cubic formula

The three roots $r_{1},r_{2},r_{3}$ of a cubic polynomial equation $x^{3}+ax^{2}+bx+c=0$ are given by

 $\displaystyle r_{1}$ $\displaystyle=$ $\displaystyle-\frac{a}{3}+\left(\frac{-2a^{3}+9ab-27c+\sqrt{(2a^{3}-9ab+27c)^{% 2}+4(-a^{2}+3b)^{3}}}{54}\right)^{1/3}$ $\displaystyle{}+\left(\frac{-2a^{3}+9ab-27c-\sqrt{(2a^{3}-9ab+27c)^{2}+4(-a^{2% }+3b)^{3}}}{54}\right)^{1/3}$ $\displaystyle r_{2}$ $\displaystyle=$ $\displaystyle-\frac{a}{3}-\frac{1+i\sqrt{3}}{2}\left(\frac{-2a^{3}+9ab-27c+% \sqrt{(2a^{3}-9ab+27c)^{2}+4(-a^{2}+3b)^{3}}}{54}\right)^{1/3}$ $\displaystyle{}+\frac{-1+i\sqrt{3}}{2}\left(\frac{-2a^{3}+9ab-27c-\sqrt{(2a^{3% }-9ab+27c)^{2}+4(-a^{2}+3b)^{3}}}{54}\right)^{1/3}$ $\displaystyle r_{3}$ $\displaystyle=$ $\displaystyle-\frac{a}{3}+\frac{-1+i\sqrt{3}}{2}\left(\frac{-2a^{3}+9ab-27c+% \sqrt{(2a^{3}-9ab+27c)^{2}+4(-a^{2}+3b)^{3}}}{54}\right)^{1/3}$ $\displaystyle{}-\frac{1+i\sqrt{3}}{2}\left(\frac{-2a^{3}+9ab-27c-\sqrt{(2a^{3}% -9ab+27c)^{2}+4(-a^{2}+3b)^{3}}}{54}\right)^{1/3}$
Title cubic formula CubicFormula 2013-03-22 12:10:25 2013-03-22 12:10:25 djao (24) djao (24) 10 djao (24) Theorem msc 12D10 cubic equation QuarticFormula GaloisTheoreticDerivationOfTheQuarticFormula FerrariCardanoDerivationOfTheQuarticFormula FundamentalTheoremOfGaloisTheory