# Cardano’s formulae

The roots (http://planetmath.org/Equation) of the (for the reducing via a Tschirnhaus transformation, see the parent (http://planetmath.org/CardanosDerivationOfTheCubicFormula) entry) cubic equation

 $\displaystyle y^{3}+py+q=0,$ (1)

with $p$ and $q$ any complex numbers, are

 $\displaystyle y_{1}=u+v,\qquad y_{2}=u\zeta+v\zeta^{2},\qquad y_{3}=u\zeta^{2}% +v\zeta,$ (2)

where  $\zeta$ is a primitive (http://planetmath.org/RootOfUnity) third root of unity (e.g. $\frac{-1+i\sqrt{3}}{2}$) and

 $\displaystyle u\,:=\,\sqrt[3]{-\frac{q}{2}+\sqrt{\left(\frac{p}{3}\right)^{3}+% \left(\frac{q}{2}\right)^{2}}},\qquad v\,:=\,\sqrt[3]{-\frac{q}{2}-\sqrt{\left% (\frac{p}{3}\right)^{3}+\left(\frac{q}{2}\right)^{2}}}.$ (3)

The values of the cube roots must be chosen such that

 $\displaystyle uv=-\frac{p}{3}.$ (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 (http://planetmath.org/Equation) of (1) depends on the sign of the radicand$\displaystyle R:=\left(\frac{p}{3}\right)^{3}\!+\!\left(\frac{q}{2}\right)^{2}$  of the above square root.  Instead of $R$ we may use the discriminant$D:=-108R$  of the equation.  As in examining the number of real roots of a quadratic equation (http://planetmath.org/QuadraticFormula), 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  $y_{1}=-2\sqrt[3]{q/2}$,  $y_{2}=y_{3}=\sqrt[3]{q/2}$.

2. 2.

$D<0$.  The square root $\sqrt{R}$ is real, and one can choose for $u$ and $v$ the real values of the cube roots (3); these satisfy (4).  Thus the root  $y_{1}=u+v$  is real, and since

 $y_{2,\,3}=-\frac{u+v}{2}\pm i\sqrt{3}\cdot\!\frac{u-v}{2},$

with  $u\neq v$, the roots $y_{2}$ and $y_{3}$ are non-real complex conjugates 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 $\varphi$ of  $u^{3}=-\frac{q}{2}+i\sqrt{-R}$  as auxiliary angle one is able to take the cube roots (http://planetmath.org/CalculatingTheNthRootsOfAComplexNumber), obtaining the trigonometric

 $y_{1}\,=\,2\sqrt{-\frac{p}{3}}\cos\frac{\varphi}{3},\qquad y_{2}\,=\,2\sqrt{-% \frac{p}{3}}\cos\frac{\varphi+2\pi}{3},\qquad y_{3}\,=\,2\sqrt{-\frac{p}{3}}% \cos\frac{\varphi+4\pi}{3}.$

This shows that the roots of (1) are three distinct real numbers.  O. L. Hölder has proved in the end of the $19^{\mathrm{th}}$ century that in this case one can not with algebraic 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 .  It actually coerced the mathematicians to begin to use non-real numbers, i.e. to introduce the complex numbers.

## References

• 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