Editing Quartic function (section)

===Nature of the roots===
Given the general quartic equation
:<math>ax^4 + bx^3 + cx^2 + dx + e = 0</math>
with real coefficients and {{math|''a'' ≠ 0}} the nature of its roots is mainly determined by the sign of its [[discriminant]] 
:<math>\begin{align} 
\Delta = {} &256 a^3 e^3 - 192 a^2 b d e^2 - 128 a^2 c^2 e^2 + 144 a^2 c d^2 e - 27 a^2 d^4 \\ 
&+ 144 a b^2 c e^2 - 6 a b^2 d^2 e - 80 a b c^2 d e + 18 a b c d^3 + 16 a c^4 e \\
&- 4 a c^3 d^2 - 27 b^4 e^2 + 18 b^3 c d e - 4 b^3 d^3 - 4 b^2 c^3 e + b^2 c^2 d^2
\end{align} </math> 
This may be refined by considering the signs of four other polynomials:
:<math>P = 8ac - 3b^2</math>
such that {{math|{{sfrac|''P''|8''a''<sup>2</sup>}}}} is the second degree coefficient of the associated depressed quartic (see [[#Converting_to_a_depressed_quartic|below]]);
:<math>R= b^3+8da^2-4abc,</math>
such that {{math|{{sfrac|''R''|8''a''<sup>3</sup>}}}} is the first degree coefficient of the associated depressed quartic; 
:<math>\Delta_0 = c^2 - 3bd + 12ae,</math>
which is 0 if the quartic has a triple root; and
:<math>D = 64 a^3 e - 16 a^2 c^2 + 16 a b^2 c - 16 a^2 bd - 3 b^4</math>
which is 0 if the quartic has two double roots.

The possible cases for the nature of the roots are as follows:<ref>{{cite journal|first= E. L.|last=Rees|title=Graphical Discussion of the Roots of a Quartic Equation|journal = The American Mathematical Monthly|volume=29|issue=2|year=1922|pages=51–55|doi=10.2307/2972804|jstor = 2972804}}</ref>

* If {{math|∆ < 0}} then the equation has two distinct real roots and two [[complex conjugate]] non-real roots.
* If {{math|∆ > 0}} then either the equation's four roots are all real or none is.
** If {{mvar|P}} < 0 and {{mvar|D}} < 0 then all four roots are real and distinct.
** If {{mvar|P}} > 0 or {{mvar|D}} > 0 then there are two pairs of non-real complex conjugate roots.<ref>{{Cite journal | last1 = Lazard | first1 = D. | doi = 10.1016/S0747-7171(88)80015-4 | title = Quantifier elimination: Optimal solution for two classical examples | journal = Journal of Symbolic Computation | volume = 5 | pages = 261–266 | year = 1988 | issue = 1–2 | doi-access = free }}</ref>
* If {{math|∆ {{=}} 0}} then (and only then) the polynomial has a [[multiplicity (mathematics)|multiple]] root. Here are the different cases that can occur:
** If {{mvar|P}} < 0 and {{mvar|D}} < 0 and {{math|∆<sub>0</sub> ≠ 0}}, there are a real double root and two real simple roots.
** If {{mvar|D}} > 0 or ({{mvar|P}} > 0 and ({{mvar|D}} ≠ 0 or {{mvar|R}} ≠ 0)), there are a real double root and two complex conjugate roots.
** If {{math|∆<sub>0</sub> {{=}} 0}} and {{mvar|D}} ≠ 0, there are a triple root and a simple root, all real.
** If {{mvar|D}} = 0, then:
***If {{mvar|P}} < 0, there are two real double roots.
***If {{mvar|P}} > 0 and {{mvar|R}} = 0, there are two complex conjugate double roots.
***If {{math|∆<sub>0</sub> {{=}} 0}}, all four roots are equal to {{math|−{{sfrac|''b''|4''a''}}}}

There are some cases that do not seem to be covered, but in fact they cannot occur.  For example, {{math|∆<sub>0</sub> > 0}}, {{mvar|P}} = 0 and {{mvar|D}} ≤ 0 is one of the cases.  In fact, if {{math|∆<sub>0</sub> > 0}} and {{mvar|P}} = 0 then  {{mvar|D}} > 0, since <math>16 a^2\Delta_0 = 3D + P^2; </math> so this combination is not possible.