Editing Cubic equation (section)

==General cubic formula==
A ''cubic formula'' for the roots of the general cubic equation (with {{math|''a'' ≠ 0}})
<math display="block">ax^3+bx^2+cx+d=0</math>
can be deduced from every variant of Cardano's formula by reduction to a [[#Depressed cubic|depressed cubic]]. The variant that is presented here is valid not only for complex coefficients, but also for coefficients {{math|''a'', ''b'', ''c'', ''d''}} belonging to any [[algebraically closed field]] of [[characteristic (algebra)|characteristic]] other than 2 or 3. If the coefficients are real numbers, the formula covers all complex solutions, not just real ones.

The formula being rather complicated, it is worth splitting it in smaller formulas.

Let
<math display="block">\begin{align}
\Delta_0 &= b^2 - 3ac,\\
\Delta_1 &= 2b^3 - 9abc + 27a^2d.
\end{align}</math>

(Both <math>\Delta_0</math> and <math>\Delta_1</math> can be expressed as [[resultant]]s of the cubic and its derivatives: <math>\Delta_1</math> is {{math|{{sfrac|−1|8''a''}}}} times the resultant of the cubic and its second derivative, and <math>\Delta_0</math> is {{math|{{sfrac|−1|12''a''}}}} times the resultant of the first and second derivatives of the cubic polynomial.)

Then let
<math display="block">C = \sqrt[3]{\frac{\Delta_1 \pm \sqrt{\Delta_1^2 - 4 \Delta_0^3}}2},</math>
where the symbols <math>\sqrt{{~}^{~}}</math> and <math>\sqrt[3]{{~}^{~}}</math> are interpreted as ''any'' square root and ''any'' cube root, respectively (every nonzero complex number has two square roots and three cubic roots). The sign "{{math|±}}" before the square root is either "{{math|+}}" or "{{math|–}}"; the choice is almost arbitrary, and changing it amounts to choosing a different square root. However, if a choice yields {{math|''C'' {{=}} 0}} (this occurs if <math>\Delta_0=0</math>), then the other sign must be selected instead. If both choices yield {{math|''C'' {{=}} 0}}, that is, if <math>\Delta_0=\Delta_1=0,</math> a fraction {{sfrac|0|0}} occurs in following formulas; this fraction must be interpreted as equal to zero (see the end of this section).
With these conventions, one of the roots is
<math display="block">x = - \frac{1}{3a}\left(b+C+\frac{\Delta_0}{C}\right).</math>

The other two roots can be obtained by changing the choice of the cube root in the definition of {{mvar|C}}, or, equivalently by multiplying {{mvar|C}} by a [[primitive root of unity|primitive cube root of unity]], that is {{math|{{sfrac|–1 ± {{sqrt|–3}}|2}}}}. In other words, the three roots are 
<math display="block">x_k = - \frac{1}{3a}\left(b+\xi^kC+\frac{\Delta_0}{\xi^kC}\right), \qquad k \in \{0,1,2\} \text{,}</math>
where {{math|''ξ'' {{=}} {{sfrac|–1 + {{sqrt|–3}}|2}}}}.

As for the special case of a depressed cubic, this formula applies but is useless when the roots can be expressed without cube roots. In particular, if <math>\Delta_0=\Delta_1=0,</math> the formula gives that the three roots equal <math>\frac {-b}{3a},</math> which means that the cubic polynomial can be factored as <math>\textstyle a(x+\frac b{3a})^3.</math> A straightforward computation allows verifying that the existence of this factorization is equivalent with <math>\Delta_0=\Delta_1=0.</math>