View source for Cardano formula

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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====
<table><TR><TD valign="top">[1]</TD>
 <TD valign="top"> A.G. Kurosh, "Higher algebra" , MIR (1972) (Translated from Russian)</TD>
</TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD>
 <TD valign="top"> B.L. van der Waerden, "Algebra" , '''1–2''' , Springer (1967–1971) (Translated from German)</TD>
</TR></table>