Talk:Quadratic equation/Archive 5

For Ardi Kule
If you think this is a bad article, try the simple english version: http://simple.wikipedia.org/wiki/Quadratic_equation --98.111.242.42 (talk) 02:12, 18 January 2010 (UTC)

Yet another derivation
What do y'all think of the following derivation? Is it sound? Should it be added to the derivation section?

Remembering that the Quadratic function maps from C -> C, write the equation as az^2 + bz + c = 0, where z = x + yi.


 * a*(x + yi)^2 + b(x + yi) + c = 0
 * ax^2 - 2axyi - ay^2 + bx + byi + c = 0

The real and imaginary terms must both sum to 0.


 * ax^2 - ay^2 + bx + c = 0
 * 2axyi + byi = 0

Solving for x and y,


 * x = -b / 2a
 * y = ±sqrt(4ac - b^2) / 2a

Finally,
 * $$z = x + yi = \frac{-b}{2a} \pm \frac{\sqrt {4ac-b^2}}{2a}i \,\!$$ —Preceding unsigned comment added by 68.199.134.93 (talk) 03:55, 22 July 2010 (UTC)

Misuse of sources
This article has been edited by a user who is known to have misused sources to unduly promote certain views (see WP:Jagged 85 cleanup). Examination of the sources used by this editor often reveals that the sources have been selectively interpreted or blatantly misrepresented, going beyond any reasonable interpretation of the authors' intent.

Please help by viewing the entry for this article shown at the cleanup page, and check the edits to ensure that any claims are valid, and that any references do in fact verify what is claimed. Tobby72 (talk) 18:58, 4 September 2010 (UTC)

Very slightly optimized implementation
In case anyone cares, you can remove one multiplication by 4 from the numerically stable implementation by pre-dividing b by 2 (or, more to the point, multiplying it by −0.5):
 * $$t := -\tfrac12 b \,\!$$
 * $$q := t + \sgn(t) \sqrt{t^2-ac} \,\!$$
 * $$x_1 := q/a \,\!$$
 * $$x_2 := c/q \,\!$$

71.41.210.146 (talk) 01:14, 31 October 2010 (UTC)


 * Copy from my response on your talk page:
 * Hi, yes, your calculation was of course entirely correct, and perhaps I should have used another template like . The rationale for my revert was that (1) you forgot to modify the text following the equation, where something is said about sgn(b), which does not occur in your equations, thus introducing a slight error of a different kind than what you thought I had in mind, (2) the cited source says something different, (3) the equivalence might be trivial for you and me, but it is beyond the "adding numbers, converting units, or calculating a person's age" of wp:CALC, and finally, (4) you replaced an inline multipication with the introduction of a new variable, which is generally less efficient. Now, you might not agree with that, and a discussion could be started about that, and perhaps sources could be found to support both viewpoints, etc... To avoid precisely that, we have policies about wp:NOR (particularly wp:CALC). So that's why I undid your edit. Cheers - DVdm (talk) 10:05, 31 October 2010 (UTC)

I notice that you added this remark (see previous time) again, so I have removed it because (1) it is not in the source, and (2) it is against wp:CALC: this not "a routine mathematical calculation, such as adding numbers, converting units, or calculating a person's age.". I have left a warning on your talk page. DVdm (talk) 10:56, 12 December 2010 (UTC)

Grade Six Math
Hello everyone. I was just wondering if, say, I don't know... a grade six child were to just come to this page, read it, and completely understand this article. Do you people think that would be weird? If you don't think it's wierd, then how young of a child would you be surprised to see repeating the quadratic formula? I'm not saying that I'm in grade six or anything... Thank you for the insightful answers. --**Najezeko**:) 03:25, 9 November 2010 (UTC) —Preceding unsigned comment added by Najzeko (talk • contribs)

In my opinion it will depend on a child's wiliness to learn such topics like quadratics at such an early age. A typical student will be introduce into quadratics in the 8th or 9th grade and slowly (maybe hastily!) progress from there. Its been argued over and over by academics that children should be introduced to topics like functions, biology, chemistry at an early age rather than taking it slow but overall it will depend on the child's capability to understand knowledge at a certain rate. YuMaNuMa (talk) 05:10, 19 June 2011 (UTC)

Quadratic equation to quadratic function
The method to get from the quadratic equation to the quadratic formula should be added in the section quadratic formula. I believe that, in spite of the fact that it's a derivation, the majority of people would still look in the section quadratic formula rather than derivations for this information. (I'm assuming here, though, that most people who need to look up the quadratic formula would not know it's a derivative of the quadratic equation. I'd also suggest adding in that section something along the lines of, "The quadratic formula is simply a derivative of the quadratic equation." — Preceding unsigned comment added by Wikigold96 (talk • contribs) 05:19, 19 December 2010 (UTC)


 * Please sign your talk page messages with four tildes ( ~ )? Thanks.
 * There are quite a few ways to derivate the formula, some of which are desribed later on in the article. I think that picking one would be add redundant information to the article. I also think that the great majority of people do not look for the derivation, bur for the result itself, which is clearly presented in the first section. The derivations are bundled together in a separate section in order to allow the interested reader to compare them. (Note: mark the difference between a derivation and a derivative.) Cheers - DVdm (talk) 10:35, 19 December 2010 (UTC)

Ok, I see your point. Now I hear it, you're right, it probably would be redundant. Although, I do think there should be some sort of reference to that the quadratic formula is a derivative of the quadratic equation. Thanks! And Happy New Year! --Wikigold96 (talk) 01:36, 2 January 2011 (UTC) Oh, well, not with UTC, but by Eastern Standard Time (where I am) it's still New Year's Day, so Happy New Year's Day!--Wikigold96 (talk) 01:38, 2 January 2011 (UTC)

Stevin priority
The quadratic formula covering all cases was first obtained by Simon Stevin in 1594, see "Principal works of Simon Stevin", p. 470. This is consistent with what is written in the history section. Descartes was after Stevin. Tkuvho (talk) 16:23, 19 December 2010 (UTC)


 * The one but last sentence (in the last paragraph) of the section says that in 1545 Gerolamo Cardano compiled the works related to the quadratic equations, which is 50 years before 1594. Could you perhaps given some more information about your source, like author, publisher, year, page, and an exact citation of what it says? DVdm (talk) 16:51, 19 December 2010 (UTC)


 * Numerous works before Stevin dealt with special cases, which is what was apparently compiled by Cardano. The principal works are online.  Also, what do you make of the claim in the current version of this page that the general solution was not found until the end of the 19th century?   Tkuvho (talk) 17:00, 19 December 2010 (UTC)


 * That 19th century claim seems to be sourced, and it talks about the "modern mathematical literature", so it seems that Cardano's, Stevin's and Descartes' works are not considered modern. Anyway, could you please provide author, publisher, year, ISBN, page, and an exact citation of what your source says, so we can insert a properly sourced something about Stevin between the Cardano and the Descartes statement? DVdm (talk) 17:11, 19 December 2010 (UTC)
 * Note that WP:BRD does not say that whoever was BOLD has to do all the technical work :) Tkuvho (talk) 12:45, 20 December 2010 (UTC)
 * I'll add the ref if you provide the data... DVdm (talk) 13:10, 20 December 2010 (UTC)


 * It's a deal: Stevin, Simon. The principal works of Simon Stevin. Vols. IIA, IIB: Mathematics.  Edited by D. J. Struik C. V. Swets & Zeitlinger, Amsterdam 1958. Vol. IIA: v+pp. 1-455 (1 plate). Vol. IIB: 1958 iv+pp. 459–976.   Tkuvho (talk) 13:57, 20 December 2010 (UTC)
 * Nice. Now, we want to be sure that it was Struik (—and not Stevin himself—) who said something like "The quadratic formula covering all cases was first obtained by Simon Stevin in 1594". So how exactly did Struik say it, and on which page? DVdm (talk) 15:13, 20 December 2010 (UTC)
 * Page 470, as I mentioned above. The text is online at http://www.historyofscience.nl/works_detail.cfm?RecordId=2702   Tkuvho (talk) 15:41, 20 December 2010 (UTC)
 * Super and ✅. DVdm (talk) 16:39, 20 December 2010 (UTC)
 * Thanks, it's a pleasure to work with you. Tkuvho (talk) 19:51, 20 December 2010 (UTC)
 * I find the end of the history section rather confusing. It would be nice if someone could state in the article in a clear way what the relationship between Ha-Nasi's, Cardano's, Stevin's und Descartes' work is. Probably it concerns the use of imaginary and negative numbers and the notation of the formula? Furthermore, I find the sentence about Heaton dubious. The citation never claims that this is the first modern appearance of the formula - it's just a peculiar method of derivation of the usual formula. Cardano (talk) 08:18, 30 March 2011 (UTC)

Rationalize Denominators?
In the Alternate Form section, denominators are not rationalized. Should this be fixed? Aero-Plex (talk) 01:00, 7 June 2011 (UTC)


 * No, that's what makes the alternate form different from the standard form. Jim.belk (talk) 18:44, 19 June 2011 (UTC)

Floating point implementation
I put back a note I added previously on computing the quadratic equation accurately with no loss of correct significant figures to double precision, and added additional references. Please do not delete this before reading the references I added! There are two forms of cancellation that can occur in computing the quadratic equation-- the first (major) one, that can lead to total loss of sig. figures, is removed by the well-known rearrangement of the formula that was given; however, there is a *second* form of cancellation that can occur which is less severe, but still can lead to loss of half the sig. figures, and which can only be removed by computing the discriminant in extended precision (or the equivalent). Brianbjparker (talk) 15:56, 8 March 2012 (UTC)


 * Well you have that problem anyway with quadratics when the discriminant is close to zero as you have to check if it is positive or negative and you can only do that accurately by going to double precision. Depending on the precision like Kahan does there is just silly I think, the user should treat the double precision as extra precision and work to a lower precision but whatever the precision Kahan complains about it and wants more because he treats the original precision as the target precision. For a maths library what he says is fine but I just don't think this sort of stuff is suitable for everybody and their dog to worry about. This is all a special case of what I was saying before about working in double the precision you want if you're working with differentials. That paper by Kahan was never published as far as I know but I think I can live with it being in the article. Dmcq (talk) 17:59, 8 March 2012 (UTC)


 * The published section in the book Accuracy and Stability of Numerical Algorithms also describes it in a succinct fashion (actually you can see it in Google books, although the page is scanned upside down).
 * It is an important practical detail for someone implementing this algorithm to know that if they use doubles they get single precision in the result-- that may be fine but one needs to be aware of it. But it is also an interesting example where extended precision is required for internal calculations and there is no (simple) numerical fix for the instability.
 * When you say that "everybody and their dog [shouldn't have] to worry about", Kahan would be in violent agreement, that was his point in that article: that it is better to just do internal computations in extended precision always by default so a typical programer doesn't need to do a full forward or backward analysis of their expressions. Brianbjparker (talk) 21:54, 8 March 2012 (UTC)
 * And as I said it is all well and fine for someone implementing a library but for a normal user they are quite liable to get a negative discriminant even when that is not theoretically possible in the problem they are dealing with when solving the quadratic occurs in the middle of something else. Depending on those last bits for anything is just ludicrous unless the user has done a very careful analysis and just doing this calculation more accurately without that is not going to help in real world calculations. The point about extended precision is that the user should not be trying to use the full precision, what Kahan is doing is continually putting things off to higher precision whatever the precision. It just does not help much in normal work to use higher precision in intermediate calculations if they are going to get shortened every so often. Just think of double precision as 47 bits plus another 6 bits of extended precision and you'll stop worrying about extended precision - and the whole calculation is then done in extended precision instead of losing chunks every time a calculation is stored. Extended precision was to help with maths library routines. Dmcq (talk) 23:44, 8 March 2012 (UTC)
 * Oh certainly, I would mostly agree with that-- if for example your application only requires about single precision for the final result then you can just calculate all intermediate results in double and you are set; or if you only need 47 bits of precision in the final result then using double for the intermediate computations would also help (although double extended would be better). It is just the principle that is important-- you need to carry extra precision for the intermediate computations and the more the better (ideally about 2x). If your application really requires around double precision for the final results then you will need either double extended or quad precision for the intermediate calculations. Quad precision would be better but it is still not available in hardware-- double extended was designed to be basically free in that it runs as fast as double. (by the way, scratch variables in a larger computation would of course also be long double (extended precision) so the intermediate results would not get shortened until the final result of the overall algorithm). Your point about getting "a negative discriminant even when that is not theoretically possible" is *exactly* the case when you do need more precision for the internal computations-- the paper by Kahan I referenced goes into depth into that issue, indeed that is the whole point and he gives examples of real-world cases where it does make a difference. Brianbjparker (talk) 05:54, 9 March 2012 (UTC)
 * The problem is that most calculations involve things like iterating time over arrays and you really do need to store the values. So how you store the values matters and it perturbs the inputs to routines like this. By the way quad precision is available in both binary and decimal on the latest IBM z Series. Binary quad has been available on a few earlier machines too but has mostly been only software implemented. You can do the stuff here almost as well as quad if the machine has fused multiply add, you only need one extra multiply. Dmcq (talk) 12:00, 9 March 2012 (UTC)
 * It strikes me that what is really needed is an article which deals with these sorts of things properly. One that starts off with the basic algorithms using IEEE floating point to do things like accurate add and multiply with two floating point results and going on to things like ad−bc or this and Kahan's summation and polynomials. Dmcq (talk) 13:15, 9 March 2012 (UTC)

The new Diagonal Sum Method for solving quadratic equations that can be factored
This article has been a thorough explanations about solving quadratic equations. To make it more complete, I suggest the author express his opinion about the new method called the Diagonal Sum Method (WikiHow.com) that has been considered as a shortcut of the factoring method. There is a new improved quadratic formula (Google search), called the quadratic formula in graphic form, that deserves the author's consideration. Nang142 (talk) 16:48, 27 March 2012 (UTC) Nang142 Nang142 (talk) 16:48, 27 March 2012 (UTC)
 * Bit of a waste of time is my opinion. The references on the web seem like self promotion, I can't see a secondary source about it which is what Wikipoedia needs. Dmcq (talk) 17:28, 27 March 2012 (UTC)

Cultural Thoughts
The article poses the quadratic equation from the common interpretation of US textbooks. An example from a German textbook (translation my own) is "Let p, q be real numbers. Then we call
 * $$x^2 +px + q = 0$$

the normal form of a quadratic equation."

This definition also requires that "equivalent transformations" be used to acquire the quadratic equation in this form. The quadratic formula is also defined in terms of p and q, following a similar procedure to the proof for the version presented in the article.

Additionally, should other cultural implications be noted in this article? In particular, there has been controversy about whether students must master solutions of the quadratic equation in their study of mathematics. Thelema418 (talk) 05:10, 7 April 2012 (UTC)


 * I think you misunderstood what they were saying. Their 'normal form' is what is called the monic form here. When they refer to a quadratic equation without qualification they usually mean one in general with abc, they don't mean one that has been converted to normal form. Dmcq (talk) 11:42, 7 April 2012 (UTC)


 * No, I would not say this is a misunderstanding, but I understand what you are saying. The German textbook only defines quadratic equations with the p, q normal form representation (or monic form). The US "general" form using a,b,c never appears; even the quadratic equation is defined in terms of p, q.  I have also seen some older math textbooks in English that define the quadratic equation with the p, q form.  I bring this up because I think the definition that appears in the introduction to this article gives strong preference to a general format and particular definition that appears in US textbooks.  It should be more general, stating that the definition appears in multiple ways.  I don't think the general reader would see that the monic form is a valid definition from reading this article and the position of the monic form explanation in its text. Thelema418 (talk) 06:21, 12 April 2012 (UTC)


 * Well I'll just point instead at, a dictionary of mathematics from the European Mathematical Society published by Springer. And how about Quadratische Gleichung which is produced by the German Wikipedia. Or how about this bit from Google books ? Dmcq (talk) 08:42, 12 April 2012 (UTC)


 * I did a search in German for "Quadratische Gleichung" on Google and it yields a significant number of definitions that use the normal form definition. While the German Wikipedia article is the first that comes up, the second and third articles define the meaning of Quadratic Equation in the monic form.  Again, my issue is that different cultures may have differing preferences in the definition of the quadratic equation, and therefore the practices of pedagogy with the quadratic equation.  The article does not give this sense.


 * Aside from the definition, I think the cultural controversies about the education of quadratic equation should be cited in this article. Most of these controversies deal with the question, why do we need to teach students the quadratic equation?  There are interesting arguments both for and against it.  Of course, these issues are tied to other cultural beliefs. Thelema418 (talk) 07:23, 13 April 2012 (UTC)


 * Well if you have a source talking about such cultural differences that would be fine I guess. I think though it might be a good idea to discuss ideas like that at Talk:Mathematics education, there should be if anything a general article talking about differences in elementary school teaching of mathematics in different countries I'd have thought and then have this article refer to it if quadratic equations are an important part of that. WikiProject Education might also be of interest to you, WT:EDUCATION would be a reasonable place to discuss how to structure information like this.Dmcq (talk) 08:59, 13 April 2012 (UTC)


 * Thanks, that is very helpful. I will check that out. Thelema418 (talk) 03:33, 3 May 2012 (UTC)

Inappropriate external link
''(Copied from my talk page User talk:DVdm) (End of copied part) - DVdm (talk)

Stigmatella aurantiaca, what are you talking about? I just tried that example and got the right answer. Try again!128.104.153.144 (talk) 19:01, 17 May 2012 (UTC)
 * The number of places carried may be browser-specific. On the browser that I am using, the results are precisely what I state. The web site uses a naive implementation of the quadratic formula that is subject to loss-of-significance-error. Try stressing the calculator a bit. Try a=0.00000000001 b=5 c=0.00000000001. x1 should never be zero. Stigmatella aurantiaca (talk) 19:12, 17 May 2012 (UTC)


 * Anon 128.104.153.*, note that it doesn't even matter whether it produces the right answer or not. The link is on an unidentified someone's personal web page, so that someone is not a recognised authority. See —again— wp:ELNO #11. - DVdm (talk) 19:16, 17 May 2012 (UTC)


 * You may have gotten an answer, but you could not possibly have gotten the right answer. Here is the javascript on that page that you are emotionally tied to. Note the rounding that exacerbates the loss of significance problem that would be present even without any rounding:
 * function solve{
 * var A = parseFloat(document.getElementById("A").value);
 * var B = parseFloat(document.getElementById("B").value);
 * var C = parseFloat(document.getElementById("C").value);
 * var D = Math.round((B*B-4*A*C)*100000)/100000;


 * if(D >= 0){
 * document.getElementById("x1").value = Math.round(((-1*B+Math.sqrt(D))*100000)/(2*A))/100000;
 * document.getElementById("x2").value = Math.round(((-1*B-Math.sqrt(D))*100000)/(2*A))/100000;
 * }


 * if(D < 0){
 * document.getElementById("x1").value = Math.round(((-1*B)*100000)/(2*A))/100000 +" + "+
 * (Math.round(((Math.sqrt(-1*(D)))*100000)/(2*A))/100000) + "i";
 * document.getElementById("x2").value = Math.round(((-1*B)*100000)/(2*A))/100000 +" - "+
 * (Math.round(((Math.sqrt(-1*(D)))*100000)/(2*A))/100000) + "i";
 * }
 * }
 * Stigmatella aurantiaca (talk) 19:52, 17 May 2012 (UTC)

Note - IP was rangeblocked. - DVdm (talk) 23:11, 17 May 2012 (UTC)


 * Why on earth would anyone want to program rounding in here? Have I missed something? JamesBWatson (talk) 23:31, 18 May 2012 (UTC)
 * Definitely very stupid. Stigmatella aurantiaca (talk) 23:34, 18 May 2012 (UTC)


 * And look at what happens with (A,B,C) = (-1,2,-2). It produces (x1,x2) = ( 1 + -1i, 1 - -1i ), whereas (A,B,C) = (-1,0,-1) produces (x1,x2) = ( 0 + -1i , 0 - -1i ). That is probably the main reason why the recognised authority of wp:ELNO #11 was put in place: to avoid us having to scrutinise the work and results of some well-meaning amateur. - DVdm (talk) 10:26, 19 May 2012 (UTC)


 * The link does not belong in WP. I tried A=0, B=1, C=15. "Answer" is x1 = NaN, x2 = -infinity. Not even trivial input checking. It is a unique resource. The criticisms above are all significant. Glrx (talk) 00:09, 21 May 2012 (UTC)

Figure legend
In the first figure on the very top right, there's no indication in either figure or caption, what the values are of the remaining 2 constants that are not the title of the plot. — Preceding unsigned comment added by 94.208.6.16 (talk) 17:59, 16 December 2012 (UTC)
 * I've expanded the description to hopefully clarify it.-- JohnBlackburne wordsdeeds 18:33, 16 December 2012 (UTC)
 * Super!

Article requires simplification
An anonymous editor added a "technical" tag with the following comment:
 * (cur | prev) 03:56, 15 April 2013‎ 108.75.41.176 (talk)‎ . . (41,148 bytes) (+30)‎ . . (see WP:TECHNICAL - even the Simple English version is difficult for even very literate and reasonably well-educated people to understand) (undo)

I have long felt exactly the same as the above anonymous editor. The article as it stands could stand some major trimming of excessive detail. Why do we need a whole section on the monic form? Why so many alternative derivations of the quadratic equation? Who is interested in the alternative parameterization ax2+2bx+c=0, which distracts from the important point made in the Other methods of root calculation section, which is that the quadratic formula frequently leads to loss of significance?

Stigmatella aurantiaca (talk) 06:41, 15 April 2013 (UTC)


 * I've just chopped out the section on the monic form, replacing it with a brief mention in the lede. Stigmatella aurantiaca (talk) 07:10, 15 April 2013 (UTC)


 * I've just chopped out the section on alternative parameterization, replacing it with a note in the text. Later on I'll try to rearrange the "Other methods of root calculation" section to emphasize the issue of catastrophic cancellation and loss of significance often associated with the usual form of quadratic formula. Stigmatella aurantiaca (talk) 12:58, 15 April 2013 (UTC)


 * Hi, James! I have to get to work, so I can't spend time right now about the disputed edit until tomorrow. I'll get back to you after midnight tonight (i.e. after I've done my taxes). Stigmatella aurantiaca (talk) 13:26, 15 April 2013 (UTC)


 * I've just moved some material on loss of significance near the presentation of the quadratic formula. The important point is that naive use of the quadratic formula can result in loss of significance errors. I feel that a brief example is necessary to make this section clearer. In adding an example, I'll be a bit at risk of violating WP:NOTTEXTBOOK, but I'll try to be careful. Stigmatella aurantiaca (talk) 04:24, 16 April 2013 (UTC)

I've cordoned off some material into an "Intermediate and advanced topics" section. Much of this is really neat stuff, but only to the right audience. Stigmatella aurantiaca (talk) 08:19, 16 April 2013 (UTC)

The "Examples of use" section strikes me as pure hodgepodge, a collection of unrelated topics that somebody or other just wanted to expound upon at some time or other. What common thread joins "Geometry", "Quadratic factorization", and "Application to higher-degree equations" (which I cordoned off into the "Intermediate and advanced topics" section)? Stigmatella aurantiaca (talk) 08:19, 16 April 2013 (UTC)

Why I don't like the sentence about the etymology of the word "quadratic" in the lede
Hi, James!

Let me first quote from the Manual of Style:
 * ....The lead serves as an introduction to the article and a summary of its most important aspects....
 * The lead should be able to stand alone as a concise overview. It should define the topic, establish context, explain why the topic is notable, and summarize the most important points—including any prominent controversies....The emphasis given to material in the lead should roughly reflect its importance to the topic, according to reliable, published sources, and the notability of the article's subject is usually established in the first few sentences. Apart from trivial basic facts, significant information should not appear in the lead if it is not covered in the remainder of the article.

Now let us examine individual sentences in the lede:
 * In mathematics, a quadratic equation is a univariate polynomial equation of the second degree.
 * Clearly defines the topic and establishes context.


 * A general quadratic equation can be written in the form $$ax^2+bx+c=0,\,$$ where x represents a variable or an unknown, and a, b, and c are constants with a ≠ 0.
 * ''Defines important terms that are used many times in the article."


 * (If a = 0, the equation is a linear equation.)
 * Necessary to understand this at several points.


 * Note: Dividing the quadratic equation by a gives the simplified monic form x2 + px + q = 0, where p = b/a and q = c/a.
 * The term "monic" is used several times in the article.


 * The constants a, b, and c are called respectively, the quadratic coefficient, the linear coefficient and the constant term or free term.
 * I don't see these particular terms used elsewhere in this article, but certainly they appear in wikilinked articles, and they are important for understanding.


 * The term "quadratic" comes from quadratus, which is the Latin word for "square".
 * Not necessary to be able to to read and understand the article. Not covered in the remainder of the article.


 * Quadratic equations can be solved by factoring, completing the square, graphing, Newton's method, and using the quadratic formula (given below).
 *  The article at present doesn't cover solving by factoring, solving by graphing, or Newton's method. Nor does the wikilink to factorization explain how to solve by factoring.

Stigmatella aurantiaca (talk) 09:13, 16 April 2013 (UTC)

I think this section should just be deleted
The following section deals with a specific numeric example and is difficult to generalize.

I suspect this is one of those sections that led an anonymous commentator to write on 5 December 2009, "You think there is at least one place where you can learn things and it turns out it is a playground for mathematicians."

Yes, it's been around for a long time, but it stinks. Sometimes less is more. Should I just delete it?

Stigmatella aurantiaca (talk) 11:15, 16 April 2013 (UTC)

Update: &mdash; Since there appeared to be no objections to my suggestion, I removed the section. Stigmatella aurantiaca (talk) 00:43, 17 April 2013 (UTC)

Application to higher-degree equations
Certain higher-degree equations can be brought into quadratic form and solved that way. For example, the 6th-degree equation in x:
 * $$x^6 - 4x^3 + 8 = 0\,$$

can be rewritten as:
 * $$(x^3)^2 - 4(x^3) + 8 = 0\,,$$

or, equivalently, as a quadratic equation in a new variable u:
 * $$u^2 - 4u + 8 = 0,\,$$

where
 * $$u = x^3.\,$$

Solving the quadratic equation for u results in the two solutions:
 * $$u = 2 \pm 2i\,.$$

Thus
 * $$x^3 = 2 \pm 2i\,.$$

Concentrating on finding the three cube roots of 2 + 2i – the other three solutions for x (the three cube roots of 2 - 2i ) will be their complex conjugates – rewriting the right-hand side using Euler's formula:
 * $$x^3 = 2^{\tfrac{3}{2}}e^{\tfrac{1}{4}\pi i} = 2^{\tfrac{3}{2}}e^{\tfrac{8k+1}{4}\pi i}\,$$

(since e2kπi = 1), gives the three solutions:
 * $$x = 2^{\tfrac{1}{2}}e^{\tfrac{8k+1}{12}\pi i}\,,~k = 0, 1, 2\,.$$

Using Eulers' formula again together with trigonometric identities such as cos(π/12) = (√2 + √6) / 4, and adding the complex conjugates, gives the complete collection of solutions as:
 * $$x_{1,2} = -1 \pm i,\,$$
 * $$x_{3,4} = \frac{1 + \sqrt{3}}{2} \pm \frac{1 - \sqrt{3}}{2}i\,$$

and
 * $$x_{5,6} = \frac{1 - \sqrt{3}}{2} \pm \frac{1 + \sqrt{3}}{2}i.\,$$

Deriving the quadratic formula "By shifting ax2"
A number of alternative derivations of the quadratic formula can be found in an internet search which are either (a) simpler than the standard completing the square method, or (b) represent interesting applications of other frequently-used techniques in algebra, or (c) offer insight into other areas of mathematics.

Deriving using Lagrange resolvents is an example of (c).

I see deriving the quadratic equation starting with the vertex form (i.e. "By shifting ax2") given as an assignment in various textbooks, but the solution presented here makes use of techniques (Vieta's formulas) that are generally presented AFTER the quadratic formula has been derived.

Rewriting this section to avoid direct use of Vieta's formulas would represent solving what seems to be a standard textbook problem. The derivation here is neither simpler than the standard completing the square method nor particularly insightful. So far as I can see, it was just somebody showing off that they could do a homework problem, only not too well.

I am definitely thinking of replacing it.

Stigmatella aurantiaca (talk) 10:27, 24 April 2013 (UTC)


 * The biggest problem with mathematical articles in Wikipedia is that they are edited by enthusiastic mathematical specialists, who see things from the point of view of serious mathematicians, and are full of adding stuff that's interesting from their standpoint, giving excessively high priority to such matters as making interesting connections with other branches of mathematics. I must have seen complaints about articles being too technical for the general reader about 500 times as often for articles on mathematical articles as on all other articles put together. Years ago, I used to fairly frequently argue on talk pages against certain mathematicians who would from time to time move in on an article, and take ownership of it, spending a large amount of time on making more and more changes to the article until they had essentially rewritten it to their own preferred version, which usually did make it more interesting to me, but made it much less accessible to the general reader of the encyclopaedia. However, I now rarely bother to do so, as it is usually a futile exercise, since such people are inevitably RIGHT and know better than anyone else what the CORRECT way to present a mathematical topic is. However, as far as the present issue is concerned, the approach which is most likely to be useful to the ordinary encyclopaedia reader is (a) the one they are most likely to come across elsewhere, and (b) the one that they are most likely to be able to understand. My experience, based on several decades of teaching mathematics, is that, whatever may seem "simpler" to me, or to other mathematicians, transformational approaches are bewildering and incomprehensible to the vast majority of people. No mathematician is ever going to turn to Wikipedia to learn about quadratic equations, but many non-mathematicians do so. By far the best change that could be made to the article would be to cut out most of the more obscure and/or more technical aspects of quadratic equations, so that it would become an article that could reasonably be read and understood by a typical reasonably well-educated and intelligent reader without a mathematician's background and without a mathematician's ability to cope with abstractions. For example, the subsection "Trigonometric solution" deals with a truly obscure topic. In fact, the whole of the section "Advanced topics" would probably be better segregated into a separate article. It has no place in a general introductory article aimed at the general public. JamesBWatson (talk) 09:42, 25 April 2013 (UTC)


 * I suppose I have to agree with you that the section on trigonometric solutions is rather obscure. The stub section had been there for years with a note that it needed to be expanded. The choice was either to delete the stub section entirely, or to expand it as requested by the note. Since I am almost old enough to remember when "computer" mean a roomful of women attacking an algorithm armed with mechanical calculators (and only a generation or so too young to remember when it meant a roomful of male clerks (sometimes female&mdash;there was no mixing of sexes, of course!) working with pencil and paper), I opted to expand, which may have been a mistake.
 * I had hoped that creating an "Advanced topics" section to provide clear warning that "Danger Lies Beyond! Proceed At Your Own Risk!" should be sufficient to make the article more accessible, but obviously not.
 * Splitting the entirety of "Advanced topics" into its own article sounds like a good idea. What do you suggest the title should be? Stigmatella aurantiaca (talk) 10:31, 25 April 2013 (UTC)


 * Yes, the article should provide approachable explanations.
 * My feelings about advanced topics are mixed. This article could contain advanced topics about the quadratic equation at the end of the article. Whether the advanced topics are complete in themselves, have a small description with a main link, or are just a see also link depends upon interest and depth of the topic. The shifting ax2 discussion is a well-hidden circular complete-the-square argument: a depressed quadratic can be solved by inspection.
 * The initial section is haphazard. The graph, which is a visual explanation of what the equation means, is buried after factoring, completing the square, and the quadratic formula. Complex coefficients(!) are brought up without any explanation of why there will be two roots. Don't worry about generalizing the problem until after the basics are covered. (Why is factoring by inspection sweating only rational solutions? b = 0.) No statement of equivalence of polynomial (&Sigma; form) and factored forms (&Pi) (which is really the heart of the solution - number of roots and any factor going to zero drives whole equation to zero). Avoiding loss of significance is not a fundamental issue; it should come after the history section (and why isn't it combined with the other floating point topic). Graphing a solution does not usually provide high accuracy. No notion of a closed form solution. Having a quadratic formula followed by quadratic factorization section seems to confuse the point of the quadratic formula.
 * History is something that many readers would find interesting. Maybe even a little context with the cubic, quartic, and quintic.
 * Glrx (talk) 16:18, 25 April 2013 (UTC)


 * On further reflection I think that having an "advanced topics" section at the end may be better than fragmenting the topic into separate articles. I also think that some of the opinions I wrote above are not all that relevant to this article. I think I got rather carried away with my thoughts based, as I said, on my general experience of how mathematical articles often go, rather than specific concerns about this particular article. I still feel very strongly that the initial presentation should be from a perspective that is accessible to the general reader, but that does not necessarily mean we can't have a more advanced treatment later, and labelling it "Advanced topics" is an excellent idea, as it lets the general reader know that they are about to step into deeper waters. JamesBWatson (talk) 15:05, 26 April 2013 (UTC)
 * Having both you and Glrx critique what I've done is really, really helpful! Thanks to both of you! I'm still working to satisfy both of your sets of criticism, but a weekend is coming up, so hopefully I'll be ready for a second round of critique by Monday. Stigmatella aurantiaca (talk) 16:54, 26 April 2013 (UTC)


 * I think that I've addressed most (but not all) of your critiques. I haven't gotten to Cedar's critique yet. Concerning closed form solution: What would you like me to write? I'm not sure discussion of closed form would be relevant in an elementary treatment of the quadratic equation. I'm also not sure I want to get into a general discussion of Sigma versus Pi forms, not at this level. Thanks! Stigmatella aurantiaca (talk) 22:46, 28 April 2013 (UTC)
 * Having twice in my professional career seen poorly conditioned math problems yield single-precision (or worse!) results where double-precision was expected (once disastrously), I have personal reason to believe that the complete lack of exposure to numerical analysis considerations in undergraduate math courses is a serious oversight in the traditional syllabus. True, neither of the disasters that I personally experienced involved the quadratic equation, but they both involved computer programs written by people who were relatively unaware of the issues involved in floating point computations. I intended to give a small taste of the problems involved in real-life computations in the elementary section. However, combining the two parts in which I have split the discussion into a single section provides far too much detail to be considered an elementary exposure to the concept of loss of significance. Stigmatella aurantiaca (talk) 01:05, 29 April 2013 (UTC)

Geometric interpretation and Quadratic factorization sections
What to do about them?

Stigmatella aurantiaca (talk) 04:03, 29 April 2013 (UTC)