Editing Discriminant (section)

===Degree 4===
[[File:Quartic Discriminant.png|thumb|The discriminant of the quartic polynomial {{math|''x''<sup>4</sup> + ''cx''<sup>2</sup> + ''dx'' + ''e''}}. The surface represents points ({{math|''c'', ''d'', ''e''}}) where the polynomial has a repeated root. The cuspidal edge corresponds to the polynomials with a triple root, and the self-intersection corresponds to the polynomials with two different repeated roots.]]
The [[quartic polynomial]]
<math> ax^4+bx^3+cx^2+dx+e\,</math>
has discriminant
:<math>\begin{align}
{} & 256a^3e^3-192a^2bde^2-128a^2c^2e^2+144a^2cd^2e \\[4pt]
& {} -27a^2d^4+144ab^2ce^2-6ab^2d^2e-80abc^2de \\[4pt]
& {} +18abcd^3+16ac^4e-4ac^3d^2-27b^4e^2+18b^3cde \\[4pt]
& {} -4b^3d^3-4b^2c^3e+b^2c^2d^2\,.
\end{align}</math>

The depressed quartic polynomial
<math> x^4+cx^2+dx+e\,</math>
has discriminant
:<math>\begin{align}
{} & 16c^4e -4c^3d^2 -128c^2e^2+144cd^2e -27d^4 + 256e^3\,.
\end{align}</math>

The discriminant is zero if and only if at least two roots are equal. If the coefficients are real numbers and the discriminant is negative, then there are two real roots and two complex conjugate roots. Conversely, if the discriminant is positive, then the roots are either all real or all non-real.