Talk:Quadratic equation/Archive 3

numerical stability

The typical numerically stable method for solving the equation is neither two-root formula, but rather a little of both. Assuming the discriminant, b²−4ac, is positive and b is nonzero, we use something like

if b < 0 then

$t:=-(b-{\sqrt {b^{2}-4ac}})/2\,\!$

else

$t:=-(b+{\sqrt {b^{2}-4ac}})/2\,\!$

$x_{1}:=t/a\,\!$

$x_{2}:=c/t\,\!$

This is well-known standard practice in NA, and really should be fixed in the article, and a citation of TOMS or whatever added. Also, the prior article notation x₊ and x₋ strikes me as a bad idea, and would be a confusing nuisance for this algorithm. --KSmrq^T 06:47, 1 July 2006 (UTC)

It was I who introduced the x₊, x_– notation to replace the old x₁, x₂. I did so to try to clear up some confusion for some user about the difference between the ± symbol and its inverted form. I was never clear on why the original notation was confusing to that user, nor whether indeed there was any confusion at all. It may be, then, that the switch in notations serves no purpose, in which case switching back to 1,2 subscripts seems like a good idea to me. -lethe ^{talk +} 07:01, 1 July 2006 (UTC)

I'm writing a quadratic solver based on this equation. Forgive me if I'm just being dense, but what happens when a = 0? Do we just assume that x1 isn't real? Doesn't exist? And what about the unlikely case of t = 0? Do we assume the second root isn't real? Doesn't exist? --Numsgil (talk) 21:17, 19 December 2007 (UTC)

t can't be 0 since the precondition stated that b is not 0. And if a is 0, you don't have a quadratic equation, but linear: bx + c = 0, which of course has only one solution. -R. S. Shaw (talk) 07:13, 20 December 2007 (UTC)

what happens when c = 0? Wouldn't you then have t = 0? --Numsgil (talk) 01:55, 21 December 2007 (UTC)

No. Try it. Plug in some values, say a=1, b=2, c=0. Then try with b=−2. -R. S. Shaw (talk) 05:45, 21 December 2007 (UTC)

I get a divide by 0. Unless I'm missing something? I realize that you could just make a bunch of special cases for when different coefficients = 0, but then it wouldn't be a good general solution. --Numsgil (talk) 05:59, 21 December 2007 (UTC)

Oh, heh, nevermind. I see that you have two different equations depending on the sign, which ensures that you don't get cancellations down to 0. So the primary catch is to check that a != 0? Is that the only restriction? --Numsgil (talk) 06:02, 21 December 2007 (UTC)

GA review

1. Well written?: Ok for a Math article.
2. Factually accurate?: Fail. Even though I'm aware most of the info in the article is accurate, it does not have any inline citations or references. See WP:WIAGA.
3. Broad in its coverage?: Pass
4. Neutral point of view?: Pass
5. Stable?: Pass

I'll put this article on hold for seven days until the inline citations are provided. Nat91 22:14, 27 January 2007 (UTC)

I added in-line citations for the history info. The article is essentially proofs which are easily verifiable by just looking at it. In WP:GA? it indicates that sources used should be cited, but it is likely much of this was written from memory. Also, WP:V indicates citations are only neccesary when a claim is likely to be disputed, which I cannot see happening.--Jorfer 03:46, 28 January 2007 (UTC)

True. I'm not a fan of embedded citations, but overall I liked this article. I didn't know the quadratic formula had a name - I guess they never mention it because it's hard to pronounce? Passed GA! Nat91 18:59, 29 January 2007 (UTC)

Bhaskarachārya's Formula

A cite for this statement would be good. I've never heard it called that, and a quick Google search for "Bhaskarachārya's Formula" turned up three hits, one of which was this page. —The preceding unsigned comment was added by 199.17.27.38 (talk) 16:47, 30 January 2007 (UTC).

Your right. Only this page asserts that so I am removing it.--Jorfer 17:43, 2 February 2007 (UTC)

I've also removed this sentence:

This formula was first given by an Indian mathematician-astronomer named Bhaskara (1114-1185) [1], also known as Bhaskara II and Bhaskara Achārya ("Bhaskara the teacher"):

If accurate this needs to be incorporated into the "History" section. However note that that section already talks about Bhaskara, and gives a more qualified history.

Paul August ☎ 00:14, 3 February 2007 (UTC)

Pop Goes the Weasel?

Would anyone like to elaborate on exactly which words are sung to Pop Goes the Weasel? Perhaps this deserves its own page/section, or should simply be removed? As is its pretty useless. Personman 15:15, 20 April 2007 (UTC)

I learned it sung to a different song. I don't know what song it went to, but the words were...

"Opposite of b,

Plus or minus,

Square root of b squared,

Minus 4ac,

All over 2a;

You remember

The quadratic equation

That way."

Again, they weren't sung to Pop Goes the Weasel. I have no idea to what song they were sung to. Maybe it could be put on the page. That song has helped me remember the formula so far, and I haven't taken Algebra in 3 years. ForestAngel 10:46, 16 August 2007 (UTC)

The words are:

x equals negative b

plus or minus ra-dical

b squared minus 4ac

all over 2a!

It's not, you know, a good song, but that's the song.

FWIW I came to this page hoping to learn about the history of the equation and couldn't find anything. Who invented it? Why? What problems were they dealing with? How has did its creation influence art and science? It would be awesome if this article had a history section, many millions of schoolchildren have been mercilessly subjected to the quadratic equation and at least a few of us would like to know why. 96.231.158.234 (talk) 15:07, 4 June 2009 (UTC)

I've removed some text which I don't see as helpful

I've removed some text from the article which I don't see as helpful.

{Start of copied text)

Furthermore, in the quadratic equation, the first differential is equal to ± the square root of the discriminant. And the first differential of the quadratic expression equals the sum of its factors.

For example,

$\displaystyle Y\colon =7x^{2}-5x-2=0$

or

$\displaystyle Y\colon =(x-1)(7x+2)=0$

Then,

$\partial Y=\pm {\sqrt {b^{2}-4ac}}$

Therefore,

$\displaystyle 14x-5=\pm {\sqrt {81}}=\pm 9$.

And so,

$\displaystyle 14x=5+9$

and,

$\displaystyle 14x=5-9$.

Hence,

$\displaystyle x=1$

and,

$\displaystyle x={\frac {-4}{14}}={\frac {-2}{7}}$.

(End of copied text)

The idea of the first sentence "Furthermore, in the quadratic equation, the first differential is equal to ± the square root of the discriminant:", while not expressed very precisely, is easilly seen to be true, since for

$y=ax^{2}+bx+c\,$

then

${\frac {dy}{dx}}=2ax+b.$

While a simple algebraic manipulation of

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

gives:

$2ax+b=\pm {\sqrt {b^{2}-4ac}}.$

What this equation is really saying is that ± the square root of the discriminant is the slope of the graph of the quadratic function at its zeroes. But I'm not sure how useful this is here. Perhaps a simple sentence to this effect, could go in the "Geometry" section. The second comment that "the first differential of the quadratic expression equals the sum of its factors", I simply don't understand.

Paul August ☎ 23:06, 20 April 2007 (UTC)

GA Delisted

This article has been removed from the GA list due to excessive amounts of jargon as well as a lack of citations in many parts of the article. If you feel that this review was in error feel free to take it to WP:GA/R. Tarret 21:04, 4 August 2007 (UTC)

Could you please be more specific. Which terms do you considered to be "jargon"? Which statements need citation? Thanks. Paul August ☎ 19:15, 4 December 2007 (UTC)

'Which terms *aren't*, Paul? 69.171.160.236 (talk) 00:39, 10 September 2009 (UTC)

Merge proposal

I recommend merging Quadratic equation and Quadratic function. Despite this page being longer, I would prefer to merge into Quadratic function since it is easy to derive an equation from a function. —Celtic Minstrel (talk • contribs) 20:47, 6 December 2007 (UTC)

Hello Celtic Minstrel, I am slightly opposed to your project to rename/merge these articles. In my experience, most school children would be happiest talking about equations, rather than functions. Besides, the article is all about solving equations; there are many other things one can do with functions, differentiating and integrating them etc, which are not and should not be listed here.

I don't know why you think quadratic functions are more easily defined than equations. To start with, you have to start being precise about domains and ranges of functions, whereas this can be left ambiguous with equations. Also, the sentence "A cubic function is one of the form ..." is not very pleasant: by some definitions, a function is just a particular set of pairs of real numbers, so it would be more accurate to say "A cubic function is a function that satisfies ..." which is also rather ugly. What do you think? Sam Staton (talk) 17:23, 11 December 2007 (UTC)

I am strongly opposed to the merge. They are distinct and as Sam Staton says, it is important for student use that they are kept separate. --Bduke (talk) 22:45, 11 December 2007 (UTC)

I oppose the merge. The two notions are distinct. Xxanthippe (talk) 23:21, 11 December 2007 (UTC).

Okay, Sam Staton has a point - an equation is something you can solve, whereas a function can be differentiated or integrated. On the other hand, they are closely related concepts, and there is some similarity between them. I don't see anything on either page which does not apply to both concepts. Some things are duplicated on both pages. So even if the merge proposal is defeated as seems likely, I feel something should be done. I'm not quite sure what, though... —Celtic Minstrel (talk • contribs) 00:49, 12 December 2007 (UTC)

I'm opposed to the merge. There is duplication but some duplication is ok. Paul August ☎ 04:14, 12 December 2007 (UTC)

I don't think they can be merged. Equations and functions are different things. For example, the equation 2x^2=0 can be written to x^2=0. But the functions y=2x^2 and y=x^2 are totally differenT! —Preceding unsigned comment added by Netking China (talk • contribs) 08:13, 15 December 2007 (UTC)

Merge - They are similar enough where the quadratic function should be merged into this article as quadratic function is more specific then the term quadratic equation.--Jorfer (talk) 19:14, 5 February 2008 (UTC)

Hello Jorfer. I'm sorry, I don't understand your argument. Also, have you seen my comments above? I would be interested to hear your rebuttals of them. Sam Staton (talk) 20:26, 5 February 2008 (UTC)

Having two article just creates more and more duplication. It is more challenging to merge similar information than to create another article when needed. I am speaking from experience. All you have to do to create a new article is cut and paste an entire section while merging usually involves deleting repetitive information and adding small pieces of text. It is better to have a small amount of high quality articles then a lot of stubs forming a disorganized mess. The quadratic function falls under the scope of this article. I will react to your comments now:

"In my experience, most school children would be happiest talking about equations, rather than functions."

Wikipedia:What Wikipedia is not#Wikipedia is not a democracy. Just because the use of functions may not be popular does not influence how the matter is presented on Wikipedia. A lot of people may not like math, but that doesn't mean that all math articles should be deleted.

"Besides, the article is all about solving equations; there are many other things one can do with functions, differentiating and integrating them etc, which are not and should not be listed here."

Why not? Wikipedia is not an elementary textbook. It contains some challenging concepts. From the article Encyclopedia:

An encyclopedia, or (traditionally) encyclopædia, is a comprehensive written compendium that contains information on all branches of knowledge or a particular branch of knowledge.

The article on compendium:

A compendium is a concise, yet comprehensive compilation of a body of knowledge.

The word comprehensive means:

1. of large scope; covering or involving much; inclusive: a comprehensive study of world affairs.

Source: http://dictionary.reference.com/browse/comprehensive

"I would prefer to merge into Quadratic function since it is easy to derive an equation from a function." —Celtic Minstrel

"I don't know why you think quadratic functions are more easily defined than equations. To start with, you have to start being precise about domains and ranges of functions, whereas this can be left ambiguous with equations. Also, the sentence "A cubic function is one of the form ..." is not very pleasant: by some definitions, a function is just a particular set of pairs of real numbers, so it would be more accurate to say "A cubic function is a function that satisfies ..." which is also rather ugly. What do you think?" -Sam Staton

Wikipedia:NPOV. Wikipedia is not to take sides on whether equations are better than equations. This goes for both of you. These articles should be merged, but not for the reason Celtic Minstrel first proposed.--Jorfer (talk) 22:30, 5 February 2008 (UTC)

Thanks for your messages, Jorfer. Correct me if I have misunderstood: I think you are advocating a merge because you think the two articles overlap a lot. In fact, they don't; the only overlap is the small section Quadratic function#Roots. The article on Quadratic Equations can be thought of as an expansion of that section. Perhaps it should be called "Quadratic Equations and their Solutions" or "Roots of Quadratic Functions". It is quite usual to have a full article that expands on the material of a small section. In the present case, we can give derivations of the formula, history of it, etc.. These would be too big a digression on the page for quadratic functions.

If the article had one of those longer titles, would you be happier? Or we could put "See main page Quadratic equation for a fuller discussion." at the top of Quadratic function#Roots.

I don't want to get into a pernickety argument, but I don't think your reference to democracy is appropriate. I am not proposing a vote about anything. See for instance WP:NC(CN): when deciding on the name of article, one should ask "What word would the average user of the Wikipedia put into the search engine?". Also, I don't think your reference to WP:NPOV is appropriate. Nobody is saying "functions are better than equations". We are saying that the two concepts are different, and deserve different articles. All the best, Sam Staton (talk) 11:47, 6 February 2008 (UTC)

I oppose the merge. Functions and equations may or may not be the same, depending on their use. Daniel —Preceding unsigned comment added by 74.161.3.161 (talk) 14:46, 6 February 2008 (UTC)

I am not saying that there is an overlap, but it is likely to happen in the future should the two stay seperate. I don't think a longer title is needed. A function can always be expressed using one or more equations (a scatter plot can be represented by multiple equations), so it is entirely appropriate for the more specific aritcle (quadratic function) to be placed as a section in the more broad article (quadratic equation). An equation is not always a function, but a function can always be expressed using equations. If the articles are not merged then WP:Summary should be used. Using this popularity checker (link blacklisted), the quadratic equation article is linked to twice, while the quadratic function article is not linked to at all, so it would be appropriate to merge quadratic function into this article. On your argument that you are not arguing whether functions are better than equation, I just want to point out that the use of the term "ugly" demonstrates subjectivity rather than objectivity and the use of "most school children would be happiest" demonstrates ad populum which carries with it the spirit of the Wikipedia rule if not the rule itself. Looking back Celtic seems to simply be advocating the placing rather than which is better, but all math can be derived from somewhere else is a ridiculous reason to merge equation into function. The more specific should go under the more broad. Most mathematicians would group these two concepts together as they are easily translatable from one form to the other, so the article should reflect that.--Jorfer (talk) 18:48, 6 February 2008 (UTC)

Hi Jorfer, I agree that we should keep an eye on the situation. I agree that quadratic functions can be expressed using equations, but I'm not sure that this means that they must be combined into one article. The last two sections on the Quadratic function don't have much to do with equations. Anyway, I've put some more links in, which I hope will clarify things for the time being. Sam Staton (talk) 13:44, 8 February 2008 (UTC)

"Most mathematicians would group these two concepts together as they are easily translatable from one form to the other, so the article should reflect that." – I guess this is the sort of thing I was thinking of when I proposed the merge. The two concepts are very closely related. I suggested merging into Quadratic function rather than merging into Quadratic equation simply because if you have a function $f\left(x\right)=3x^{2}+12$ (or any other algebraic expression), you can derive an equation by fixing something. For example, you could set $f\left(x\right)=0$ and then you get the equation $3x^{2}+12=0$. Or you could set $f\left(x\right)=y$ to get $y=3x^{2}+12$. Is there a similarly easy way to derive a function from an equation? I suppose it wouldn't really be all that hard... –Celtic Minstrel (talk • contribs) 14:09, 8 February 2008 (UTC)

Actually, when you set y as 0 you get an imaginary number, but I understand your point. Your thinking as if all equations are derived from functions. You would not be able to derive $y=\pm {\sqrt {x}}$ directly from a function due to the strict definition of a function. $3x^{2}+12=12$ (not using your example due to imaginary number) could be a result of $f\left(x\right)=3x^{2}+12$, but it could also be the result of $9x^{2}+36=y$ where y=36, or it could just be a way of expressing x=0. If you have a function on the other hand, it is always the result of an equation. $f\left(x\right)=3x^{2}+12$ can be derived from $y=3x^{2}+12$, $y-12=3x^{2}$, $z-12=3x^{2}$, $3t-36=9x^{2}$, and so on. In fact $f\left(x\right)=3x^{2}+12$ is an equation, while $12=3x^{2}+12$ is not a function.--Jorfer (talk) 00:09, 9 February 2008 (UTC)

Does anyone object if we close this merge proposal now, and leave the pages as they are - for the time being at least? Sam Staton (talk) 13:21, 15 February 2008 (UTC)

Good idea. -R. S. Shaw (talk) 20:01, 15 February 2008 (UTC)

I've removed the "merge" tags from both articles. Paul August ☎ 04:50, 14 March 2008 (UTC)

Don't merge Quadratic equation is about the roots of a quadratic function. In that sense quadratic equation can be grouped into quadratic function, but the quadratic equation (and the formula for its solution, plus completing the square) are so important in their own right, they need their own article. Quadratic function should be about much more, than just the roots. --345Kai (talk) 19:46, 3 April 2008 (UTC)

I learnt more about what a quadratic equation is from this discussion than from the article

I think that says something about the quality of the article and its obsession with the use of technical terms at the expense of enlightenment for the masses of non mathematicians. By all means include the technical stuff, but how about a bit of English for the rest of us —Preceding unsigned comment added by 81.187.233.172 (talk) 14:03, 25 October 2008 (UTC)

I totally agree. Besides, within the 2nd section Quadratic_equation#Discriminant It already stops making sense. 64.231.121.118 (talk) 03:09, 5 November 2008 (UTC)

This article is far to difficult for a layman to understand. The very first 'example' shows a quadratic equation with some other complex maths stuff added in. Why can't articles be written for ordinary folk to get to grips with and then further expand into the more difficult to understand stuff? —Preceding unsigned comment added by 195.59.180.153 (talk) 10:10, 20 November 2008 (UTC)

I fully agree with this. The problem with mathematics explained by mathematicians is that they don't seem to explain the use of formulas. To non-mathematicians this all seems like it must be a game like sudoku with no redeeming real-world applications. An encyclopedia should contain information that is accessible to all comers, not some kind of proof that the authors know some esoterica. Without giving a real-world context, all you end up with is logical meaninglessness. General Ludd (talk) 02:20, 8 July 2010 (UTC)