Editing Cubic equation (section)

==Cardano's formula==
[[Gerolamo Cardano]] is credited with publishing the first formula for solving cubic equations, attributing it to [[Scipione del Ferro]] and [[Niccolo Fontana Tartaglia]]. The formula applies to depressed cubics, but, as shown in {{slink||Depressed cubic}}, it allows solving all cubic equations.

Cardano's result is that if 
<math display="block">t^3+pt+q=0</math>
is a cubic equation such that {{mvar|p}} and {{mvar|q}} are [[real number]]s such that <math> \frac{q^2}4+\frac{p^3}{27}</math> is positive (this implies that the [[discriminant]] of the equation is negative) then the equation has the real root
<math display="block">\sqrt[3]{u_1}+\sqrt[3]{u_2},</math>
where <math>u_1</math> and <math>u_2</math> are the two numbers <math>-\frac q2 + \sqrt{\frac{q^2}4+\frac{p^3}{27}}</math> and <math>-\frac q2 - \sqrt{\frac{q^2}4+\frac{p^3}{27}}.</math>

See {{slink||Derivation of the roots}}, below, for several methods for getting this result.

As shown in {{slink||Nature of the roots}}, the two other roots are non-real [[complex conjugate]] numbers, in this case. It was later shown (Cardano did not know [[complex number]]s) that the two other roots are obtained by multiplying one of the cube roots by the [[primitive root of unity|primitive cube root of unity]] <math>\varepsilon_1=\frac{-1+i\sqrt 3} 2,</math> and the other cube root by the other primitive cube root of the unity <math>\varepsilon_2=\varepsilon_1^2=\frac{-1-i\sqrt 3} 2.</math> That is, the other roots of the equation are <math> \varepsilon_1\sqrt[3]{u_1}+\varepsilon_2 \sqrt[3]{u_2}</math> and <math>\varepsilon_2\sqrt[3]{u_1}+\varepsilon_1 \sqrt[3]{u_2}.</math><ref>{{Cite web|title=Solution for a depressed cubic equation|url=https://uniteasy.com/post/1287/|access-date=2022-11-23|website=Math solution|language=en}}</ref> 

If <math>4p^3+27q^2 < 0,</math> there are three real roots, but [[Galois theory]] allows proving that, if there is no rational root, the roots cannot be expressed by an [[algebraic expression]] involving only real numbers. Therefore, the equation cannot be solved in this case with the knowledge of Cardano's time. This case has thus been called ''[[casus irreducibilis]]'', meaning ''irreducible case'' in Latin.

In ''casus irreducibilis'', Cardano's formula can still be used, but some care is needed in the use of cube roots. A first method is to define the symbols <math>\sqrt{{~}^{~}}</math> and <math>\sqrt[3]{{~}^{~}}</math> as representing the [[principal value]]s of the root function (that is the root that has the largest real part). With this convention Cardano's formula for the three roots remains valid, but is not purely algebraic, as the definition of a principal part is not purely algebraic, since it involves inequalities for comparing real parts. Also, the use of principal cube root may give a wrong result if the coefficients are non-real complex numbers. Moreover, if the coefficients belong to another [[field (mathematics)|field]], the principal cube root is not defined in general.

The second way for making Cardano's formula always correct, is to remark that the product of the two cube roots must be {{math|−''p'' / 3}}. It results that a root of the equation is 
<math display="block">C-\frac p{3C}\quad\text{with}\quad C=\sqrt[3]{-\frac q2+\sqrt{\frac{q^2}4+\frac{p^3}{27}}}. </math>
In this formula, the symbols <math>\sqrt{{~}^{~}}</math> and <math>\sqrt[3]{{~}^{~}}</math> denote any square root and any cube root. The other roots of the equation are obtained either by changing of cube root or, equivalently, by multiplying the cube root by a primitive cube root of unity, that is <math>\textstyle \frac{-1\pm \sqrt{-3}}2.</math>

This formula for the roots is always correct except when {{math|1=''p'' = ''q'' = 0}}, with the proviso that if {{math|1=''p'' = 0}}, the square root is chosen so that {{math|''C'' ≠ 0}}. However, Cardano's formula is useless if <math>p=0,</math> as the roots are the cube roots of <math>-q.</math> Similarly, the formula is also useless in the cases where no cube root is needed, that is when the cubic polynomial is not [[irreducible polynomial|irreducible]]; this includes the case <math>4p^3+27q^2=0.</math>

This formula is also correct when {{mvar|p}} and {{mvar|q}} belong to any [[field (mathematics)|field]] of [[characteristic (algebra)|characteristic]] other than 2 or 3.