PlanetMath: Cardano's formulae

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Cardano's formulae

(Topic)

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

$\displaystyle y^3+py+q = 0,$

(1)

with

and

any complex numbers, are

$\displaystyle y_1 = u+v, \quad y_2 = u\zeta+v\zeta^2, \quad y_3 = u\zeta^2+v\zeta,$

(2)

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

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

(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 and are real. The number of the real roots 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 we may use the discriminant 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):

. This is possible only when either or . Then we get the real roots $y_1 = -2\sqrt[3]{q/2}$ , $y_2 = y_3 = \sqrt[3]{q/2}$ .
. The square root $\sqrt{R}$ is real, and one can choose for and the real values of the cube roots (3); these satisfy (4). Thus the root is real, and since
$\displaystyle y_{2,\,3} = -\frac{u+v}{2}\pm i\sqrt{3}\cdot\!\frac{u-v}{2},$
with $u \neq v$ , the roots and are non-real complex conjugates of each other.
. This requires that 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, obtaining the trigonometric presentation
$\displaystyle y_1 \,=\, 2\sqrt{-\frac{p}{3}}\cos\frac{\varphi}{3},\quad y_2 \,=... ...varphi+2\pi}{3},\quad y_3 \,=\, 2\sqrt{-\frac{p}{3}}\cos\frac{\varphi+4\pi}{3}.$
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 casus irreducibilis. It actually coerced the mathematicians to begin to use non-real numbers, i.e. to introduce the complex numbers.