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单词 QuarticFormula
释义

quartic formula


The four roots r1,r2,r3,r4 of a quartic polynomial equationx4+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.

Titlequartic formula
Canonical nameQuarticFormula
Date of creation2013-03-22 12:12:29
Last modified on2013-03-22 12:12:29
Ownerdjao (24)
Last modified bydjao (24)
Numerical id7
Authordjao (24)
Entry typeTheorem
Classificationmsc 12D10
Synonymbiquadratic formula
Synonymquartic equation
Synonymbiquadratic equation
Related topicGaloisTheoreticDerivationOfTheCubicFormula
Related topicCubicFormula
Related topicCardanosDerivationOfTheCubicFormula
Related topicFundamentalTheoremOfGaloisTheory
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更新时间:2025/5/4 19:23:08