Quadratic equation was one of the good articles, but it has been removed from the list. There are suggestions below for improving the article to meet the good article criteria. Once these issues have been addressed, the article can be renominated. Editors may also seek a reassessment of the decision if they believe there was a mistake.
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January 29, 2007 Good article nominee Listed
August 4, 2007 Good article reassessment Delisted
April 27, 2013 Good article nominee Not listed
Current status: Delisted good article
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## Current state of the article

I've been a bit too focused on this article and need some critiques and advice on where to go with it. For instance, the "Geometric interpretation" and "Quadratic factorization" sections seem out of place. Thanks for any suggestions! Stigmatella aurantiaca (talk) 23:10, 1 May 2013 (UTC)

### Floating-point implementation & related issues

It would be very useful to merge the issues related to numerical computation of the roots on a standard computer into one simple section. Please avoid vague statements such as "the code will be something like the following". Why not give code for a working and robust implementation (link to Kahan paper?) 193.157.242.60 (talk) 12:23, 30 May 2013 (UTC)

The discussion is split up because the first part discusses a computation that can easily be verified with a cheap pocket calculator. Loss of significance is an important topic that is rarely emphasized at K-12 level, and I believe that the revised form of quadratic equation presented here is elementary enough — and the illustrated loss of significance dramatic enough — to be in the "simple introductory" part of the article. On the other hand, the second part of the numerical analysis discussion cannot be verified except with extended precision arithmetic, considerably beyond the capabilities of simple pocket calculators.
I will certainly change the vague wording, which was inherited from earlier versions of this article. Thanks! Stigmatella aurantiaca (talk) 19:45, 30 May 2013 (UTC)
If I were to consolidate the two sections into a single section, where would I put it? It would be too long and too detailed to put into the "simple introductory" part of the article, so it would have to go into the "advanced topics" section. But that would mean there would be no mention of numerical analysis issues whatsoever in the "simple introductory" part of the article. Now, of course, many would be OK with not mentioning these issues in the "simple introductory" part of the article, but I personally am uncomfortable with the relative neglect of this topic in K-12 textbooks. Thanks for any additional thoughts you may have on this! Stigmatella aurantiaca (talk) 04:17, 31 May 2013 (UTC)

I have figured out how I might use the "collapse top/collapse bottom" templates to make a readable combined section that could fit into the "simple introductory" part of the article, but the explanation of this template at Template:Collapse top discusses use of this template only on talk pages. Is it acceptable to use this template in the main/article space? Stigmatella aurantiaca (talk) 12:48, 1 June 2013 (UTC)

Sure. Use {{cot|description of what's here}} and {{cob}}, but make sure you don't include (sub)section headers. For instance:
Longwinding derivation
bla
bla
bla
Good idea. - DVdm (talk) 12:59, 1 June 2013 (UTC)
I'm opposed to including detailed numerical issues early. The motivation is phrased as a problem with K-12 education and implies trying to fix that problem. It also detracts from the initial challenge of solving the equation, and numerical accuracy a full subject in its own right (naive sinh(1.0e-9)). Maybe include a sentence in the lead to make the reader aware, but keeping the numerical topics together seems appropriate (and the GA review suggested it).
I don't like the collapsible box idea at all. In this case, it seems to be a rationalization for placing material in a more prominent position (e.g., within introductory material) when it does not belong there. A collapsible box is used when the material is too detailed; I don't think the numerical material is too detailed (if that is the target); it belongs in the article, but it should not be prominent. Use a hyperlink to another (lower down) section or articles that discuss the topic in detail. It is an advanced topic; most readers of encyclopedia don't have to worry about floating point errors. Glrx (talk) 16:06, 1 June 2013 (UTC)
Is it an advanced topic to teach people not to trust numbers coming out of a computer just because they came out of a computer? Still, your suggestion of making a brief mention of the issue in the intro section along with inserting a link to a unified discussion in the advanced section is also good, just somewhat opposed to mine. Let's see what comes out in way of a consensus. I'll follow what people think is the best solution. Thanks! Stigmatella aurantiaca (talk) 16:26, 1 June 2013 (UTC)
I'm not convinced we need to go into too much detail here. We may be getting close to WP:UNDUE weight. I'd be happy with it just appearing in the advanced topic section as its an issue not many people have to deal with. I'd also be wary of too much reliance on the Kahan paper as it only seems to have 8 citations.[1] I generally wary of saying this is how this should be calculated, or is actually calculated. For the most part it is better handled by the specialist literature.--Salix (talk): 19:05, 1 June 2013 (UTC)
I am not relying solely on Kahan. Higham (2002) also covers this subject in considerable detail, but without the specific numerical examples presented in Kahan (which I've double-checked using a computer program). The entire book is downloadable from the web (probably illegally, so I'm not giving a link). I will certainly bear in mind WP:UNDUE. However, the complete opposite of giving a topic undue weight is the complete neglect of a topic despite its being easy to understand at an elementary level. I hope to find some compromise acceptable to all.
From Higham: "Unfortunately, there is a more pernicious source of cancellation: the subtraction b2 − 4ac. Accuracy is lost here when b2 ≈ 4ac (the case of nearly equal roots) , and no algebraic rearrangement can avoid the cancellation. The only way to guarantee accurate computed roots is to use extended precision (or some trick tantamount to the use of extended precision) in the evaluation of b2 − 4ac."
Stigmatella aurantiaca (talk) 03:16, 2 June 2013 (UTC)

## Don't indiscriminately italicize

Don't indiscriminately italicize everything in non-TeX mathematical notation. One italicizes variables but NOT parentheses and NOT digits and NOT things line sin, log, max, det, etc.

Right: ax2 + bx + c = 0
Wrong: ax2 + bx + c = 0
Right: y = f(x)
Wrong: y = f(x)

This is codified in WP:MOSMATH and is consistent with TeX style. Michael Hardy (talk) 18:22, 3 May 2013 (UTC)

Thanks for the heads-up on this! It will be a few days before I can get to this matter, but after I finish making the changes, I'll probably need you to double-check my work. Stigmatella aurantiaca (talk) 18:40, 3 May 2013 (UTC)
Oh, wait, I see you've been working on this already! Stigmatella aurantiaca (talk) 18:41, 3 May 2013 (UTC)

## Screen readers do rather horribly with mathematical articles

With good intentions, Neil Soiffer started converting some HTML markup into TeX using the rationale that TeX is better read and interpreted by screen readers.

I tried downloading two free screen readers, NVDA and Thunder, and found that "better" is still pretty awful. They do OK with regular text, but math is a different story. I'm going to look into making audio versions of this and other technical articles that I've been heavily involved with. My speaking voice isn't exactly the greatest, but it's still going to be a lot better than the utter mess that screen readers have trying to interpret mathematics markup.

Does anybody have any experience with this sort of thing to point me in the right direction to get started?

Thanks, Stigmatella aurantiaca (talk) 04:51, 10 July 2013 (UTC)

I read over Wikipedia:WikiProject Spoken Wikipedia and listened to a sample recording. This is going to be a pretty major effort, but I already own some of the equipment that I'm going to need to make decent recordings. Stigmatella aurantiaca (talk) 05:14, 10 July 2013 (UTC)

Just wanted to say to you, Neil, I really appreciate your pointing out this issue! Stigmatella aurantiaca (talk) 22:11, 10 July 2013 (UTC)

Neil Soiffer: I'm not sure the right way to reply, but here goes -- feel free to move this to the appropriate place...

Yes, screen readers don't natively do well with math (they don't do anything). We (Design Science) have a free plug-in for IE called MathPlayer that makes the math sounds reasonable. It works with MathJax to make the pages accessible to assistive technology.

Although creating the audio is a good intention, I caution against doing it. First off, as someone else remarked, you have tens of thousands of expressions to deal with. More significantly, one "size" doesn't work for everyone. People who are blind need to know where 2d notations start and end. The standard is to add pauses and also start/end [fraction, script, ...]. However, people who have dyslexia and other vision related learning impairments find those extra words confusing. The later group uses tools from TextHelp or Kurzweil and that sync highlights the words they hear. MathPlayer does the same (seamlessly) for math when used with those tools.

A limitation of MathPlayer is that it is currently IE only. However, we have a version undergoing testing that works with all browsers. Although we are way ahead of others in our support for math accessibility, we are not the only ones working on it. ChromeVox recently added math support to Chrome. Others are working on it also. The key for Wikipedia is to get the math cleaned up in pages so it doesn't use HTML markup and instead uses TeX or MathML. That allows speech to be generated based on user needs, and it also allows for braille generation on refreshable braille displays. Wikipedia is such a key resource for learning, getting all the math to be accessible is a huge win for accessibility in education and elsewhere. — Preceding unsigned comment added by Neil Soiffer (talkcontribs) 22:56, 10 July 2013 (UTC)

My impression is the current WP bias is to avoid math markup in text and use html markup instead. Math markup is converted to a picture and looks bad. The math markup discussion needs to happen at a more global level. Glrx (talk) 02:15, 11 July 2013 (UTC)
1. It wasn't a WP consensus that led me to use HTML, but my own choice, and I don't mind being overruled on a matter as important as this. Actually, I've been very much a minority in using HTML. HTML is quite limiting in what it can do.
2. Looking at the source, we see that the MathPlayer software must be reading the alt: <img class="tex" alt="\left(x+\frac{b}{2a}\right)^2=\frac{b^2-4ac}{4a^2}." src="//upload.wikimedia.org/math/6/e/e/6ee0ac4d5fad8cdfa2ddd45a843e7541.png" />
3. I'll start by restoring Neil's edits, and together Neil and I should be able to get quadratic equation fixed up to where it is accessible to TeX-enabled readers.
4. By simply using the "textstyle" directive, we can get the TeX so that it doesn't always look grossly oversized, i.e. ${\displaystyle \textstyle \left(x+{\frac {b}{2a}}\right)^{2}={\frac {b^{2}-4ac}{4a^{2}}}}$ versus ${\displaystyle \left(x+{\frac {b}{2a}}\right)^{2}={\frac {b^{2}-4ac}{4a^{2}}}}$. The directive doesn't always fix things up, but it can make a noticeable difference in a significant number of cases.
5. In other words, I opt for both the use of TeX markup and audio recording so that there are as many different ways for these pages to be accessible as possible. MathPlayer sounds pretty awesome.
Stigmatella aurantiaca (talk) 06:22, 11 July 2013 (UTC)
Another thing is that I've been uniformly lazy in providing alt text for my images all over Wikipedia. I need to correct this. Stigmatella aurantiaca (talk) 06:46, 11 July 2013 (UTC)

Well, I've finally started recording. Any advice how I would pronounce the matrix ${\displaystyle \left({\begin{smallmatrix}1&1\\1&-1\end{smallmatrix}}\right),}$ by the way? And why does TeX use parentheses instead of square brackets? Stigmatella aurantiaca (talk) 12:31, 24 July 2013 (UTC)

How about "the two by two matrix whose elements in left to right, top to bottom order are one, one, one, negative one"? Any better ways, or more standard ways? Stigmatella aurantiaca (talk) 12:35, 24 July 2013 (UTC)

I put the quadratic formula in the lead. It's an incredibly important formula, and is discussed repeatedly in the body of the article. I made a few other changes too. Looks like a section is needed on applications, or at least an image showing an application. I have inserted an image at the top showing a very well-known application.Anythingyouwant (talk) 20:01, 11 September 2013 (UTC)

You are starting to put far too much information into the lede. Be a little careful here. Stigmatella aurantiaca (talk) 04:32, 12 September 2013 (UTC)
I'm not sure the history material belongs in the lede. Think of what 99% of high school or college students would want to see in the first few seconds that they read through this article. Certainly not the fact that the ancient Babylonians knew how to solve special cases of the quadratic equation! Stigmatella aurantiaca (talk) 04:55, 12 September 2013 (UTC)
Per your suggestion, I have cut most of the historical material from the lead, but left in a brief mention that this stuff dates back to 2000 BC.Anythingyouwant (talk) 06:21, 12 September 2013 (UTC)
Thanks! Stigmatella aurantiaca (talk) 14:25, 12 September 2013 (UTC)

The quadratic formula has been removed from the lead, and I disagree with that. The quadratic formula gives the solution to the quadratic equation. No other formula does that. Moreover, "The Quadratic Formula is one of the most important formulas in algebra, and you should memorize it."[1]. It is "the most useful method."[2]

The lead presently gives undue weight to other methods: "Quadratic equations can be solved by a process known in American English as factoring, or 'factorising' in other English, by completing the square, using the quadratic formula, or by graphing." This statement in the lead makes it seem like the methods are of comparable value, which is grossly misleading. For example, "graphing" is almost never used for this purpose. Same goes for completing the square. Here's what some reliable sources say:

"While the method of completing the square may be used to solve quadratic equations, it is more involved than the quadratic formula, and is seldom used in practical work."[3]

"You can use the quadratic formula, as well as completing the square, to solve any quadratic equation. However, you will find that the quadratic formula is easier to use."[4]

Completing the square "can be a rather long and complicated procedure and is seldom used in practical applications."[2]

As for factoring, "The factoring method has limited application. Only certain quadratic equations can be solved by factoring."[2]. I therefore propose to introduce this information in the body of the article, and reinsert the quadratic formula in the lead.

[1]Larson, R. and Hodgkins A. College Algebra with Applications for Business and Life Sciences, p. 104 (Cengage Learning 2009).

[2]Smith, R. and Peterson, J. Introductory Technical Mathematics, pp. 408-409 (Cengage Learning 2006).

[3]Payne, M. Intermediate Algebra, p. 289 (West Publishing 1985).

[4]Davis, L. Technical Mathematics, p. 174. (Merrill Publishing 1990).

Anythingyouwant (talk) 16:53, 13 September 2013 (UTC)

I had disagreed with your lede edits when I saw them, but I refrained from reverting them because you obviously have very good intentions towards this article, which I admire.
But now that JamesBWatson has made many of the same edits that I refrained from performing myself, I will have to say that I agree with his judgement.
Factoring and completing the square are taught in schools not because they are generally applicable or highly efficient methods, but because they teach important skills. Likewise, the same graphing methods that seem wasted on solving the quadratic equation are immediately applicable to finding the real roots of equations that have no analytic solution.
Stigmatella aurantiaca (talk) 18:40, 13 September 2013 (UTC)
Thanks. Of course, I'm not suggesting that various methods should not be taught in school, or should not be mentioned in this article. I'm not even suggesting to reinsert info about the discriminant into the lead. No one disagrees with the notion that the quadratic formula is one of the most important formulas in algebra, so I strongly disagree with burying it in the body of this Wikipedia article. Cheers.Anythingyouwant (talk) 18:47, 13 September 2013 (UTC)

If there are no objections, I would like to insert this brief material into the lead:

While several methods of solving the quadratic equation teach important skills, the method that is most efficient and widely applicable is using the quadratic formula, which is among the most important equations in algebra:

${\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}}$

That's it.Anythingyouwant (talk) 19:04, 13 September 2013 (UTC)

That doesn't sound too bad to me. What do you think, James? Stigmatella aurantiaca (talk) 21:13, 13 September 2013 (UTC)
Thanks for your input, I went ahead just now and put it in.Anythingyouwant (talk) 14:16, 15 September 2013 (UTC)
• Oppose. Superlatives are not in body. No claim to most efficient method. There is a widely used claim in body. An important equation to students of algebra (stmt in body) rather than most important equations in algebra (how important is QF to modern algebra?). The statement should just state the QF widely used. Glrx (talk) 00:25, 16 September 2013 (UTC)
The quotations from the cited sources are explicitly given in the previous section of this talk page. I will put them into the footnotes of the Wikipedia article. They seem to support the wording of the Wikipedia article. Nevertheless, I rephrased a little bit.Anythingyouwant (talk) 00:35, 16 September 2013 (UTC)
• Agree with the objections stated by Glrx. Also, many students of algebra learn another formula for solving the quadratic equation: First reduce the equation to the monic form ${\displaystyle x^{2}+px+q=0}$ and then use the formula ${\displaystyle x=-{\frac {p}{2}}\pm {\sqrt {\left({\frac {p}{2}}\right)^{2}-q}}}$. Other students may have encountered the quadratic formula in a slightly different form:
${\displaystyle x=-{\frac {b}{2a}}\pm {\sqrt {{\frac {b^{2}}{(2a)^{2}}}-{\frac {c}{a}}}}}$ Isheden (talk) 09:42, 16 September 2013 (UTC)
I agree that Glrx had some good points, so I edited the article accordingly, and I'll be curious to know whether that takes care of the problem. Regarding the procedure of reducing the quadratic equation and then giving the quadratic formula for a=1, I am not aware of any textbook that takes that approach without also giving the full quadratic formula; I'd be very interested to know about such a textbook, so that we can use it as a reference. Regarding your formula using "p" and "q", I don't think it is used often, especially if "p" is not an even number.[2] I think readers will understand that the full quadratic formula can be rewritten many different ways, for example by using different letters, and perhaps that would be worth pointing out explicitly in the body of the Wikipedia article. Cheers.Anythingyouwant (talk) 13:17, 16 September 2013 (UTC)
Actually I don't know what is standard in English textbooks, but it seems to be known as the "pq formula" or something similar in several other languages including German and Swedish. However, I found an English textbook [3] which refers to it as "the following well-known quadratic formula", specifying that it solves the monic quadratic equation. Probably it is not worthwhile to mention it in the lead, but I guess it should be mentioned somewhere in the article. I'm not sure how often it is used in teaching, but I think it is the simplest formula available for quadratic equations reduced to monic form. Isheden (talk) 15:54, 16 September 2013 (UTC)
Interesting, thanks for the info.Anythingyouwant (talk) 19:49, 16 September 2013 (UTC)

1. The lead should be a general introduction to the subject of the article. Details such as specific methods of solution should not be in the lead.
2. Things such as "most important" are subjective, and don't belong anywhere in the article. Yes, of course you can find published opinions that say that, but it is still subjective.
3. "Most efficient"? By what measure of efficiency?
4. Factorisation and completing the square are far more important than the formula in terms of providing information, in terms of techniques which have further applications, and so on. The only area in which the formula can be reasonably considered to have priority is as a practical method of finding solutions to equations (and even there factors are more convenient in those cases where they can be easily spotted). By not only putting the formula in the lead, but also accompanying it by a manifesto about how much better and more "efficient" it is than the others, the article has been made to be heavily biased towards covering quadratic equations from the point of view of treating them as practical tools, marginalising the very great importance that they, and the accompanying methods, have in algebraic theory. This looks, in fact, very much like the view of mathematics that I have often seen from many physicists, engineers, etc, who view mathematics only as a set of tools. JamesBWatson (talk) 21:36, 16 September 2013 (UTC)
Thanks for your comments, James. I'll respond to your first comment now, and the rest tomorrow. The thing about the quadratic formula is that it's not merely a method or an algorithm; it's the general solution. Wikipedia articles sometimes state a problem in the lead as well as the solution. See, for example: System of linear equations, Legendre's differential equation, Basel problem, Gaussian integral, et cetera.Anythingyouwant (talk) 23:22, 16 September 2013 (UTC)
James' comments, of course, overlap many of the points that I had made earlier, only I also put graphing techniques right up there with factoring and completing the square in terms of what they teach the student. I suppose that comes from the pretty strong numerical analysis bias in my background. Mathcad, Maple, Mathematica etc. have revolutionized the way that that complex math problems are approached by engineers and scientists alike. So what really is the most important concept that students need to get out of an algebra course? It's pretty hard to say. Stigmatella aurantiaca (talk) 23:52, 16 September 2013 (UTC)
Yes, I agree. Although I didn't mention it before, I think that the graphical approach has great importance, not only because of its relevance to the sorts of software you refer to, but also for a number of other reasons, including: its pedagogical importance in teaching the concepts involved in quadratic equations; the fact that the method is applicable to other types of equations which are not susceptible to solution by algebraic algorithms, but can best be introduced in the simple quadratic case; the fact that the graphical approach provides important information about such issues as how solutions of the equation relate to the properties of the quadratic function, how changing the parameters of that function change the solutions, and so on and so on. The complete dismissal of the graphical approach as unworthy of consideration appears to be based on the very narrow view that the only thing that is of any significance in the subject of quadratic equations is how a mathematically competent person can most easily find solutions to particular equation. (I say "a mathematically competent person" because years of teaching algebra have taught me that there are many people who can manage to draw a quadratic graph and use it to find at least tolerable estimates of the solutions of an equation, but who, no matter how much help and tuition they are given, cannot cope with substituting numbers into the quadratic formula and calculating the solutions.) JamesBWatson (talk) 01:07, 17 September 2013 (UTC)
Against my better judgment, I have removed the quadratic formula from the lead, at least for now. However, I urge you folks to consider that I have produced several reliable sources attesting that it is the most efficient and widely applicable method of solution, whereas no sources have been produced that suggest otherwise. Instead, reliable sources have simply been dismissed as mere "opinion", so it seems futile for me to produce more of the same. The quadratic formula is among the most widely used mathematical formulas in existence. It is the general solution to the quadratic equation. 'Nuf said. Cheers.Anythingyouwant (talk) 00:21, 17 September 2013 (UTC)
No doubt without intending to, you are misrepresenting what I said. I did not say that "reliable sources [are] mere opinion". I said that what is "most important" is subjective, and that giving reliable sources does not change that. A reliable source may indicate that a particular subjective view is held by a reliable author, that it is a very widely held view, or even that it is an almost universally held view, but that does not stop it from being subjective. Also, you seem to be missing the point of some of the things that have been said in this discussion. You emphasise that you have produced reliable sources that the formula is "the most ... widely applicable method of solution", and seem to have completely failed to notice that nobody has denied that it is, and, more importantly, you seem to have failed to notice that it has been questioned whether being the most widely applicable method of solution is so overwhelmingly important as to give it priority over anything else. Not only are you still treating the issue as though it goes without saying that finding solutions to particular numerical examples is overwhelmingly more important than every other aspect of the subject, but you also don't seem to have even noticed that that view has been questioned. JamesBWatson (talk) 00:44, 17 September 2013 (UTC)
Nope, sorry, I do not believe that finding solutions to particular numerical examples is important in the lead. The quadratic formula is the general solution, not a particular numerical example. If you're objecting now to the image at the top, please feel free to remove it, though I think it spruces up the article nicely. Cheers.Anythingyouwant (talk) 01:14, 17 September 2013 (UTC)
I am totally bewildered as to how you can possibly read what I wrote as meaning that. My one last attempt to clarify what I have been saying, which you again and again seem to fail to read, follows. I did not suggest that the quadratic formula was "a particular numerical example", nor that I thought you had suggested "that finding solutions to particular numerical examples is important in the lead". What I did suggest was that the arguments that you have put forward to justify giving the quadratic solution formula high priority over other aspects of the subject are based entirely on its usefulness as a method of finding solutions to particular numerical examples, as though you are blind to all other aspects of the subject, and think that use to find such numerical solutions is the only purpose of a method of solution. JamesBWatson (talk) 01:36, 17 September 2013 (UTC)
Well, it seems that we both find this conversation bewildering. The material that I put into the lead specifically said that "several methods of solving the quadratic equation teach important skills". Moreover, nothing I put into the lead made any reference to particular examples. The quadratic formula is justly famous for furnishing the solutions of any quadratic equations whatsoever, and thus is not limited to particular examples. I will be moving on from this article now. Cheers.Anythingyouwant (talk) 01:55, 17 September 2013 (UTC)

## Someone made a merge proposal

FYI, there seems to be a proposal to merge material into this article. See here.Anythingyouwant (talk) 14:24, 15 September 2013 (UTC)

Alas, I've had to scrap what I've done so far in the audio recording project. I few discrepancies I could tolerate, but it sounds like the article will be getting a major overhaul. Let me know when, in your estimate, the article has reached some level of stability. Thanks! Stigmatella aurantiaca (talk) 16:48, 15 September 2013 (UTC)
Hi Stigmatella. I'm all through. Just wanted to get a good image at the top, plus the quadratic formula in the lead (with support in the body of the article). If I manage to get quadratic polynomial and quadratic function merged into a single article, then I will modify the hatnote of this article. I don't think either of them should be merged into this article. Cheers.Anythingyouwant (talk) 17:41, 15 September 2013 (UTC)

## Illustration at the top

It's not very clear how the illustration at the top is supposed to be related to the quadratic equation. In this case the times and corresponding distances are given as

t s
0 0
0.05 9
0.10 20
0.15 33
0.20 48
0.25 65
0.30 84

and one might expect that the problem is to find the uniform acceleration a and initial velocity u that best fit the experimental results in the table. Perhaps one could find a better suited initial example or at least explain the illustration more clearly? Isheden (talk) 16:58, 16 September 2013 (UTC)

I could adjust the image so that it gives an explicit value for the initial velocity "u" and an explicit value for "a" (9.8 m/s/s).Anythingyouwant (talk) 17:06, 16 September 2013 (UTC)
Okay, I have adjusted the image and caption. It seems more clear now. The reader can confirm that the experimental results are consistent with the formula in the caption, and can also predict how long it will take ("t") for the ball to travel any other distance ("d").Anythingyouwant (talk) 19:22, 16 September 2013 (UTC)

## Summarizing

I started a new article called "quadratic formula", and accordingly have trimmed that material in this article, per WP:Summary style.Anythingyouwant (talk) 18:43, 12 October 2013 (UTC)

Good idea. Stigmatella aurantiaca (talk) 20:08, 12 October 2013 (UTC)
Thanks. Would you like to discuss particular edits?Anythingyouwant (talk) 23:50, 12 October 2013 (UTC)

## Disagree with latest edits on derivation of quadratic formula

### Version as of 22:22, 11 October 2013‎‎

Completing the square can be used to derive the quadratic formula.[1]

Dividing the quadratic equation ${\displaystyle ax^{2}+bx+c=0}$ by a, which is allowed because a is non-zero, gives

${\displaystyle x^{2}+{\frac {b}{a}}x+{\frac {c}{a}}=0}$

or

${\displaystyle x^{2}+{\frac {b}{a}}x=-{\frac {c}{a}}.}$

The quadratic equation is now in a form to which the method of completing the square can be applied. To "complete the square" is to add a constant to both sides of the equation such that the left hand side becomes a complete square:

${\displaystyle x^{2}+{\frac {b}{a}}x+\left({\frac {1}{2}}{\frac {b}{a}}\right)^{2}=-{\frac {c}{a}}+\left({\frac {1}{2}}{\frac {b}{a}}\right)^{2},}$

which produces

${\displaystyle \left(x+{\frac {b}{2a}}\right)^{2}=-{\frac {c}{a}}+{\frac {b^{2}}{4a^{2}}}.}$

The right side can be written as a single fraction with common denominator ${\displaystyle 4a^{2}}$. This gives

${\displaystyle \left(x+{\frac {b}{2a}}\right)^{2}={\frac {b^{2}-4ac}{4a^{2}}}.}$

Taking the square root of both sides yields

${\displaystyle x+{\frac {b}{2a}}=\pm {\frac {\sqrt {b^{2}-4ac\ }}{2a}}.}$

Isolating x gives

${\displaystyle x=-{\frac {b}{2a}}\pm {\frac {\sqrt {b^{2}-4ac\ }}{2a}}={\frac {-b\pm {\sqrt {b^{2}-4ac\ }}}{2a}}.}$

The plus-minus symbol "±" indicates that both

${\displaystyle x={\frac {-b+{\sqrt {b^{2}-4ac}}}{2a}}\quad {\text{and}}\quad x={\frac {-b-{\sqrt {b^{2}-4ac}}}{2a}}}$

are solutions of the quadratic equation.[2]

### Version as of 22:52, 12 October 2013

The general formula for solving the quadratic equation ${\displaystyle ax^{2}+bx+c=0}$ is called the quadratic formula:

${\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}}$.

Of course, it follows that the solution to the reduced quadratic equation ${\displaystyle x^{2}+px+q=0}$ is:

${\displaystyle x=-{\frac {p}{2}}\pm {\sqrt {\left({\frac {p}{2}}\right)^{2}-q}}}$.

Completing the square can be used to derive these formulae,[3] and this procedure will now be briefly summarized. It can easily be seen, by polynomial expansion, that the following equation is equivalent to the quadratic equation:

${\displaystyle \left(x+{\frac {b}{2a}}\right)^{2}={\frac {b^{2}-4ac}{4a^{2}}}.}$

Taking the square root of both sides, and isolating x, gives:

${\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac\ }}}{2a}}.}$

The plus-minus symbol "±" indicates that there are typically two solutions of the quadratic equation.[4]

1. ^ Rich, Barnett; Schmidt, Philip (2004), Schaum's Outline of Theory and Problems of Elementary Algebra, The McGraw-Hill Companies, ISBN 0-07-141083-X, Chapter 13 §4.4, p. 291
2. ^ Sterling, Mary Jane (2010), Algebra I For Dummies, Wiley Publishing, p. 219, ISBN 978-0-470-55964-2
3. ^ Rich, Barnett; Schmidt, Philip (2004), Schaum's Outline of Theory and Problems of Elementary Algebra, The McGraw-Hill Companies, ISBN 0-07-141083-X, Chapter 13 §4.4, p. 291
4. ^ Sterling, Mary Jane (2010), Algebra I For Dummies, Wiley Publishing, p. 219, ISBN 978-0-470-55964-2

### Discussion of these recent edits‎‎

The typical middle school student will see the words "It can easily be seen, by polynomial expansion, that the following equation is equivalent to the quadratic equation:" and get a feeling like the critic in Sidney Harris' famous cartoon "Then a Miracle Occurs".

Refactoring the article so that most discussion on the quadratic formula is in its own article was a good idea. But these last edits have gone too far.

What do other editors think? Stigmatella aurantiaca (talk) 00:10, 13 October 2013 (UTC)

The full derivation is at Quadratic formula, so it doesn't seem necessary to give the whole thing twice, per WP:Summary style. I've taken the liberty of tweaking this article so it says: "Completing the square can be used to derive these formulae, and the mathematical proof will now be briefly summarized." (Citation omitted).Anythingyouwant (talk) 00:43, 13 October 2013 (UTC)
Yes, I can see that the full derivation is at Quadratic formula, but a certain repetition of content is desirable in certain circumstances so that the article can be read as an autonomous, standalone reference. Think of the primary audience for this article.
Between the version of April 15 and the version of July 25 I made two sets of major revisions to this article, the first set of revisions being to split the article into an introductory section accessible to students in the middle-to-senior high school range and an "advanced" section, the second set of revisions being to make the article more accessible to screen readers.
While critiquing my contributions, James B. Watson wrote:
The article is currently very different than it was when James made his comment to me, but the spirit of his comments continue to drive me not just in this article, but in other articles that I've edited. As James wrote, "No mathematician is ever going to turn to Wikipedia to learn about quadratic equations, but many non-mathematicians do so." Also, "...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."
So, what do non-mathematicians want?
I like most of what you've done with this article, and I recognize that I may be misguided in my present sentiments. So I will defer to your judgment. But I would appreciate it if other editors would offer their opinions on this matter.
Stigmatella aurantiaca (talk) 11:13, 13 October 2013 (UTC)
Completely agree with Stigmatella aurantiaca. An article about quadratic equations needs to derive the quadratic formula in a way that is easily accessible. In an older version of the article [4], the derivation was easy to read, but the actual formula was mentioned only at the end. The present version basically starts by stating the formula and then ask the reader to check for himself is he's not convinced. If I have the choice between these two versions I'd prefer going back the first one. Isheden (talk) 13:37, 13 October 2013 (UTC)
I can either put it back now, or else wait for further comments. Note that the article about elementary algebra doesn't derive the quadratic formula, so I'm not sure why this article must do that, as long as there's a prominent link to the full derivation. If the full derivation does go back in here, then I think the QF ought to be provided in the lead, since it's such a huge part of the article.Anythingyouwant (talk) 13:54, 13 October 2013 (UTC)
I think the most straightforward way of presenting the general quadratic formula is to apply the steps 1-6 outlined in the section "Completing the square" to the general quadratic equation. It can probably be shortened a bit compared to the old version, and e.g. the explanation of the plus/minus symbol could be moved to the example in the previous section. Then we would have:
We illustrate the general algorithm by solving ${\displaystyle ax^{2}+bx+c=0}$
1. ${\displaystyle x^{2}+{\frac {b}{a}}x+{\frac {c}{a}}=0}$
2. ${\displaystyle x^{2}+{\frac {b}{a}}x=-{\frac {c}{a}}}$
3. ${\displaystyle x^{2}+{\frac {b}{a}}x+\left({\frac {b}{2a}}\right)^{2}=\left({\frac {b}{2a}}\right)^{2}-{\frac {c}{a}}}$
4. ${\displaystyle \left(x+{\frac {b}{2a}}\right)^{2}=\left({\frac {b}{2a}}\right)^{2}-{\frac {c}{a}}}$
5. ${\displaystyle x+{\frac {b}{2a}}=\pm {\sqrt {\left({\frac {b}{2a}}\right)^{2}-{\frac {c}{a}}}}}$
6. ${\displaystyle x=-{\frac {b}{2a}}\pm {\sqrt {\left({\frac {b}{2a}}\right)^{2}-{\frac {c}{a}}}}}$
This can be rearranged to give ${\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}}$.
One could also use the letters p and q as introduced earlier in the derivation. Isheden (talk) 19:08, 13 October 2013 (UTC)
Isheden, textbooks sometimes leave derivation of the quadratic formula as an exercise for the student.[5] Indeed, once students understand how to complete the square, they are equipped with all the necessary knowledge to derive the QF themselves, which is an exciting exercise. Nevertheless, I suggest also giving students a wikilink to the full derivation, just in case they don't want to experience the thrill of rediscovery (this is a lot more than leaving the problem as an exercise). That full derivation is the centerpiece of the new article on the quadratic formula, and it seems very adequate in this article on the quadratic equation to explicitly prove (in an extremely brief and understandable way) that the QF is the solution of the QE. If we have to include the full derivation here in the present article, then I would be tempted to just delete the new article that I started on the quadratic formula. That's my view, anyway.Anythingyouwant (talk) 22:14, 13 October 2013 (UTC)
Wikipedia is not a textbook, but an encyclopedic reference that presents facts. Your argument was that the derivation takes up a lot of space, but since we can build on the method outlined in the section before, the derivation is short and easy to follow and would work as a summary of the "stand-alone" derivation in quadratic formula. Probably an equal amount of space is needed for stating the explicit formula, hinting at the method of derivation and proving that the QF is indeed a solution to the QE. I think a separate article about the quadratic formula with alternative proofs is motivated anyway, cf. Proof that π is irrational, Pythagorean theorem, in order to keep the size of this article reasonable. Isheden (talk) 09:23, 14 October 2013 (UTC)

(Outdent) I agree that we should have an encyclopedic summary here. Repeating the whole derivation in quadratic formula is not the same as summarizing, though. An excellent summary is here:

Himonas, Alex. Calculus for Business and Social Sciences, p. 64 (Richard Dennis Publications, 2001).

Anythingyouwant (talk) 16:59, 14 October 2013 (UTC)

I can only see a snippet view of the book. Would you please copy the summary here? Isheden (talk) 18:46, 14 October 2013 (UTC)

(Outdent) Sure.

Himonas, Alex. Calculus for Business and Social Sciences, p. 64 (Richard Dennis Publications, 2001).Anythingyouwant (talk) 20:00, 14 October 2013 (UTC)

Yes, "The details are left as an exercise." To a competent mathematician, the "exercise" is too trivial to even merit the title "exercise"; perhaps for the business and social science students for whom the book is intended it is a reasonable exercise, perhaps not; but for a very large proportion of the people who might be reading this article to find out about quadratic equations, "it can easily be seen, by polynomial expansion" is likely to be complete gobbledygook. The vast majority of the people learning how to solve quadratic equations have no idea even what the expression "polynomial expansion" means, let alone the ability to complete the "exercise". This is a first class example of the kind of thing that I was referring to in the passage which is quoted above: there is absolutely no point whatever in an article on a fairly elementary topic such as quadratic equations which contains material that is incomprehensible to the kind of person who is most likely to consult such an article. JamesBWatson (talk) 19:51, 15 October 2013 (UTC)
Unlike the arcane and advanced topic of "floating point implementation" (which you are urging to reinsert back into this article), polynomial expansion is an elementary algebra skill that any high school student must learn. That's what the following operation is: ${\displaystyle (x+y)^{2}=x^{2}+2xy+y^{2}}$. However, if it would smooth out any disagreements, I would be glad to remove the term polynomial expansion from the article, for the sake of world peace and happy coexistence. The very same paragraph which now mentions polynomial expansion also says this: "Completing the square can be used to derive these formulae", and I do not understand why that is inadequate.Anythingyouwant (talk) 20:07, 15 October 2013 (UTC)
You have latched onto the minor point of use of the words "polynomial expansion" as though that were the whole issue. Even if you replace it by some other wording, such as "multiplying out the brackets" or something, the other points would still stand. JamesBWatson (talk) 20:20, 15 October 2013 (UTC)
I wish you would acknowledge that there is an extremely prominent wikilink to the section of quadratic formula that presents the detailed derivation. Per WP:Summary style: "The parent article should have general summary information and the more detailed summaries of each subtopic should be in child articles and in articles on specific subjects."Anythingyouwant (talk) 20:24, 15 October 2013 (UTC)
(edit conflict) One more thought. Given a choice between, on the one hand, a derivation, and, on the other hand, simply presenting the result and inviting the reader to verify it, other things being equal the derivation is preferable, because, as well as confirming that it is a correct result, it also shows where the result comes from. Of course, other things are not always equal, but this is one more factor to take into account in deciding whether to prefer the restoration of the original content. JamesBWatson (talk) 20:16, 15 October 2013 (UTC)

No, neither James nor I are urging to reinsert "floating point implementation" back into the article. When I was working on the article, there was a modest plurality against this section, but since it was only a modest plurality, I went with my gut feelings. Currently we have a clear consensus against this section, and I can't deny consensus.
Perhaps a compromise would be to write something like this:
A detailed derivation of the quadratic formula may be found at its main article. In summary, application of steps 1 through 4 of completing the square yields
${\displaystyle \left(x+{\frac {b}{2a}}\right)^{2}=\left({\frac {b}{2a}}\right)^{2}-{\frac {c}{a}}}$
Taking the square root of both sides and rearranging the terms yields
${\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}}$.
Stigmatella aurantiaca (talk) 20:28, 15 October 2013 (UTC)
Sure, that would work. Incidentally, in the next section of this talk page there is discussion of the floating point stuff, which is where I got the idea that James supports reinsertion.Anythingyouwant (talk) 20:33, 15 October 2013 (UTC)
Seems we were all editing simultaneously and having edit conflicts. Stigmatella aurantiaca (talk) 21:48, 15 October 2013 (UTC)
Mind you, I don't particularly like the compromise, but let's show the world that we are a more functional group than Congress. :-) Stigmatella aurantiaca (talk) 22:58, 15 October 2013 (UTC)

(Outdent) I fully second JamesBWatson's arguments above. In addition, in my view a summary of a derivation only makes sense if the steps left out are trivial to the reader of the article. Instead of speculating on what steps might be considered trivial, one could simply look for this derivation in a few textbooks at this level. Here is one example: [6] I find the proposed compromise better than the current version because it refers to the detailed steps in the previous section. Nonetheless, I still don't understand why some steps have to be left out, especially since the quadratic formula is considered the most important tool for solving quadratic equations. Another possible compromise could be to derive the quadratic formula for the monic equation ${\displaystyle x^{2}+px+q=0}$ here and leave the derivation of the general quadratic formula for the main article. Isheden (talk) 08:29, 16 October 2013 (UTC)

(to Anything) Those would be my sentiments as well. Only in my case, I'd add that if you want to save space in this section, why bother dealing with the monic form at all? The quadratic formula for the monic is not general, nobody should waste brain cells memorizing it, and if you learn the general quadratic formula, you immediately know how to handle the monic. The verbiage saved by deleting the discussion of quadratic formula for the monic makes up for a fair proportion of that would be gained by restoring the full derivation, and the reader does not have to migrate away from the page if that is their main interest.
Another point is that the steps of the full derivation for the quadratic formula, as presented by Isheden, are carefully correlated with the steps for solving the quadratic equation by completing the square. This may be contrasted with the intermediate step presented in the Himonas textbook that you cited, which does not correspond to any intermediate step in any of my children's old textbooks or in the textbooks that I used when tutoring.
The version of the derivation as I had left it did not use numbered steps, but was nevertheless carefully correlated with the steps presented in the completing the square section. I remember debating with myself whether using numbered steps to relate to the completing the square presentation was better, or use of explanatory text. At the time, I felt that using explanatory text was better, but it was a tough decision, and Isheden's proposal has a lot of merit. It's certainly less verbose than my version. Stigmatella aurantiaca (talk) 12:31, 16 October 2013 (UTC)
Regarding the monic form: I wouldn't mind deleting it from the section in question. I just think it should be mentioned somewhere in the article. How about moving the section "Reduced quadratic equation" down and include the monic form of the formula there? Isheden (talk) 13:11, 16 October 2013 (UTC)
Regarding the intermediate step in the Himonas textbook: You'd get that by completing the square of the quadratic polynomial ${\displaystyle ax^{2}+bx+c}$ on the left hand side to ${\displaystyle a\left(x+{\frac {b}{2a}}\right)^{2}+{\frac {4ac-b^{2}}{4a}}}$. But that's not how completing the square is typically taught in the case of a quadratic equation. Isheden (talk) 13:27, 16 October 2013 (UTC)
Yes, of course. By the way, in case in wasn't clear from my previous writing, although several months back I omitted step numbering in the derivation so that I could add additional text, at this point in time I definitely prefer your approach. Stigmatella aurantiaca (talk) 19:49, 16 October 2013 (UTC)

I think the section title "Quadratic formula and its derivation" promises too much. Either include the derivation (in my view preferable) or change the title. Isheden (talk) 18:55, 25 October 2013 (UTC)

Agree. I think including Isheden's derivation is best. The derivation in Quadratic formula will not be rendered redundant by including Isheden's derivation here, because the Quadratic formula version includes additional useful commentary. Second choice (significantly inferior) would be a compromise such as I offered above. Stigmatella aurantiaca (talk) 19:42, 25 October 2013 (UTC)

## Floating-point implementation

I removed this section for two reasons. First it was not on topic: it was not on the quadratic equation so much as on numeric methods for dealing with algebraic equations. Second it was a long how to, and WP is not a how-to guide. The content specific to the quadratic equation is already covered in the Avoiding loss of significance section, so a lengthy worked example also on this is unnecessary.--JohnBlackburnewordsdeeds 18:26, 13 October 2013 (UTC)

This removal inadvertently breaks a reference in the Loss of significance article, so I will transfer the removed content to that article. I would disagree with you on whether the removed content belongs in Quadratic equation or not (I was responsible for much of it), but the general consensus appears to be on your side. Cheers! Stigmatella aurantiaca (talk) 19:48, 13 October 2013 (UTC)
I thought it was appropriate to have it in Quadratic equation. Where is the evidence that consensus is against it? In the sub-section of this page "Floating-point implementation & related issues" I see consensus for changing the presentation, and giving it less prominence, but far from seeing consensus for removing it from the article, I see a clear consensus for keeping it in, at least in some form. JamesBWatson (talk) 19:58, 15 October 2013 (UTC)
I have not examined Loss_of_significance#Instability_of_the_quadratic_equation very carefully, but does it contain all of the removed material? If so, then that would seem to be a further reason in support of removal (i.e. interested readers of this article can go get details at that article).Anythingyouwant (talk) 20:18, 15 October 2013 (UTC)
I did a crude insert of the deleted material into that article. Later when I have time, I will rewrite to reduce undesirable redundancy. Stigmatella aurantiaca (talk) 20:42, 15 October 2013 (UTC)
(to James) We are keeping in a brief mention of the loss-of-significance issue, but without any detailed examples. Stigmatella aurantiaca (talk) 20:42, 15 October 2013 (UTC)

## Too abstract for the intended audience

The section Solving the quadratic equation starts out like this: "A quadratic equation with real or complex coefficients has two solutions, called roots." In view of the intended audience, I think the article should focus on real roots in the beginning and discuss complex roots and coefficients later, perhaps in the subsection Discriminant. Complex coefficients are otherwise first mentioned in the Section Generalization of quadratic equation. Isheden (talk) 08:11, 17 October 2013 (UTC)

I agree 100%. JamesBWatson (talk) 12:10, 17 October 2013 (UTC)
Agreed. Even for someone familiar with complex numbers, it could be confusing. What's the advantage of saying a quadratic equation has two roots that are not distinct, instead of just saying it has one root?Anythingyouwant (talk) 15:28, 17 October 2013 (UTC)
A mathematician would prefer to be able to say, "An algebraic equation of degree n has n roots." Otherwise, s/he would have to say something stoopid like "A polynomial equation of degree n has n roots, except when it has fewer. But it's always going to have at least one, I think. Oh, you want me to say when? Duh...." :-) Stigmatella aurantiaca (talk) 20:31, 25 October 2013 (UTC)
Right, you and I know that but the reader might not, so something like your explanation might be good in the article.Anythingyouwant (talk) 20:48, 25 October 2013 (UTC)
Yep. - DVdm (talk) 20:15, 25 October 2013 (UTC)

The first statement is:

"Pythagoras and Euclid used a strictly geometric approach, and found a general procedure to solve the quadratic equation."

. There is indeed a similarity with [7], which says:

"The first attempts to find a more general formula to solve quadratic equations can be tracked back to geometry (and trigonometry) top-bananas Pythagoras (500 BC in Croton, Italy) and Euclid (300 BC in Alexandria, Egypt), who used a strictly geometric approach, and found a general procedure to solve the quadratic equation."

I have reworded the statement to:

"With a purely geometric approach Pythagoras and Euclid created a general procedure to find solutions of the quadratic equation."

The second statement is:

"In his work Arithmetica, the Greek mathematician Diophantus solved the quadratic equation, but giving only one root, even when both roots were positive."

This statement seems to technically legitimately backed by the cited source on the specified page ([8]):

"In solving quadratic equations Diophantus used only one root, even where both are positive."

I think there's no copyright problem here, so I have kept the sentence but cast the source into a proper template. - DVdm (talk) 09:54, 14 April 2014 (UTC)

## Factoring by inspection

I have reverted a recent edit by Matteo Rodolfo, and, by the way, copyedited the section. The old version was confusing for several reason: firstly the notation changed between first and second paragraph, with p and q denoting different things in the two paragraphs. Secondly, it was not clearly stated that the second paragraph was about seeking a factorization. The edit of Matteo Rodolfo, instead of clarifying the method add tot he confusion, by changing what is sought: it is no more a factorization, but directly the roots. Also, the edit changed "vieta's rule" into Vieta's formulas, without any explanation. Although the link is correct, it is of a much higher level than this elementary article. Therefore, such a link has to be provided with an explanation. On the other hand, I do not know if the term of "Vieta's rule" is standard. Therefore, I have tagged it as {{citation needed}}.

I believe that I have corrected most issues of the old version as well as of Matteo's one. If I am wrong, explain here what is wrong in my edits and do not start an edit war. Thanks. D.Lazard (talk) 22:06, 27 May 2014 (UTC)

## Weak Examples

Like most mathematical entries, this page is pretty fine, but the "Examples" section, which happens to be first, seems weak in comparison. The subject is very important, so relevant and captivating examples should be presented to introduce the concept. Physics books are full of them.

189.130.254.47 (talk) 17:45, 2 June 2015 (UTC) baden k.

## On completing the square

I think that this issue is settled. I just would like to say that I believe for the reader the suggested edits of October 2013 would be easier to deal with than the instructions given in English on this page. I am willing to admit that this is only an opinion and that I can follow the derivation as stated on the page.Nicodemus (talk) 22:18, 23 May 2017 (UTC)

Please, don't use caps in your section titles. See MOS:SECTIONCAPS. Thanks. - DVdm (talk) 06:22, 24 May 2017 (UTC)

## An Angle On The Modular Quadratic Equation (according to Gauss)

Says p207 of Dickson's "Theory of Numbers" vol 1: [1]

The congruence ${\displaystyle a*x^{2}+b*x+c{\bmod {m}}\equiv 0}$ is reduced (art. 152 [of Gauss' work]) to ${\displaystyle y^{2}{\bmod {4*a*m}}\equiv b^{2}-4*a*c}$. For each root ${\displaystyle y}$, it remains to solve ${\displaystyle 2*a*x+b=y{\bmod {4*a*m}}}$.

If ${\displaystyle y^{2}{\bmod {4*a*m}}\equiv b^{2}-4*a*c}$ is a natural square then the modular quadratic equation is solvable. In any case this above formula of Gauss changes the modular quadratic equation to a modular square root equation. Endo999 (talk) 05:35, 5 October 2017 (UTC)

... exactly in the same way as the quadratic formula changes the quadratic equation into a square root equation. D.Lazard (talk) 07:30, 5 October 2017 (UTC)

References

1. ^ "History of the Theory of Numbers" Volume 1 by Leonard Eugene Dickson, p207 url=https://ia800209.us.archive.org/12/items/historyoftheoryo01dick/historyoftheoryo01dick.pdf