Editing Cubic equation (section)

===Discriminant===
The [[discriminant]] of a [[polynomial]] is a function of its coefficients that is zero if and only if the polynomial has a [[multiple root]], or, if it is divisible by the square of a non-constant polynomial. In other words, the discriminant is nonzero if and only if the polynomial is [[square-free polynomial|square-free]].

If {{math|''r''{{sub|1}}, ''r''{{sub|2}}, ''r''{{sub|3}}}} are the three [[root of a function|roots]] (not necessarily distinct nor [[real number|real]]) of the cubic <math>ax^3+bx^2+cx+d,</math> then the discriminant is
<math display="block">a^4(r_1-r_2)^2(r_1-r_3)^2(r_2-r_3)^2.</math>

The discriminant of the depressed cubic <math>t^3+pt+q</math> is 
<math display="block">-\left(4\,p^3+27\,q^2\right).</math>

The discriminant of the general cubic <math>ax^3+bx^2+cx+d</math> is 
<math display="block">18\,abcd - 4\,b^3d + b^2c^2 - 4\,ac^3 - 27\,a^2d^2.</math>
It is the product of <math>a^4</math> and the discriminant of the corresponding depressed cubic. Using the formula relating the general cubic and the associated depressed cubic, this implies that the discriminant of the general cubic can be written as 
<math display="block">\frac{4(b^2-3ac)^3-(2b^3-9abc +27 a^2d)^2}{27a^2}.</math>

It follows that one of these two discriminants is zero if and only if the other is also zero, and, if the coefficients are [[real number|real]], the two discriminants have the same sign. In summary, the same information can be deduced from either one of these two discriminants.

To prove the preceding formulas, one can use [[Vieta's formulas]] to express everything as polynomials in {{math|''r''{{sub|1}}, ''r''{{sub|2}}, ''r''{{sub|3}}}}, and {{mvar|a}}. The proof then results in the verification of the equality of two polynomials.