# Ferrari-Cardano derivation of the quartic formula

Given a quartic equation^{} ${x}^{4}+a{x}^{3}+b{x}^{2}+cx+d=0$, apply the Tchirnhaus transformation $x\mapsto y-\frac{a}{4}$ to obtain

$${y}^{4}+p{y}^{2}+qy+r=0$$ | (1) |

where

$p$ | $=$ | $b-{\displaystyle \frac{3{a}^{2}}{8}}$ | ||

$q$ | $=$ | $c-{\displaystyle \frac{ab}{2}}+{\displaystyle \frac{{a}^{3}}{8}}$ | ||

$r$ | $=$ | $d-{\displaystyle \frac{ac}{4}}+{\displaystyle \frac{{a}^{2}b}{16}}-{\displaystyle \frac{3{a}^{4}}{256}}$ |

Clearly a solution to Equation (1) solves the original, so we replace the original equation with Equation (1). Move $qy+r$ to the other side and complete^{} the square on the left to get:

$${({y}^{2}+p)}^{2}=p{y}^{2}-qy+({p}^{2}-r).$$ |

We now wish to add the quantity ${({y}^{2}+p+z)}^{2}-{({y}^{2}+p)}^{2}$ to both sides, for some unspecified value of $z$ whose purpose will be made clear in what follows. Note that ${({y}^{2}+p+z)}^{2}-{({y}^{2}+p)}^{2}$ is a quadratic in $y$. Carrying out this addition, we get

$${({y}^{2}+p+z)}^{2}=(p+2z){y}^{2}-qy+({z}^{2}+2pz+{p}^{2}-r)$$ | (2) |

The goal is now to choose a value for $z$ which makes the right hand side of Equation (2) a perfect square^{}. The right hand side is a quadratic polynomial in $y$ whose discriminant^{} is

$$-8{z}^{3}-20p{z}^{2}+(8r-16{p}^{2})z+{q}^{2}+4pr-4{p}^{3}.$$ |

Our goal will be achieved if we can find a value for $z$ which makes this discriminant zero. But the above polynomial^{} is a cubic polynomial in $z$, so its roots can be found using the cubic formula^{}. Choosing then such a value for $z$, we may rewrite Equation (2) as

$${({y}^{2}+p+z)}^{2}={(sy+t)}^{2}$$ |

for some (complicated!) values $s$ and $t$, and then taking the square root of both sides and solving the resulting quadratic equation in $y$ provides a root of Equation (1).

Title | Ferrari-Cardano derivation of the quartic formula |
---|---|

Canonical name | FerrariCardanoDerivationOfTheQuarticFormula |

Date of creation | 2013-03-22 12:37:21 |

Last modified on | 2013-03-22 12:37:21 |

Owner | djao (24) |

Last modified by | djao (24) |

Numerical id | 8 |

Author | djao (24) |

Entry type | Proof |

Classification | msc 12D10 |

Related topic | CardanosDerivationOfTheCubicFormula |