Talk:Quartic function

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Untitled[edit]

In the section "Converting to a depressed quartic", it mentions "expanding" an expression; this ought to be changed, as expansion is an entirely different operation, far surpassing even tetration in size... Timeroot (talk)


It's writen "If β ≠ 0 then α + 2y ≠ 0" I can't see why this sentence is right. Thank you —Preceding unsigned comment added by Avavrin (talkcontribs) 14:13, 28 January 2010 (UTC)

History of the Quartic Formula[edit]

A previous version of the page included the claim that the Spanish Inquisitor Torquemada had a Spanish mathematician, Paolo Valmes, killed claiming to have solved the quartic equation. The previous wiki author cited only one original source, Beckmann's "History of Pi"; the other source cited, http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2329002/pdf/iowaorthj00021-0206.pdf, is clearly a comic article satirizing medical coding procedures; it cites Beckmann as its own source. Peter Zoll recounts, however (American Mathematical Monthly, v. 96, pp. 709-710), that Beckmann only got the story from a Russian textbook Tales about Mathematics by one Ivan I. Depman. Zoll's inquiries to Depman's publisher and to historians of the Spanish Inquisition found no documentation confirming this story. In short, this is part of the Black Legend, an undocumented tale that feeds both on anti-Catholic prejudice and on the supposed conflict between faith and reason. (This animus, incidentally, informs Beckmann's writing.) Without further documentation, this spurious tale ought not be included in Wikipedia. —Preceding unsigned comment added by 138.129.125.121 (talk) 13:54, 3 September 2010 (UTC)

Even if it is a spurious tale, it should be mentioned as such in Wikipedia. It is clearly notable (taking into account the aforementioned publications) and would prevent adding it as a fact by other editors. Maxal (talk) 15:26, 1 November 2010 (UTC)

Please! What is this confusion all about?[edit]

We call this a quartic function in the article, but the article of the higher grade is called quintic equation! Shouldn't we call them "equations" altogether, bringing a sort of red thread into the article series? -andy 77.7.100.60 (talk) 03:47, 16 October 2010 (UTC)

Actually Quintic equation redirects to Quintic function. I've changed the see-also link to the direct link. Now they're all called "... function." Duoduoduo (talk) 16:56, 3 August 2011 (UTC)

An error in Equation (8')?[edit]

It seems to me there is a sign error in Equation (8'): the two ± signs with an s subscript should not be similar! In the simplification operation, -(±) should turn it upside down. This is important, since the two signs are related. Could the author of this article, who did an excellent job anyway, have a look at this issue?Gemb47 (talk) 07:08, 14 March 2013 (UTC)

Simpler Formulas for Quartic Roots[edit]

The general quartic equation

x^4 + ax^3 + bx^2 + cx + d = 0

has the roots

x_{1,2} = -\frac{a}{4} \pm \frac{\sqrt{u\ +\ v}}{2} - \frac{\sqrt{2u\ -\ v\ \mp\ w}}{2}
x_{3,4} = -\frac{a}{4} \pm \frac{\sqrt{u\ +\ v}}{2} + \frac{\sqrt{2u\ -\ v\ \mp\ w}}{2}

where

u = \frac{a^2}{4} - \frac{2}{3}b . . . and . . . v = \frac{t}{3} + \frac{\Delta_0}{3t} . . . and . . . w = \frac{a^3 - 4ab + 8c}{4\sqrt{u\ +\ v}}

and

t = \sqrt[3]{\frac{\Delta + \sqrt{\Delta^2 - 4\Delta_0^3}}{2}} . . . and . . . \Delta_0 = b^2 - 3ac + 12d
\Delta = 2b^3 - 9abc + 27c^2 + 27a^2 d - 72bd79.113.213.130 (talk) 22:11, 16 March 2013 (UTC)
Yes, that is perfect. Unless someone objects (—see also Talk:Cubic function#Simpler Formulas for Cubic Roots—), feel free to add it to the article. Make sure you mention the talk page in your edit summary. - DVdm (talk) 10:38, 23 April 2013 (UTC)
I already went ahead. See [1]. - DVdm (talk) 11:29, 23 April 2013 (UTC)
I think there needs to be a bit more information on the nature of the roots. What does the discriminant say about the roots? Pokajanje|Talk 18:42, 4 May 2013 (UTC)
Let's find sources. - DVdm (talk) 20:41, 7 May 2013 (UTC)
The three most dreaded words in the English language ! :-) — 79.113.240.179 (talk) 23:31, 8 May 2013 (UTC)
I found this PDF as the first result for Googling "quartic formula". On page 4 it describes reduction to a cubic whose discriminant can then determine the nature of the roots. But I'm not sure that will work with the current general formula. Pokajanje|Talk 01:49, 9 May 2013 (UTC)
Here's another one, see page 21 (Ferrari) and 22 (Lagrange). — 79.113.240.179 (talk) 02:00, 9 May 2013 (UTC)
Are we searching for the same thing? I'm not seeing anything about the discriminants on that page. Would it count as original research if something is self-evident and correct? Regarding that, I tested your discriminant formula with the coefficients for the simple equation (x + 1)4 and got 1460, while Wolfram Alpha says it's 0.
Keep in mind that there are several possible cases for the nature of the roots — I think I read somewhere about nine. Pokajanje|Talk 04:48, 9 May 2013 (UTC)
I honestly doubt that you tested my discriminant, since you seem to think that this is \Delta instead \Delta^2 - 4 \Delta_0^3 \over m \cdot a^n , where m and n are some integers whose values I don't remember (n is definitely either 3 or 4, and m might be negative, but I'm not sure). (The article has \Delta_1 instead of \Delta for precisely this reason, but it seems that the notation I used on this talk page has thrown you off. The article itself doesn't give any formula for any actual discriminant, in case you haven't noticed yet). Also, the formulas in question have been verified with Mathematica. Try calculating it again. — 79.113.240.179 (talk) 08:39, 9 May 2013 (UTC)
The PDF was for DVdm. It contains two historic formulas for solving the quartic, which aren't mentioned at all in this encyclopedic article, but probably should. (The article does contain however another approach, belonging to the same Ferrari, but using a depressed quartic instead of a general one). — 79.113.240.179 (talk) 08:44, 9 May 2013 (UTC)
By the way, for (x+1)4=0 the above Δ (aka the article's Δ1) is indeed 0, not 1460. But so are u, Δ0 and t, resulting in both v and w being undefined.
That Zhao thesis looks like a good source, but writing out —and testing!— the entire scheme of its Theorem 4.1.1 on pages 60–61, is a bit of a burden, since it needs that r (the number of distinct real roots), which in turn needs the discriminant sign list and the reversed sign list (see page 22) to be construced. Of course, the testing should not be our job. Is that thesis cited/used somewhere else? After, we shouldn't use wp:primary sources, should we? ;-)
Who said it would be simple? - DVdm (talk) 09:57, 9 May 2013 (UTC)
Quite ! :-) — 79.113.240.179 (talk) 10:12, 9 May 2013 (UTC)
The thesis in question cites Ferrari and Lagrange... Otherwise I wouldn't even have bothered bringing it up in the first place. Are you saying we should find the original works of these two, and quote them also ? :-\ — 79.113.240.179 (talk) 10:08, 9 May 2013 (UTC)
No... I was kinda joking. It looks excellent to me, but quite a job taking it onboard here.

But in the mean while I think it would be a good idea to put a little proviso in the text: in stead of writing "The roots in terms of these 4 coefficients are given by...", perhaps we should write "Unless the value of t is zero, the roots in terms of these 4 coefficients are given by...". - DVdm (talk) 10:23, 9 May 2013 (UTC)

I have a large screen, which has no problems with porperly displaying the picture to the right... It fits rather very nicely onto the screen... But when you reduce the size of the browser and make it about 1,000 pixels in width (as in the standard 1024x768 resolution, which many of our readers might still have), it looks rather ugly, intruding over the writing. Also, the formula with Q1 + Q2 might present problems of its own, so I'm not sure whether it's good to delete the old one with Q2 and Delta0. — 79.113.240.179 (talk) 11:50, 9 May 2013 (UTC)
Ok, centering is fine as well.

We can re-add the old with the proviso "unless Q=0", and also add a little remark for w like "(or w=0 if a^3-4ab+8c=0 and u+v=0)". - DVdm (talk) 12:00, 9 May 2013 (UTC)

Either centering, or we move it to the end of the sub-section... Also, I have NO idea what happens to w when u + v = 0. Are you absolutely positively 100% sure it's 0 in that case, and not something else ? (Could you also explain why ?) :-( — 79.113.240.179 (talk) 12:05, 9 May 2013 (UTC)
By looking at the expressions of the roots, it is clear that w must be zero for (x+whatever)4=0. But that's the only infinity of pathologiocal cases I have checked. So, I'm just absolutely positively 99% sure it's zero :-) - DVdm (talk) 12:22, 9 May 2013 (UTC)
Maybe that case of a quadruple root is not compatible with u + v = 0 ? :-\ — 79.113.240.179 (talk) 14:26, 9 May 2013 (UTC)

──────────────────────────────────────────────────────────────────────────────────────────────────── Despite the rewrite it would require, I think that we should change the general formula to Wolfram Alpha's method. This method does not require dividing the whole equation by a4 (after all, suppose a is a very large number and the other coefficients much smaller; calculation would be difficult), and it gives us at least one highly reliable source. Pokajanje|Talk 14:59, 26 May 2013 (UTC)

I disagree: Mathematica or Maple are reliable sources to verify that a formula is correct. They are reliable source for a formula only if, when substituting the value of the roots in the equation, the result may be simplified to zero. Otherwise their formula are simply the transcript of something published elsewhere and the transcription may have introduced errors. Moreover, these programs are unable to a clever grouping of the terms in order to have a formula that is as simple as possible. Without such a human simplification, the formula is unreadable and thus not useful for the reader (unless it is a computer :-)). On the other hand, if you have a verification by a program of the formula this may be a reliable source for the exactness of the formula (this is a personal opinion, I am not sure that there is a consensus about it). D.Lazard (talk) 15:37, 26 May 2013 (UTC)
(Have a paragraph break. I'm buying.)
Their formula is actually very similar to the one we already have, with the useful exception of not having to divide by a. Part of the simplification you request is:
ax^4 + bx^3 + cx^2 + dx + e = 0, a \neq 0
\Delta_0 = c^2 - 3bd + 12ae, \Delta_1 = 2c^3 - 9bcd + 27b^2e + 27ad^2 - 72ace
The discriminant remains the same, -\frac{\Delta_1^2 - 4\Delta_0^3}{27}. The expanded form can be seen in discriminant.
Q = \sqrt[3]{\Delta_1 + \sqrt{\Delta_1^2 - 4\Delta_0^3}}
u = \frac{3b^2 - 8ac}{12a^2}, v = \frac{Q + \Delta_0\sqrt[3]{2}}{3aQ}, w = \frac{b^3 - 4abc + 8a^2d}{4\sqrt{u+v}} (Not sure about v)
I can't make more progress until I can understand the point of 3a\sqrt[3]{2}. Pokajanje|Talk 18:17, 26 May 2013 (UTC)
No offense guys, but where do you think that the PlanetMath site (which is used as a source for this article's formula) got it from in the first place ? :-) It's exactly what Mathematica and other mathematical software display when asked to solve a general quartic equation. Unfortunately for us, they didn't use the general formula but the normalized one (with a = 1), and their site does qualify as a source under Wikipedia standards, whereas a computer software doesn't. — Yeah, twisted logic, I know, but I don't make the rules, I just follow them... :-) So unless you can't find at least one other "trustworthy" site, mentioning the exact same formula, our hands are tied. (Personally I like the current simplified formula better, because it has a nicer, less complicated expression for w then the more general one does, in case you haven't noticed). — 79.113.227.125 (talk) 08:41, 27 May 2013 (UTC)
Of course, I use the word "unfortunately" with a certain sense of humor: I personally haven't seen this formula being quoted or mentioned anywhere else, so we should be rather grateful we managed to find at least that one source, as "imperfect" as it might be, because otherwise we would have had nothing to offer that doesn't use either "depressed" quartics (I can only picture those sad and "depressed" variables pushing Prozac) or, even "better" (!), formulas that use parameters whose values are determined by solving an intermediary cubic equation... as if solving a quartic isn't enough, and we have to make hell warmer for ourselves :-) — 79.113.227.125 (talk) 08:56, 27 May 2013 (UTC)
Can you make head or tail of WA's confused notation there? Pokajanje|Talk 15:53, 27 May 2013 (UTC)
Studying the "Quartic formula written out in full" image from PlanetMath, it actually appears identical to that of Wolfram Alpha, save that it uses a monic polynomial. Indeed, I think it's a bit silly to have the article's main formula differ from the image. Pokajanje|Talk 20:13, 27 May 2013 (UTC)
I think I am very close to a simplified (i.e. non-monic) formula here. I tried using the above expressions using the methods of the existing formula and the equation 4x^4 - 20x^3 + 5x^2 + 50x + 25 (expanded form of (2x^2 - 5x - 5)^2). I got some promising results, including the correct discriminant of 0 and w equaling 0. The final result, however, was the incorrect \frac{10 \pm \sqrt{130} - \sqrt{130}}{8}. The correct answer is \frac{5 \pm \sqrt{65}}{4}, which is actually fairly close to my results.
The values I obtained were \Delta_0 = 4225, \Delta_1 = 549250, u = \frac{260}{3}, Q = 65, v = \frac{130}{3}. I would appreciate it if someone could check these calculations and/or figure out the way to obtain a non-monic formula. Pokajanje|Talk 20:46, 27 May 2013 (UTC)

──────────────────────────────────────────────────────────────────────────────────────────────────── The general quartic equation

ax^4 + bx^3 + cx^2 + dx + e = 0

has the roots

x_{1,2} = -\frac{b}{4a} \pm \frac{\sqrt{u\ +\ v}}{2} - \frac{\sqrt{2u\ -\ v\ \mp\ w}}{2}
x_{3,4} = -\frac{b}{4a} \pm \frac{\sqrt{u\ +\ v}}{2} + \frac{\sqrt{2u\ -\ v\ \mp\ w}}{2}

where

u = \frac{b^2}{4 a^2} - \frac{2c}{3a} . . . and . . . v = \frac{1}{3a}\left(t + \frac{\Delta_0}{t}\right) . . . and . . . w = \frac{b^3 - 4abc + 8 a^2 d}{4a^3\sqrt{u\ +\ v}}

and

t = \sqrt[3]{\frac{\Delta + \sqrt{\Delta^2 - 4\Delta_0^3}}{2}} . . . and . . . \Delta_0 = c^2 - 3bd + 12ae
\Delta = 2c^3 - 9bcd + 27 d^2 a + 27 b^2 e - 72ace79.113.211.75 (talk) 11:03, 28 May 2013 (UTC)
That's the ticket! I'll add that to the article straight away. (I'm not sure how to add references, though, and now we have both PlanetMath and WA on our side; could someone take care of this?) Pokajanje|Talk 15:10, 28 May 2013 (UTC)
79.x, you have edited the discriminant to be \frac{\Delta_1^2 - 4\Delta_0^3}{-27a^3}, but the expansion is then this incorrect expression. I don't think this is right. Pokajanje|Talk 16:53, 28 May 2013 (UTC)
Corrected! — 79.113.221.225 (talk) 17:21, 28 May 2013 (UTC)
Please do not invert the ± signs ! It may be "easier", as you say, but it's wrong ! The only way to make "plus go with plus" is to put a minus in front of both fractions and then change the signs OUTSIDE the radicals, NOT inside them ! — 79.113.221.225 (talk) 17:26, 28 May 2013 (UTC)

Complex numbers in the expression of real roots[edit]

I formed a simple quartic equation having roots 1, 2, 3 and 4. The resulting equation has coefficients: 1, -10, 35, -50, 24. The formula does not work because the value of t is derived as a complex number and this is because the difference of the two numbers inside the squareroot of the expression for t is negative. It should not happen as the four roots are clearly 1, 2, 3 and 4. — Preceding unsigned comment added by Jainsohan (talkcontribs) 09:53, 30 April 2013 (UTC)

The general formula for the roots, which is given, is intended to be valid for all equations, or, at least for almost all, because it may be not valid when two roots are equal (\Delta_1^2 - 4\Delta_0^3=0 ) or when \Delta_0=0 (I have not verified if it is true or not in these spacial cases where simpler formulas are available). As the formula is intended for a general case, it should not be a surprise that, in special cases, like integer roots, the formula may be replaced by a simpler formula, which may not be extended to the general case. However, it may be proven that if all roots are real and distinct (as in your example), then the discriminant \Delta_1^2 - 4\Delta_0^3 is always negative, and the formula uses non-real numbers to express the real roots. Unfortunately, it may also be proven that it does not exists any general formula for this case of four real roots, which expresses the roots in real terms. Thus the strange fact that you have remarked is unavoidable. D.Lazard (talk) 11:02, 30 April 2013 (UTC)
I think the general formula does exist, but it includes trigonometric functions. El Selenita (talk) 19:17, 27 July 2013 (UTC)
It's the same thing as with the cubic: several special cases require a particular discussion, especially these three: when t, and/or Delta, and/or u + v are 0. — 79.113.234.47 (talk) 00:49, 3 May 2013 (UTC)
Jainsohan, the imaginary parts will eventually cancel each other out. — 79.113.225.171 (talk) 12:03, 3 May 2013 (UTC)
While that's fine, it would be nice to know the simpler formula for when division by zero occurs. Pokajanje|Talk 17:55, 5 May 2013 (UTC)
In no case can a division by zero occur in the formula for the roots of the quartic equation. Lklundin (talk) 19:23, 5 May 2013 (UTC)
Suppose u + v = 0. Then w is undefined. Or perhaps if t or the discriminant (listed earlier on this talk page as 2b^3 - 9abc + 27c^2 + 27a^2 d - 72bd) is 0, then v is undefined. Pokajanje|Talk 19:22, 7 May 2013 (UTC)
u is the unknown of the depressed quartic and v is the unknown of the depressed cubic. One first solves for v and then that solution is used to solve for u. Supposing that u + v = 0 is non-sense. Go to the section "Summary of Ferrari's method" and you will see that in no case can a division by zero occur. There is therefore no place for a 'simpler formula for when division by zero occurs'. Lklundin (talk) 22:19, 7 May 2013 (UTC)
Are we talking about the same method? I'm referring to the one that an anonymous editor just added to the article. Pokajanje|Talk 21:23, 8 May 2013 (UTC)

Discriminant and "nature of the roots"[edit]

Several posts in the thread #Simpler Formulas for Quartic Roots mention the discriminant and the nature of the roots. This deserves a new thread. A first remark is that the article is lacking of a section "Nature of the roots]] to be placed before the general formula. Although this section should be similar in nature to that of cubic function, it must contain more material, I'll try to summarize the information that is lacking:

  • The discriminant of the quartic is -\frac{\Delta_1^2-4 \Delta_0^3}{27} (ommitted factor 4 added after the remark of user:Pokajanje, below (19:35, 9 May 2013 (UTC)) D.Lazard (talk) 17:52, 10 May 2013 (UTC))
  • If it is positive, the four roots are either all real or all non-real. If it is zero, there is a multiple root. If it is negative, there are two real roots and two non-real ones.
  • In the case of a positive discriminant, the nature of the roots may be decided by the sign of another polynomial (just now, I do not remember which one).
  • The variable v is the root of a cubic equation called cubic resolvent (in the case of the depressed quartic, this is the equation (4) of the article, up to the scaling of the roots by a constant factor to have integer coefficients)
  • The cubic resolvent has the same discriminant (up to a constant factor) as the quartic
  • The roots may be expressed without cubic roots if and only the cubic resolvent has a rational root, which occurs if and only if  \Delta_1=0 (the "only if" has to be checked, which is more of less equivalent to verify that  \Delta_1 is a resolvent for the dihedral group of four elements).

D.Lazard (talk) 11:51, 9 May 2013 (UTC)

We've modified the formula for v a bit in the article itself, so as to avoid the case where t (which is now called Q, from quartic) is 0. But maybe this modification presents its own problems ? What do you think ? — 79.113.240.179 (talk) 12:12, 9 May 2013 (UTC)
I've changed Q and \bar Q to Q1 and Q2. This is otherwise fantastic stuff, but let's find sources, of course (Also, 79.113.x, it's considered bad etiquette to edit others' posts. While making certain text bold isn't too bad, it would still be appreciated if you refrained from it.) Pokajanje|Talk 14:32, 9 May 2013 (UTC)
It's considered bad etiquette to edit others' posts — Yes, I know. I just didn't want to turn this into a blog-or-forum-like comment, where one copy-pastes the text once again, and then answers to it. Sort of like I did just now. :-) — 79.113.240.179 (talk) 14:54, 9 May 2013 (UTC)
I've changed Q and \bar Q to Q1 and Q2 — I don't know if that was such a good idea, since their cubic counterparts are marked as C and \bar C ... — 79.113.240.179 (talk) 14:54, 9 May 2013 (UTC)
Let's find sources — The formulas in question are from the Planet Math article, which is apparently an acceptable source per Wiki standards. If you meant sources for the discussion concerning Delta and the roots, I tried, but couldn't find any yet... — 79.113.240.179 (talk) 15:08, 9 May 2013 (UTC)

──────────────────────────────────────────────────────────────────────────────────────────────────── I am not happy with the present state of the formula. Firstly, like for the cubic case, \bar Q or Q2 should not appear (otherwise, there is the same problem as in the cubic case). Instead, we should have A/Q, where A is a rational function of the coefficients. Secondly, there are too many ± in the formula: this is not evident for most readers, which signs are to be chosen for each root and that the same has to be taken for the two occurrences of \sqrt{u+v}. To solve this, the best would be to not giving a name to v, and, instead, to give a name, say u' to \sqrt{u+v}. This would give the formula

x_{1,2} = -\frac{a}{4} + \frac{u'}{2} \pm \frac{\sqrt{2u - v + w}}{2}
x_{3,4} = -\frac{a}{4} - \frac{u'}{2} \pm \frac{\sqrt{2u - v - w}}{2}.

This makes it clear that there are only four roots. My third remark is deeper. The quantity u' is a quadratic function of the root. It is not clear which one has been chosen in this formula. The most usual choices are x_1x_3-x_2x_4 or (x_1+a/4)(x_3+a/4)-(x_2+a/4)(x_4+a/4). I am quite sure that one of these choices would give a formula which is not more complicated and on which the action of the dihedral group on the roots is more clear. D.Lazard (talk) 16:11, 9 May 2013 (UTC)

Like for the cubic case, \bar Q or Q2 should not appear (otherwise, there is the same problem as in the cubic case) — That's what I was thinking as well... But until someone doesn't find out what happens when Q = 0 (because Q' = Deltao / Q), I don't see what is to be done... It's as if the formulas are doomed either way... :-( Also, introducing yet another variable, u', seems too much... unless we replace 2u + v with say v' ? But then the formulas for u' and v' would be more "crowded" than those for u and v, I think... Also, unless I'm mistaken, the formulas for the roots would have to be:
x_{1,2} = -\frac{a}{4} {\color{red}+} \frac{u'}{2} \pm \frac{\sqrt{2u - v {\color{red}-} w}}{2}
x_{3,4} = -\frac{a}{4} {\color{red}-} \frac{u'}{2} \pm \frac{\sqrt{2u - v {\color{red}+} w}}{2} . — 79.113.240.179 (talk) 16:28, 9 May 2013 (UTC)
Your are probably correct, I have not verified my formula. This shows how unclear may be the use of several ∓ or ∓ before the same quantity. About the case Q=0, it may occur only when \Delta_0=0 (the case where no cubic root is needed to solve). I suggest: Let
Q = \sqrt[3]{\frac{\Delta_1 \pm \sqrt{\Delta_1^2 - 4\Delta_0^3}}{2}},
where ± stands for either + or -, except that, when \Delta_0=0 the sign must chosen for having Q≠0.
This solves the problem of the division by Q.
The case of u+v=0: It depends on the choice of the cubic resolvent (the cubic polynomial whose root is Q+\bar Q) and the expression of its root as a function of the roots of the quartic: the cubic resolvent has to be chosen in order that if u+v=0, then the quartic has a factor with rational coefficients. I do not remember the details, but this exists in the literature, I do not remember where.
D.Lazard (talk) 18:07, 9 May 2013 (UTC)
Done for Delta, done for roots, done for Q. :-) — 79.113.240.179 (talk) 18:30, 9 May 2013 (UTC)
There's a possible contradiction here with the article discriminant, which has a sourced statement saying that a quartic discriminant has 16 terms, and also mentions it later in the article using five instead of four coefficients. If you multiply everything out in the discriminant D.Lazard gave, you'll see it has 14 terms. Pokajanje|Talk 19:35, 9 May 2013 (UTC)
Thank you for the observation! I've added the word "general" to the article in question, and I've also made it clearer that the quartic used in the formula for roots section is `monic`, not `general`. — 79.113.208.159 (talk) 01:58, 10 May 2013 (UTC)
If "no radical is needed to represent the roots", how should they be represented? Pokajanje|Talk 20:34, 16 May 2013 (UTC)
If "no radical is needed to represent the roots", this means that the polynomial is a product of linear factors. More generally, if the quartic is not irreducible (that is if it is a product of two polynomials of lower degree with coefficients in the same field as the coefficients of the quartic) some care is needed, and, as polynomial factorization is not very difficult, it is common to discard this case when giving a formula. The general formula applies probably, but, as the reducible case is usually discarded, a verification is needed that the proof of the general formula applies also to this case. If there are four rational roots and if the general formulas is correct in this case, then it expresses the roots as the sum of two complex conjugate numbers, which is not really what is desired. D.Lazard (talk) 21:37, 16 May 2013 (UTC)
What I meant by that question is, in that case is there a way to represent the roots as they are represented in cubic function? Pokajanje|Talk 21:54, 16 May 2013 (UTC)
Precisely... and now if only someone could find out just what that simpler formula is... :-) — 79.113.209.71 (talk) 12:39, 22 May 2013 (UTC)

──────────────────────────────────────────────────────────────────────────────────────────────────── For the moment, the formula is WP:OR, being not sourced, nor related to any of the methods described in the article. Myself, I am unable to decide to which classical method it is a variant. Thus, before to improve the formula, it is important to rewrite the remainder of the article to make it understandable, notationally coherent and to clarify the relationship between the different methods. This is required for improving the formula in the special cases, because such an improvement is strongly related with Galois theory, and applying Galois theory here needs to interpret every auxiliary variable as a function of the roots. D.Lazard (talk) 12:59, 22 May 2013 (UTC)

Myself, I am unable to decide to which classical method it is a variant — I didn't say that you should write it directly in the article... Write it here, so that others might see it, and be able to help you in finding out a source for it.
it is important to rewrite the remainder of the article to make it understandable — I personally don't possess the knowledge needed to do that. Perhaps it would be better to discuss here the specific major changes we plan on doing to the article before we actually do them ? — 79.113.209.71 (talk) 20:26, 22 May 2013 (UTC)
I don't think we need to cite the formula so long as it works. If it works, it's verifiable and it can be included. Pokajanje|Talk 20:57, 22 May 2013 (UTC)
How do you know for sure that it works? How does the casual reader verify? If it is not sourced, many readers will feel tempted to verify it. That should not be necessary. Just look how much time has been wasted on the article and here on the talk page. This thing desperately needs sources. - DVdm (talk) 21:18, 22 May 2013 (UTC)
I agree with DVdm that sources are needed. Probably, they do not exist for this exact formula, but for a variant such that one passes easily from one formula to the other by a routine computation, like simple change of auxiliary variables grouping the terms differently. IMO, up to such routine computations, there are very few possible formulas, each corresponding to one or several named methods (in the cubic case, all the methods give essentially the same formula). For the moment we do not know how the formula has been derived and the different methods that are described are difficult to understand, as their main ideas are hidden by over detailed trivial explanations and incoherent notations. It is thus very difficult to rely the formula to existing methods, and thus to source it. @DVDM: sources are more desperately needed for the method of the derivation of the formula than for the verification: with modern computer algebra system, this is a routine computation to verify the formula, by substituting it in the equation and calling the simplifier. D.Lazard (talk) 21:52, 22 May 2013 (UTC)
Are examples acceptable on Wikipedia? If so, we could solve an example equation to show that the formula works. Pokajanje|Talk 03:15, 23 May 2013 (UTC)
No, of course not. I mean, just imagine someone would say or write: The general formula for the roots of a quartic is x1 = x2 = x3 = x4 = 0, and then use the individual case x^4 = 0 to "proof" that it is true! :-) — 79.113.227.125 (talk) 08:59, 27 May 2013 (UTC)
Just look how much time has been wasted on the article and here on the talk page — The hunt is better than the catch... :-) Joke aside, I'm currently searching for quartic equation formula roots special cases on Google: still no luck so far... :-( — 79.113.224.229 (talk) 07:42, 23 May 2013 (UTC)

u + v = 0[edit]

According to Mathematica, the denominator of w is 0 when:

1. the numerator of w is also 0 ⇔ a3 - 4ab + 8c = 0 ⇔ w = %
2. Δ0 = 0 ⇔ b2 - 3ac + 12d = 0
3. |Q| = Infinity Impossible

79.113.245.147 (talk) 20:15, 23 May 2013 (UTC)

Cross out case 1 as well; I've encountered several cases with a 0 numerator and the formula has still returned the correct results. Number 2 will be solved if we can find a simpler representation of the roots as discussed in the previous section. Pokajanje|Talk 02:57, 24 May 2013 (UTC)
The formula breaks down when #1 and #2 are both satisfied. If you don't believe me, try the following input combination: x4 + x3 + x2 + (3/8) x + (1/96) = 0. :-) Please also note that this case is not covered in the article's Special cases section, since Q ≠ 0 and the four roots are not equal (xi ≠ xj, i ≠ j). — 79.113.225.20 (talk) 09:37, 24 May 2013 (UTC)
Wow, that's unfortunate. Here are the results I get: u = -\frac{5}{12}, \Delta_0 = 0, \Delta_1 = \frac{61}{64}, Q = \frac{\sqrt[3]{61}}{4}, v = \frac{1}{3}, w = 0 and all four expressions for the roots are imaginary. Pokajanje|Talk 14:44, 24 May 2013 (UTC)
One of Murphy's laws says that 90% of the work takes 10% of the time, and the remaining 10% of work take up 90% of the time: this is basically what we're doing here, trying to "patch up" the particular cases of a general formula. :-) — 79.113.226.12 (talk) 18:52, 24 May 2013 (UTC)
And of course, I made some errors (this is a very error-prone formula). Not all of the above constants are correct. Even so, w = \frac{0}{0}, so it could be any real number. For the record, the correct answers are \frac{-3 \pm \sqrt{30\sqrt{3} - 45}}{12}. Pokajanje|Talk 19:16, 24 May 2013 (UTC)
Which is the same as saying w = ( 5/6 ) √3. — 79.113.226.12 (talk) 22:40, 24 May 2013 (UTC)

──────────────────────────────────────────────────────────────────────────────────────────────────── I have edited the beginning of Ferrari's method to make it understandable. This shows that, if the general formula is derived from Ferrari's method, then the case of a division by zero may occur only if the associated depressed quartic is bi-quadratic, which may be solved without cubic root extraction. I have also indicated how to make the formula valid for all quartics that are not a fourth power. Thus a really general formula seems to be hard to deduce from Ferrari's method. D.Lazard (talk) 13:38, 25 May 2013 (UTC)

"All quartics that are not a fourth power"? Would that not just be cubics, which has already been perfectly solved? Pokajanje|Talk 16:10, 25 May 2013 (UTC)
Yes and no: In the two methods that I have clarified, if a division by zero occurs for all the three roots of the cubic resolvent then these three roots are equal, and the four roots of quartic are also equal. To be accurate, a quartic which is not a fourth power is a quartic that can not be factored as (x-a)^4. D.Lazard (talk) 16:49, 25 May 2013 (UTC)
I think that I can fully clarify the case u+v=0: if u+v=0, then the numerator of w is zero. Otherwise, a small variation of the coefficients would provide monic equations with bounded coefficients, with arbitrarily large roots (in absolute value); this is impossible. A purely algebraic proof is possible, using Galois theory, but I should think harder to find such an algebraic proof. As the numerator of w is zero if and only if the associated depressed quartic is bi-quadratic, one may eliminate this case by considering separately the quartics whose depressed form is bi-quadratic.
If one want a truly general formula, one may choose another determination of the cubic root defining Q. If the three determinations give u+v=0, this implies that all the coefficients of the associate depressed cubic are zero (to be verified, but I am pretty sure), thus the quartic is (x+\frac{a}{4})^4. D.Lazard (talk) 17:46, 25 May 2013 (UTC)

Section "Ferrari's solution in the special case of real coefficients"[edit]

I have removed this section with the edit summary "Remove a misplaced section, which is WP:OR". An editor ask me on my talk page to restore it, arguing that it is sourced. I will explain in more detail why I have removed it and why I consider as non suitable to restore it as it was. Firstly, the source is a primary source published in a non-notable journal, probably not peer-reviewed. Thus it is original research for Wikipedia criteria. But this is not the main reason for the removing. The aim of the section is obscure. It could be to explain that Ferrari did not know of complex numbers. Thus, such a section could be devoted to describe the cases where the solution does not involve complex numbers (that is when there are two real and two complex roots); this is not the case. One conclusion of this section is that a quadratic polynomial over the real can always be factorized into two quadratic factors with real coefficients. This easy consequence of Fundamental theorem of algebra is misplaced in the solving section. One other conclusion of the section is that one of the real roots of the cubic resolvent provide such a real factorization (this may be wrong in the case of bi-quadratic equations). This is not specific to Ferrari's method, and applies to the other solving methods with a simpler proof than the one which is given (the three roots of the resolvent provide the three ways to factorize into two quadrics, thus at least one must give a real factorization). Thus, if something has to be restored, this should be independent of the solving method, placed in a separate section either at the end of the solving section or after it, and the text should be completely different. Personally, I have other priorities than writing this new section, that I consider of a minor importance. D.Lazard (talk) 09:25, 27 May 2013 (UTC)

To do[edit]

I have almost finished to restructure the article in order to obtain better readability and providing usable formulas. What remains to do is not mathematically deep but is time consuming. I hope that somebody else will do it. The remaining tasks are (please add tasks if I have omitted some):

  • Harmonizing the notation between sections
  • Expand, by back substitution of the change of variables, the formulas described in sections 3.6 and 3.7 (they are the same) in order to obtain a formula which is structured like the "general formula"
  • If the result is as simple as the general formula (I guess that), then replace it and add something like "this formula results from the formulas of the following sections by changing back the variables and expanding the formula for the cubic equation"

D.Lazard (talk) 16:22, 27 May 2013 (UTC)

Application to computer graphics[edit]

" Quartics often arise in computer graphics and during ray-tracing against surfaces such as quadric or tori surfaces"

The phrase "in computer graphics and during ray-tracing" is a bit odd. Most people would regard ray tracing as a part of computer graphics, so the "and" seems out of place.

Ray tracing of quadric surfaces gives rise to quadratic equations, not quartics. Ray tracing of tori gives rise to quartics.

"which are the next level beyond the sphere and developable surfaces."

I recommend removing this entirely. It somehow implies a progression in difficulty or poularity from spheres/developables to quadrics. This progression doesn't exist, as far as I know. The "hierarchy" of geometric forms is far more complex than this. — Preceding unsigned comment added by 58.32.209.2 (talk) 07:18, 28 May 2013 (UTC)

Yes check.svg Done D.Lazard (talk) 07:49, 28 May 2013 (UTC)

Invariance under 3° root choice[edit]

In all the methods for finding the roots of the quartic we have to consider a root from a cubic. How is it that the triple choice we have at that point does not yield extra roots (In general: one would naively expect 12 roots. (Though I know that in any field a quartic is supposed to have at most 4 roots))? I know that the talk-page is not supposed to be a forum, but I guess many people are stuck with the same question. So shouldn't it be addressed in this article? 193.190.253.144 (talk) 18:14, 30 May 2013 (UTC)

Changing the root of the cubic amounts to permute the roots: the three roots correspond to the three ways of partitioning the four roots into two pairs. Similarly the change of the sign of the first square root amounts to exchange the pairs in the partition defined by the first choice. Thus in any case, the formula gives only four roots corresponding to the choice of the signs of the last square roots. D.Lazard (talk) 21:45, 30 May 2013 (UTC)

Oh, I see now. I suggest moving the "factoring the quartic into quadratics"-method perhaps a little upward. It shows in an elementary way (= understandable for someone being unfamiliar with Galois theory) what you just explained to me. 193.190.253.144 (talk) 09:02, 31 May 2013 (UTC)

To IP user 79.113....[edit]

Hi,

It would be much simpler for everybody if you would create an account and use it for editing. This would allows you to use useful tools like your watchlist. This would also allow you to receive messages on your user talk page, which is presently not possible because you IP address changes every day.

This message is misplaced here, because it does not concern the general discussion of this article. If you had a talk page, I would have post there what follows.

Why do you insist for using {\color{white}.} instead of ~ (tilde) for inserting spaces at the end of the formulas? They have the same effect and the latter is simpler.

You have changed S into S2. IMHO, my version has the advantage to make clear that the same square root is used everywhere, in w and in the value of the roots. By the way, the value of w is presently wrong. Note that I have used capital S (the initial of square root) to insist that it is, like Q a radical.

I thought it came from Sum (of u and v). Either way, I'm not the one who re-baptized it with the name v. — 79.113.232.75 (talk) 17:38, 31 May 2013 (UTC)

In the "nature of the root", you have suppressed the bullets from the items of the lists and added a period at the end. This is not conform to the manual of style WP:BULLETLIST.

Please revert your edits if you agree with my objections or explain why you disagree.

Finally, we desperately would need in this talk page a description on the way you have derived this formula and the starting point you have used. This would allow to source the formula as "This formula is derived from ... through simple calculations".

I didn't "derive" anything, the guys at Planet Math did, I just made some notations, that's all... (And, before you ask, I'm not the one who replaced the monic formula with the general one). — 79.113.232.75 (talk) 17:38, 31 May 2013 (UTC)

Sincerely D.Lazard (talk) 14:42, 31 May 2013 (UTC)

Another off-topic message from myself. I fully agree with the above. Anon 79.113, please take a username.
An on-topic message: can we please add sourced content here? I was happy to participate, but this has become —sorry— ridiculous, so I stopped checking the changes to the article, and even had removed the article from my watchlist for a while.
In my opinion, this, on page 60, is the perfect source. The only thing that needs a bit of work, is to incorporate the calculation of r (the number of distinct real roots), which is handled on page 22. -DVdm (talk) 16:11, 31 May 2013 (UTC)
In my opinion — NPOV ! ;-) — 79.113.232.75 (talk) 17:52, 31 May 2013 (UTC)

I think that some of your concerns are probably better addressed at Pokajanje than at myself... Secondly, I never got upset at anyone.

Why do you insist for using {\color{white}.} instead of ~ (tilde) for inserting spaces at the end of the formulas? They have the same effect and the latter is simpler. — If this is so, my computer mysteriously fails to comply... Tilda has exactly zero effect on my monitor... (But now that you've told me, I won't re-change it).

As for the bulleted list, I'm sorry for the bullets, the only modification I wanted to make was to render the sub-cases and sub-sub-cases more visually pleasing in their on-screen display, as well as less redundant and less repetitive. I'm sorry if I've upset anyone, I won't modify the article one way or the other ever again.

As for the S (whose notation I left untouched, except for removing the radical), you made an error yourself: it was supposed to be S2 under the radical of the third and last term in each of the two root-formulas. So I changed S to S2 in those two places, then I thought that it is redundant baptize the radical with S when we're using its square anyway... Anyway, as I said, I won't re-modify anything. And since I'm "removing myself from the [quartic] equation", I guess I won't need an account anyway :-) Take care ! — 79.113.232.75 (talk) 17:29, 31 May 2013 (UTC)

All I'm trying to do is simplify fractions... sorry... Pokajanje|Talk 01:52, 1 June 2013 (UTC)
One final request: please re-order the three terms in the numerator of v so that the term in Q2 comes first, the one only in Q second, and the free term last, as if it were a quadratic polynomial in Q. — 79.113.227.162 (talk) 11:15, 1 June 2013 (UTC)
One final observation: You're hardly "simplifying" anything: I mean, remember the quadratic formula: \tfrac{-b \pm \sqrt\Delta}{2a} , and notice that if we were to break it up into a sum of fractions, none of these fractions (ie, -\tfrac{b}{2a} and \pm \tfrac{\sqrt\Delta}{2a} ) would be simplifiable, which is obviously not the case here with the quartic. — 79.113.227.162 (talk) 11:15, 1 June 2013 (UTC)

Simplifications and names of the variables in the formula[edit]

Apparently, user:Pokajanje believe that transforming a sum of fractions in a single fraction is always a simplification. This is clearly true if the operands of the fractions are rational numbers. If the operands of the fractions are polynomials, this is frequently true, but not always. For example, is A is a polynomial, transforming A+\frac{1}{A} into  \frac{A^2+1}{A} produces a more complicated expression involving 3 operations instead of 2 (and a much more complicated expression if the numerator is expanded), which hides the symmetry of the expression. For expressions involving radicals, like here, this transformation of a sum into a fraction is not convenient for the following reasons:

In the expression of the roots, this complicates the formula and induces, in the resulting formula, unnecessary operations (multiplication by a followed by a division by a (because a remains explicitly before the radical). Therefore a must appear only in the first term. For the numerical denominators, this could be discussed, but the formula is visually simpler by distributing them.

In the definition of v (or S), transforming the sum into a single fraction has, as in the above example, the drawback of hiding the symmetry. Also this introduces implicitly the symbol \left(\sqrt[3]{.}\right)^2, which is not particularly simple. Moreover, keeping Q+\frac{\Delta_0}{Q} recalls the formula for the cubic equation and has the advantage of suggesting to the reader that a cubic equation is involved here (it would be much better to write it down explicitly somewhere).

Name of the variables: To make readable such a complicated formula, it is useful to use some semantics for the name of the variables. Here Greek capitals denote the fundamental invariants of the quartic. In my edit that has been reverted, I used Latin capitals for radicals and Latin lower case for other variables. Moreover, the fact of naming the radicals, which induce the choice of a fixed root, is useful for insisting that this choice must be done once for all the formula. Therefore, I insists to name S the square root of what is called presently v. This induces another choice: either to write v = \cdots, S=\sqrt{v} or to write directly S=\sqrt{\cdots}. The latter choice has the drawback of introducing S^2 in the final value of the roots. The former has the drawback of augmenting the number of auxiliary variables. Personally, I prefer the latter, which is, in the whole, visually simpler.

I will edit the formula in this way. Please, before reverting me, partially of totally, discuss here the reasons of the change you want to do. D.Lazard (talk) 12:28, 1 June 2013 (UTC)

I personally have no problem either way... The only thing I truly care about is that the formulas are both correct and visually pleasing: and they are. I personally prefer S without the radical, and the fractions "unsimplified", but that's just my personal preference, not something "vital".
Furthermore, the reason I named the entire radical Q is because the quantity in question ALWAYS appears UNDER the radical, so there would've been no pragmatic purpose in doing otherwise. It wasn't for any "philosophic" reason :-)
I also think that -for visual reasons alone- the \Delta_0 from the "Nature of roots" section should be either changed back to T (fom triple), or at least minimized in size (by adding "\scriptstyle" in front of it inside the "math" tags).
I also think that u and w should be renamed to something else. (My initial notation used the four consecutive alphabet letters: t, u, v, w: but with two of them changed, I think the other two are sort of out-of-place there, notation-wise). 79.113.238.141 (talk) 14:36, 1 June 2013 (UTC)
I agree with the two last suggestions. If I have not done the last one, it is because too much simultaneous changes is an efficient method to introduce errors :-) D.Lazard (talk) 15:09, 1 June 2013 (UTC)
I'm now finding the formula very complicated in the absence of u + v. Pokajanje|Talk 15:42, 1 June 2013 (UTC)
The preceding post is a personal opinion. I have not the same.
About the names of the auxiliary variables, I would suggest
  • to define p=-\frac{3u}{2} and to replace every occurrence of u by -\frac{2p}{3}. This is a minor change of the appearance of the formula and has the advantage that p is the coefficient of degree two of the associated depressed quartic, and it has this name in the section where the depressed quartic is defined.
  • to rename w as q: this is the letter that follows p. Also Sw is the coefficient of degree 1 of the depressed quartic, which is denoted q elsewhere.
  • to add comments explaining the relation between these variables and the coefficients of the associated depressed quartic.
This would have the advantage of materializing that the formula is derived from that of the associated depressed quartic. D.Lazard (talk) 16:28, 1 June 2013 (UTC)

──────────────────────────────────────────────────────────────────────────────────────────────────── Uhm... Replace them both with their exact relationship with the actual p and q ! Replace u with -\tfrac23\ p , and w with \tfrac2S\ q . — 79.113.238.141 (talk) 18:15, 1 June 2013 (UTC)

At the risk of this becoming a debate, which I dearly wish to avoid as you can see on my main userpage, I will no longer be editing this article or discussing it here. I hope that I have at least been of some help in improving it. Pokajanje|Talk 23:51, 1 June 2013 (UTC)
I don't deny that the innitial formula was somewhat simpler and in a more direct link to the Planet Math formulas from the image to the left... but it is ultimately less connected to the rest of the article, and also fails to show from where the various variables appeared. Besides, math-challenged people don't really go around poking their noses into cubic formulas, let alone quartic ones! :-) Either way, even this "complicated" formula is a thousand times simpler than the one directly in coefficients, which occupies several screens... :-) — 79.113.213.196 (talk) 01:44, 2 June 2013 (UTC)
I changed the order of p and q inside the radical so that they might follow directly one after another; I also extracted the predominating minus sign from underneath the radical and turned it into an i placed in front of it, which also provides the advantage of keeping the signs in front of "S" and "q/S" the same, rather than having them opose each other (you might change that back if you like, but please keep the order of p and q intact); and I made the Delta0 small font in the "Nature of roots section" (you might change it back into a T if you like, as long as it stays small font). Also changed "numerator of w" to "numerator of q" in the "Special cases" section. — 79.113.213.196 (talk) 01:19, 2 June 2013 (UTC)
I find that the i outside the radical may be misleading for the reader: non-real numbers appear in the formula, even when the roots are real. Also, as i is implicitly a square root, this does not keep to the minimum the number of square roots appearing in the formula. Thus I'll reinsert the minus signs in the square roots, without changing the order of the terms. D.Lazard (talk) 06:18, 2 June 2013 (UTC)
More misleading than the cubic formula, where i appears regardless of the nature of roots ? And couldn't it be also argued that its lack is equally misleading ? Shouldn't people be educated or made to understand that the visual presence of all-plus-signs doesn't "mean" or "guarantee" anything, since the quantities being "added" could just as well be either negative or complex ? Just a thought. — 79.113.216.15 (talk) 14:32, 2 June 2013 (UTC)
You are right. May be it could be added a remark that, when the four roots are real, p is always negative. May be by adding at the end of the section a sentence like "The fact that all the terms inside one of the square roots have a minus sign does not mean that the square root is not real. On the contrary, when all the roots are real, p is negative and all square roots are real, except the one which is inside Q." D.Lazard (talk) 15:45, 2 June 2013 (UTC)
Perhpaps the inclusion of such a short, clear, and concise explanation would be very helpful for both cubic and quartic articles... :-\ — 79.113.216.15 (talk) 17:12, 2 June 2013 (UTC)

Pseudocode[edit]

I came to this article as a reader, lacking expertise, wanting to write code (in PostScript — forgive my sins) that finds real solutions of a real quartic using only real arithmetic. (PostScript has reals that are only single precision, and no derived types, so doing anything in complex arithmetic would be very painful.) For this reader, and I suspect for others, pseudocode, or links thereto, would be very welcome. Thank you. JDAWiseman (talk) 18:25, 20 June 2013 (UTC)

You are apparently mixing two different problems: The "formula for the roots" is intended for an exact representation of the roots involving square and cubic roots. When the four roots are real, there cannot exists such a representation that is purely real (this is a theorem). Apparently you are not interested by such an exact representation of the roots, but by a numerical approximation, as the "reals" of Postscript (and every programming language) are not the reals of mathematics, but a small subset of the rational numbers, well suited for approximating real numbers. For computig such approximations of the roots, the algorithms that are specific to degree four are only small optimization of the general algorithms. Thus the answer of your question should be in Root-finding algorithm, or more specifically into Newton's method, which is probably the method that is the easiest to implement.D.Lazard (talk) 09:08, 22 June 2013 (UTC)

Using Symmetric Notations for Polynomial Coefficients[edit]

If the polynomial coefficients would have symmetrical notations

a x^4\ +\ b x^3\ +\ c x^2\ +\ b' x\ +\ a'\ =\ 0

then expressions of the two Deltas would be easier to remember

\Delta_0\ =\ \ {\color{white}.} c^2 -\ 3\,\Big(bb' - 4aa'\Big)
\Delta_1\ =\ 2c^3 - 9c\,\Big(bb' + 8aa'\Big) + 27\,\Big(ab'^2 + a'b^2\Big)

79.113.220.62 (talk) 17:41, 14 August 2013 (UTC)

Quartic formula[edit]

What kind of quartic formula is this article supposed to have?? It looks like one for solving quartics whose leading coefficient is 1. Can anyone replace it with one for solving quartics in general?? Georgia guy (talk) 01:38, 29 November 2013 (UTC)

I do not understand: The sections "Nature of the roots" and "General formula for roots" provide formulas for a general leading coefficient. The four methods that have been provided for deriving a formula suppose a monic (and depressed) equation, but are preceded by section "Converting to a depressed quartic", which shows how to convert the results into a general formula. D.Lazard (talk) 08:48, 29 November 2013 (UTC)

Nature of roots seems incomplete[edit]

Using the classification of roots according to the discriminants  \Delta , p and D, I noticed that some cases are missing, that can also occur. An example is p < 0 and D > 0 leading to two equal real roots and two imaginary. In my eyes, a better and complete characterisation of the roots is developed in [1], whose system should be adopted or at least the article cited.

78.43.175.215 (talk) 09:46, 27 January 2014 (UTC) buonshi

The case "P < 0 and D > 0" is covered in the article as a subcase of "P < 0 or D < 0". Otherwise, the classification is essentially the same. The main difference is that, in the WP article the cases which correspond to the same nature of the roots are regrouped by mean of an "or" condition. Nevertheless, thanks for providing a source for this section. D.Lazard (talk) 13:55, 27 January 2014 (UTC)

Then I would also replace the formulation below \Delta = 0: "P < 0, D < 0" by "P < 0 and D < 0" for the sake of clarity. Glad that I could contribute to improving the article with a source. 78.43.175.215 (talk) 15:17, 27 January 2014 (UTC) buonshi