# Cardano’s derivation of the cubic formula

To solve the cubic polynomial equation ${x}^{3}+a{x}^{2}+bx+c=0$ for $x$, the first step is to apply the Tchirnhaus transformation $x=y-\frac{a}{3}$. This reduces the equation to ${y}^{3}+py+q=0$, where

$p$ | $=$ | $b-{\displaystyle \frac{{a}^{2}}{3}}$ | ||

$q$ | $=$ | $c-{\displaystyle \frac{ab}{3}}+{\displaystyle \frac{2{a}^{3}}{27}}$ |

The next step is to substitute $y=u-v$, to obtain

$${(u-v)}^{3}+p(u-v)+q=0$$ | (1) |

or, with the terms collected,

$$(q-({v}^{3}-{u}^{3}))+(u-v)(p-3uv)=0$$ | (2) |

From equation (2), we see that if $u$ and $v$ are chosen so that $q={v}^{3}-{u}^{3}$ and $p=3uv$, then $y=u-v$ will satisfy equation (1), and the cubic equation^{} will be solved!

There remains the matter of solving $q={v}^{3}-{u}^{3}$ and $p=3uv$ for $u$ and $v$. From the second equation, we get $v=p/(3u)$, and substituting this $v$ into the first equation yields

$$q=\frac{{p}^{3}}{{(3u)}^{3}}-{u}^{3}$$ |

which is a quadratic equation in ${u}^{3}$. Solving for ${u}^{3}$ using the quadratic formula, we get

${u}^{3}$ | $=$ | $\frac{-27q+\sqrt{108{p}^{3}+729{q}^{2}}}{54}}={\displaystyle \frac{-9q+\sqrt{12{p}^{3}+81{q}^{2}}}{18}$ | ||

${v}^{3}$ | $=$ | $\frac{27q+\sqrt{108{p}^{3}+729{q}^{2}}}{54}}={\displaystyle \frac{9q+\sqrt{12{p}^{3}+81{q}^{2}}}{18}$ |

Using these values for $u$ and $v$, you can back–substitute $y=u-v$, $p=b-{a}^{2}/3$, $q=c-ab/3+2{a}^{3}/27$, and $x=y-a/3$ to get the expression for the first root ${r}_{1}$ in the cubic formula. The second and third roots ${r}_{2}$ and ${r}_{3}$ are obtained by performing synthetic division^{} using ${r}_{1}$, and using the quadratic formula on the remaining quadratic factor.

Title | Cardano’s derivation of the cubic formula |
---|---|

Canonical name | CardanosDerivationOfTheCubicFormula |

Date of creation | 2013-03-22 12:10:28 |

Last modified on | 2013-03-22 12:10:28 |

Owner | djao (24) |

Last modified by | djao (24) |

Numerical id | 12 |

Author | djao (24) |

Entry type | Proof |

Classification | msc 12D10 |

Related topic | FerrariCardanoDerivationOfTheQuarticFormula |