# quartic formula

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

 $\displaystyle r_{1}$ $\displaystyle=$ $\displaystyle{\frac{-a}{4}-\frac{1}{2}{\sqrt{\frac{a^{2}}{4}-\frac{2b}{3}+% \frac{2^{\frac{1}{3}}\left(b^{2}-3ac+12d\right)}{3{\left(2b^{3}-9abc+27c^{2}+2% 7a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^% {2}+27a^{2}d-72bd\right)}^{2}}}\right)}^{\frac{1}{3}}}+\left(\frac{{2b^{3}-9% abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^% {3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2}}}}}{54}\right)^{\frac{1}{3}}}}-% \frac{1}{2}{\sqrt{\frac{a^{2}}{2}-\frac{4b}{3}-\frac{2^{\frac{1}{3}}\left(b^{2% }-3ac+12d\right)}{3{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^% {2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2}}}% \right)}^{\frac{1}{3}}}-\left(\frac{{2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{% -4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd% \right)}^{2}}}}}{54}\right)^{\frac{1}{3}}-\frac{-a^{3}+4ab-8c}{4{\sqrt{\frac{a% ^{2}}{4}-\frac{2b}{3}+\frac{2^{\frac{1}{3}}\left(b^{2}-3ac+12d\right)}{3{\left% (2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{% \left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2}}}\right)}^{\frac{1}{3}}}+% \left(\frac{{2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d% \right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2}}}}}{54}% \right)^{\frac{1}{3}}}}}}}}$ $\displaystyle r_{2}$ $\displaystyle=$ $\displaystyle{\frac{-a}{4}-\frac{1}{2}{\sqrt{\frac{a^{2}}{4}-\frac{2b}{3}+% \frac{2^{\frac{1}{3}}\left(b^{2}-3ac+12d\right)}{3{\left(2b^{3}-9abc+27c^{2}+2% 7a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^% {2}+27a^{2}d-72bd\right)}^{2}}}\right)}^{\frac{1}{3}}}+\left(\frac{{2b^{3}-9% abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^% {3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2}}}}}{54}\right)^{\frac{1}{3}}}}+% \frac{1}{2}{\sqrt{\frac{a^{2}}{2}-\frac{4b}{3}-\frac{2^{\frac{1}{3}}\left(b^{2% }-3ac+12d\right)}{3{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^% {2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2}}}% \right)}^{\frac{1}{3}}}-\left(\frac{{2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{% -4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd% \right)}^{2}}}}}{54}\right)^{\frac{1}{3}}-\frac{-a^{3}+4ab-8c}{4{\sqrt{\frac{a% ^{2}}{4}-\frac{2b}{3}+\frac{2^{\frac{1}{3}}\left(b^{2}-3ac+12d\right)}{3{\left% (2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{% \left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2}}}\right)}^{\frac{1}{3}}}+% \left(\frac{{2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d% \right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2}}}}}{54}% \right)^{\frac{1}{3}}}}}}}}$ $\displaystyle r_{3}$ $\displaystyle=$ $\displaystyle{\frac{-a}{4}+\frac{1}{2}{\sqrt{\frac{a^{2}}{4}-\frac{2b}{3}+% \frac{2^{\frac{1}{3}}\left(b^{2}-3ac+12d\right)}{3{\left(2b^{3}-9abc+27c^{2}+2% 7a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^% {2}+27a^{2}d-72bd\right)}^{2}}}\right)}^{\frac{1}{3}}}+\left(\frac{{2b^{3}-9% abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^% {3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2}}}}}{54}\right)^{\frac{1}{3}}}}-% \frac{1}{2}{\sqrt{\frac{a^{2}}{2}-\frac{4b}{3}-\frac{2^{\frac{1}{3}}\left(b^{2% }-3ac+12d\right)}{3{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^% {2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2}}}% \right)}^{\frac{1}{3}}}-\left(\frac{{2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{% -4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd% \right)}^{2}}}}}{54}\right)^{\frac{1}{3}}+\frac{-a^{3}+4ab-8c}{4{\sqrt{\frac{a% ^{2}}{4}-\frac{2b}{3}+\frac{2^{\frac{1}{3}}\left(b^{2}-3ac+12d\right)}{3{\left% (2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{% \left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2}}}\right)}^{\frac{1}{3}}}+% \left(\frac{{2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d% \right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2}}}}}{54}% \right)^{\frac{1}{3}}}}}}}}$ $\displaystyle r_{4}$ $\displaystyle=$ $\displaystyle{\frac{-a}{4}+\frac{1}{2}{\sqrt{\frac{a^{2}}{4}-\frac{2b}{3}+% \frac{2^{\frac{1}{3}}\left(b^{2}-3ac+12d\right)}{3{\left(2b^{3}-9abc+27c^{2}+2% 7a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^% {2}+27a^{2}d-72bd\right)}^{2}}}\right)}^{\frac{1}{3}}}+\left(\frac{{2b^{3}-9% abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^% {3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2}}}}}{54}\right)^{\frac{1}{3}}}}+% \frac{1}{2}{\sqrt{\frac{a^{2}}{2}-\frac{4b}{3}-\frac{2^{\frac{1}{3}}\left(b^{2% }-3ac+12d\right)}{3{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^% {2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2}}}% \right)}^{\frac{1}{3}}}-\left(\frac{{2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{% -4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd% \right)}^{2}}}}}{54}\right)^{\frac{1}{3}}+\frac{-a^{3}+4ab-8c}{4{\sqrt{\frac{a% ^{2}}{4}-\frac{2b}{3}+\frac{2^{\frac{1}{3}}\left(b^{2}-3ac+12d\right)}{3{\left% (2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{% \left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2}}}\right)}^{\frac{1}{3}}}+% \left(\frac{{2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d% \right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2}}}}}{54}% \right)^{\frac{1}{3}}}}}}}}$

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