Cardano formula

A formula for finding the roots of the general cubic equation over the field of complex numbers $$x^3 + px + q = 0.\label{1}$$ Any cubic equation can be reduced to the above form. The Cardano formula for the roots of (1) has the form:

$\def\radix#1#2{{}^{{}^{\scriptstyle{#1}}}\kern-6pt\sqrt{#2}}$ $$x= \radix{3}{-\frac{q}{2}+\sqrt{\frac{q^2}{4}+\frac{p^3}{27}}} + \radix{3}{-\frac{q}{2}-\sqrt{\frac{q^2}{4}+\frac{p^3}{27}}}.$$

In this formula one must choose, for each of the three values of the cube root $$\alpha = \radix{3}{-\frac{q}{2}+\sqrt{\frac{q^2}{4}+\frac{p^3}{27}}},$$ that value of the cube root $$\beta = \radix{3}{-\frac{q}{2}-\sqrt{\frac{q^2}{4}+\frac{p^3}{27}}}$$ for which the relation $\alpha\beta = -p/3$ holds (such a value of $\beta$ always exists). In the Cardano formula, $p$ and $q$ are arbitrary complex numbers. In the case of real coefficients $p$ and $q$, the property of the roots being real or imaginary depends on the sign of the discriminant of the equation,

$$D= -27q^2 -4 p^3 = -108\Big(\frac{q^2}{4} + \frac{p^3}{27}\Big).$$ When $D>0$ all three roots are real and distinct. However, according to Cardano's formula, the roots are expressed in terms of cube roots of imaginary quantities. Although in this case both the coefficients and the roots are real, the roots cannot be expressed in terms of the coefficients by means of radicals of real numbers; for this reason, the above case is called irreducible. When $D=0$, all roots are real; when $p$ and $q$ are both non-zero, there is one double and one single root; and when $p$ and $q$ are both zero, there is one triple root. When $D<0$, all three roots are distinct, one of them being a real number and the other two — conjugate complex numbers.

The Cardano formula is named after G. Cardano, who was the first to publish it in 1545.

References

[1]	A.G. Kurosh, "Higher algebra" , MIR (1972) (Translated from Russian)

References

[a1]	B.L. van der Waerden, "Algebra" , 1–2 , Springer (1967–1971) (Translated from German)