A cubic equation is a Polynomial equation of degree three. Given a general cubic equation

(1) |

(2) |

(3) | |

(4) | |

(5) |

(6) |

(7) | |||

(8) | |||

(9) |

so equation (1) becomes

(10) |

(11) |

(12) |

(13) | |||

(14) |

then allows (12) to be written in the standard form

(15) |

(16) |

(17) |

(18) |

(19) |

where and are sometimes more useful to deal with than are and . There are therefore six solutions for (two corresponding to each sign for each Root of ). Plugging back in to (17) gives three pairs of solutions, but each pair is equal, so there are three solutions to the cubic equation.

Equation (12) may also be explicitly factored by attempting to pull out a term of the form from the cubic
equation, leaving behind a quadratic equation which can then be factored using the Quadratic Formula. This process
is equivalent to making Vieta's substitution, but does a slightly better job of motivating Vieta's ``magic''
substitution, and also at producing the explicit formulas for the solutions. First, define the intermediate variables

(20) | |||

(21) |

(which are identical to and up to a constant factor). The general cubic equation (12) then becomes

(22) |

(23) |

(24) |

(25) |

(26) |

(27) |

(28) |

(29) |

(30) | |||

(31) |

Plugging and into the left side of (28) gives

(32) |

(33) |

(34) |

These can be simplified by defining

(35) | |||

(36) |

so that the solutions to the quadratic part can be written

(37) |

(38) | |||

(39) | |||

(40) |

where is the Discriminant (which is defined slightly differently, including the opposite Sign, by Birkhoff and Mac Lane 1965) then gives very simple expressions for and , namely

(41) | |||

(42) |

Therefore, at last, the Roots of the original equation in are then given by

(43) | |||

(44) | |||

(45) |

with the Coefficient of in the original equation, and and as defined above. These three equations giving the three Roots of the cubic equation are sometimes known as Cardano's Formula. Note that if the equation is in the standard form of Vieta

(46) |

(47) | |||

(48) | |||

(49) |

The equation for in Cardano's Formula does not have an appearing in it explicitly while and do,
but this does not say anything about the number of Real and Complex Roots
(since and are themselves, in general, Complex). However, determining which
Roots are Real and which are Complex can be accomplished by
noting that if the Discriminant , one Root is Real
and two are Complex Conjugates; if , all Roots are Real and at least two are equal; and if , all Roots are Real and unequal. If
, define

(50) |

(51) | |||

(52) | |||

(53) |

This procedure can be generalized to find the Real Roots for any equation in the standard form (46) by using the identity

(54) |

(55) |

(56) |

(57) |

(58) |

(59) |

(60) |

(61) |

(62) |

(63) |

(64) |

An alternate approach to solving the cubic equation is to use Lagrange Resolvents.
Let
, define

(65) | |||

(66) | |||

(67) |

where are the Roots of

(68) |

(69) |

(70) |

(71) |

(72) |

(73) | |||

(74) |

The solutions satisfy Newton's Identities

(75) | |||

(76) | |||

(77) |

In standard form, , , and , so we have the identities

(78) | |||

(79) | |||

(80) |

Some curious identities involving the roots of a cubic equation due to Ramanujan are given by Berndt (1994).

**References**

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

Berger, M. §16.4.1-16.4.11.1 in *Geometry I.* New York: Springer-Verlag, 1994.

Berndt, B. C. *Ramanujan's Notebooks, Part IV.* New York: Springer-Verlag, pp. 22-23, 1994.

Beyer, W. H. *CRC Standard Mathematical Tables, 28th ed.* Boca Raton, FL: CRC Press, pp. 9-11, 1987.

Birkhoff, G. and Mac Lane, S. *A Survey of Modern Algebra, 3rd ed.* New York: Macmillan, pp. 90-91,
106-107, and 414-417, 1965.

Dickson, L. E. ``A New Solution of the Cubic Equation.'' *Amer. Math. Monthly* **5**, 38-39, 1898.

Dickson, L. E. *Elementary Theory of Equations.* New York: Wiley, pp. 36-37, 1914.

Dunham, W. ``Cardano and the Solution of the Cubic.'' Ch. 6 in
*Journey Through Genius: The Great Theorems of Mathematics.*
New York: Wiley, pp. 133-154, 1990.

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

Jones, J. ``Omar Khayyám and a Geometric Solution of the Cubic.'' http://jwilson.coe.uga.edu/emt669/Student.Folders/Jones.June/omar/omarpaper.html.

Kennedy, E. C. ``A Note on the Roots of a Cubic.'' *Amer. Math. Monthly* **40**, 411-412, 1933.

King, R. B. *Beyond the Quartic Equation.* Boston, MA: Birkhäuser, 1996.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. ``Quadratic and Cubic Equations.'' §5.6 in
*Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed.* Cambridge, England:
Cambridge University Press, pp. 178-180, 1992.

Spanier, J. and Oldham, K. B. ``The Cubic Function and Higher Polynomials.''
Ch. 17 in *An Atlas of Functions.* Washington, DC: Hemisphere, pp. 131-147, 1987.

van der Waerden, B. L. §64 in *Algebra.* New York: Frederick Ungar, 1970.

© 1996-9

1999-05-25