13-(2b3-9abc+27c2+27a2d-72bd+-4(b2-3ac+12d)3+(2b3-9abc+27c2+27a2d-72bd)254)13--a3+4ab-8c4a24-2b3+213(b2-3ac+12d)3(2b3-9abc+27c2+27a2d-72bd+-4(b2-3ac+12d)3+(2b3-9abc+27c2+27a2d-72bd)2)13+(2b3-9abc+27c2+27a2d-72bd+-4(b2-3ac+12d)3+(2b3-9abc+27c2+27a2d-72bd)254)13
r3 = -a4+12a24-2b3+213(b2-3ac+12d)3(2b3-9abc+27c2+27a2d-72bd+-4(b2-3ac+12d)3+(2b3-9abc+27c2+27a2d-72bd)2)13+(2b3-9abc+27c2+27a2d-72bd+-4(b2-3ac+12d)3+(2b3-9abc+27c2+27a2d-72bd)254)13-12a22-4b3-213(b2-3ac+12d)3(2b3-9abc+27c2+27a2d-72bd+-4(b2-3ac+12d)3+(2b3-9abc+27c2+27a2d-72bd)2)13-(2b3-9abc+27c2+27a2d-72bd+-4(b2-3ac+12d)3+(2b3-9abc+27c2+27a2d-72bd)254)13+-a3+4ab-8c4a24-2b3+213(b2-3ac+12d)3(2b3-9abc+27c2+27a2d-72bd+-4(b2-3ac+12d)3+(2b3-9abc+27c2+27a2d-72bd)2)13+(2b3-9abc+27c2+27a2d-72bd+-4(b2-3ac+12d)3+(2b3-9abc+27c2+27a2d-72bd)254)13
r4 = -a4+12a24-2b3+213(b2-3ac+12d)3(2b3-9abc+27c2+27a2d-72bd+-4(b2-3ac+12d)3+(2b3-9abc+27c2+27a2d-72bd)2)13+(2b3-9abc+27c2+27a2d-72bd+-4(b2-3ac+12d)3+(2b3-9abc+27c2+27a2d-72bd)254)13+12a22-4b3-213(b2-3ac+12d)3(2b3-9abc+27c2+27a2d-72bd+-4(b2-3ac+12d)3+(2b3-9abc+27c2+27a2d-72bd)2)13-(2b3-9abc+27c2+27a2d-72bd+-4(b2-3ac+12d)3+(2b3-9abc+27c2+27a2d-72bd)254)13+-a3+4ab-8c4a24-2b3+213(b2-3ac+12d)3(2b3-9abc+27c2+27a2d-72bd+-4(b2-3ac+12d)3+(2b3-9abc+27c2+27a2d-72bd)2)13+(2b3-9abc+27c2+27a2d-72bd+-4(b2-3ac+12d)3+(2b3-9abc+27c2+27a2d-72bd)254)13

The formulas for the roots are much too unwieldy to be used for solving quartic equationsMathworldPlanetmath by radicalsPlanetmathPlanetmath, even with the help of a computer. A practical algorithmMathworldPlanetmath 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