A general quartic equation (also called a Biquadratic Equation) is a fourth-order Polynomial of the form

(1) |

(2) | |

(3) | |

(4) | |

(5) |

Ferrari was the first to develop an algebraic technique for solving the general quartic.
He applied his technique (which was stolen and published by Cardano) to the equation

(6) |

The term can be eliminated from the general quartic (1) by making a substitution
of the form

(7) |

(8) |

(9) |

(10) |

(11) | |||

(12) | |||

(13) |

Adding and subtracting to (10) gives

(14) |

(15) |

(16) |

(17) |

(18) |

(19) | |||

(20) | |||

(21) |

Let be a Real Root of the resolvent Cubic Equation

(22) |

(23) |

(24) | |||

(25) | |||

(26) | |||

(27) |

where

(28) | |||

(29) | |||

(30) |

Another approach to solving the quartic (10) defines

(31) | |||

(32) | |||

(33) |

where use has been made of

(34) |

(35) | |||

(36) |

Comparing with

(37) | |||

(38) | |||

(39) |

gives

(40) |

**References**

Abramowitz, M. and Stegun, C. A. (Eds.).
*Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.*
New York: Dover, pp. 17-18, 1972.

Berger, M. §16.4.1-16.4.11.1 in *Geometry I.* New York: Springer-Verlag, 1987.

Beyer, W. H. *CRC Standard Mathematical Tables, 28th ed.* Boca Raton, FL: CRC Press, p. 12, 1987.

Birkhoff, G. and Mac Lane, S. *A Survey of Modern Algebra, 3rd ed.* New York: Macmillan, pp. 107-108, 1965.

Ehrlich, G. §4.16 in *Fundamental Concepts of Abstract Algebra.* Boston, MA: PWS-Kent, 1991.

Faucette, W. M. ``A Geometric Interpretation of the Solution of the General Quartic Polynomial.''
*Amer. Math. Monthly* **103**, 51-57, 1996.

Smith, D. E. *A Source Book in Mathematics.* New York: Dover, 1994.

van der Waerden, B. L. §64 in *Algebra, Vol. 1.* New York: Springer-Verlag, 1993.

© 1996-9

1999-05-25