quartic formula
The four roots r1,r2,r3,r4 of a quartic polynomial equation x4+ax3+bx2+cx+d=0 are given by
r1 | = | -a4-12√a24-2b3+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)13-12√a22-4b3-213(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-8c4√a24-2b3+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 | = | -a4-12√a24-2b3+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+12√a22-4b3-213(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-8c4√a24-2b3+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+12√a24-2b3+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)13-12√a22-4b3-213(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-8c4√a24-2b3+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+12√a24-2b3+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+12√a22-4b3-213(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-8c4√a24-2b3+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 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 |