Cardano’s formulae

The roots ( of the (for the reducing via a Tschirnhaus transformation, see the parent ( entry) cubic equation

y3+py+q=0, (1)

with p and q any complex numbersMathworldPlanetmathPlanetmath, are

y1=u+v,y2=uζ+vζ2,y3=uζ2+vζ, (2)

where  ζ is a primitive ( third root of unityMathworldPlanetmath (e.g. -1+i32) and

u:=-q2+(p3)3+(q2)23,v:=-q2-(p3)3+(q2)23. (3)

The values of the cube roots must be chosen such that

uv=-p3. (4)

Cardano’s formulae, essentially (2) and (3), were first published in 1545 in Geronimo Cardano’s book “Ars magna”.  The idea of (2) and (3) is illustrated in the entry example of solving a cubic equation.

Let’s now assume that the coefficients p and q are real.  The number of the real roots ( of (1) depends on the sign of the radicandR:=(p3)3+(q2)2  of the above square root.  Instead of R we may use the discriminantPlanetmathPlanetmathPlanetmathPlanetmathD:=-108R  of the equation.  As in examining the number of real roots of a quadratic equation (, we get three different cases also for the cubic (1):

  1. 1.

    D=0.  This is possible only when either  p<0  or  p=q=0.  Then we get the real roots  y1=-2q/23,  y2=y3=q/23.

  2. 2.

    D<0.  The square root R is real, and one can choose for u and v the real values of the cube roots (3); these satisfy (4).  Thus the root  y1=u+v  is real, and since

    y2, 3=-u+v2±i3u-v2,

    with  uv, the roots y2 and y3 are non-real complex conjugatesMathworldPlanetmath of each other.

  3. 3.

    D>0.  This requires that p is negative.  The radicands of the cube roots (3) are non-real complex conjugates.  Using the argument φ of  u3=-q2+i-R  as auxiliary angle one is able to take the cube roots (, obtaining the trigonometric

    y1= 2-p3cosφ3,y2= 2-p3cosφ+2π3,y3= 2-p3cosφ+4π3.

    This shows that the roots of (1) are three distinct real numbers.  O. L. Hölder has proved in the end of the 19th century that in this case one can not with algebraicMathworldPlanetmath means eliminate the imaginarity from the Cardano’s formulae (2), but “the real roots must be calculated via the non-real numbers”.  This fact has been known already much earlier and called the casus irreducibilisMathworldPlanetmath.  It actually coerced the mathematicians to begin to use non-real numbers, i.e. to introduce the complex numbers.


  • 1 K. Väisälä: Lukuteorian ja korkeamman algebran alkeet.  Tiedekirjasto No. 17.   Kustannusosakeyhtiö Otava, Helsinki (1950).
Title Cardano’s formulae
Canonical name CardanosFormulae
Date of creation 2014-11-27 15:52:49
Last modified on 2014-11-27 15:52:49
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 26
Author pahio (2872)
Entry type Topic
Classification msc 12D10
Synonym solution of cubic equation
Synonym Cardanic formulae
Related topic CubicFormula
Related topic ATrigonometricCubicFormula
Related topic Complex
Related topic GaloisGroupOfTheCubic
Related topic CasusIrreducibilis
Related topic QuadraticResolvent
Related topic SimpleAnalyticDiscussionOfTheCubicEquation
Related topic GoniometricSolutionOfCubicEquation