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

(1)

(2)

(3)

(4)

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 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, we get three different cases also for the cubic (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. $D < 0$ . 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. $D > 0$ . 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, obtaining the trigonometric presentation $$y_1 \,=\, 2\sqrt{-\frac{p}{3}}\cos\frac{\varphi}{3},\quad y_2 \,=\, 2\sqrt{-\frac{p}{3}}\cos\frac{\varphi+2\pi}{3},\quad y_3 \,=\, 2\sqrt{-\frac{p}{3}}\cos\frac{\varphi+4\pi}{3}.$$ 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.

Bibliography

1

K. V¨AISÄLÄ: Lukuteorian ja korkeamman algebran alkeet. Tiedekirjasto No. 17. Kustannusosakeyhtiö Otava, Helsinki (1950).