Talk:Quadratic formula/Archive 1

This is an archive of past discussions about Quadratic formula. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page.

Archive 1

Untitled

So how do you solve quadratics in characteristic 2? AxelBoldt 19:38, 11 February 2003 (UTC)

No longer a redirect

More than a decade later, there's plenty for a full article instead of a redirect.Anythingyouwant (talk) 06:26, 5 October 2013 (UTC)

Revert to old black-and-white image

I'd like to start by saying that I, and I'm sure many others really appreciate User:Anythingyouwant's editing on this page recently. That being said, I would like to revert to the previous image used on the page. It was changed into an orange scroll from its former, simpler image. ([1]) I'd love to hear some feedback before going ahead and changing it.

Jdc1197 - (talk · userpage · contributions) 17:48, 15 October 2013 (UTC)

Sure, either way is fine with me. And thanks for the star.Anythingyouwant (talk) 18:06, 15 October 2013 (UTC)

Motivation

Until yesterday, the article said this about Brahmagupta:

“

What motivated him? Mathematics Professor Elizabeth Stapel answers that question:[2]:

Somebody (possibly in seventh-century India) was solving a lot of quadratic equations by completing the square. At some point, he noticed that he was always doing the exact same steps in the exact same order for every equation. Taking advantage of one of the great powers and benefits of algebra (namely, the ability to deal with abstractions, rather than having to muck about with the numbers every single time), he made a formula out of what he'd been doing....

”

This seems like useful stuff to me. The edit summary deleting it said: "Brahmagupta's motivation unimportant to discussion of quadratic formula. Belongs in historical discussion of Brahmagupta personally." But this isn't merely biographical material about that person; it's an explanation of why he made his discovery. Motivation for a discovery is often as interesting as the discovery itself. So, I will restore it, a bit lower in the section, along with some of the other deletions made without any explanation.Anythingyouwant (talk) 16:45, 13 December 2013 (UTC)

Order and signs of solutions

Here is a table with properties of the real solutions that I find useful.

With $x_{1}={\frac {-b+{\sqrt {b^{2}-4ac}}}{2a}},x_{2}={\frac {-b-{\sqrt {b^{2}-4ac}}}{2a}},d=b^{2}-4ac$ :

Order of solutions
x1 > x2 if a > 0 && d > 0
x1 < x2 if a < 0 && d > 0
x1 == x2 if a != 0 && d == 0

Sign of x1 (assuming d >= 0)
x1 > 0 if
(a < 0 && b > 0 && c < 0) || ((b < 0 || c < 0) && a > 0)
x1 < 0 if
(a < 0 && (b < 0 || c > 0)) || (a > 0 && b > 0 && c > 0)
x1 == 0 if
c == 0 && a != 0 && b >= 0

Sign of x2 (assuming d >= 0)
x2 > 0 if
(a < 0 && (b > 0 || c > 0)) || (a > 0 && b < 0 && c > 0)
x2 < 0 if
(a < 0 && b < 0 && c < 0) || ((c < 0 || b > 0) && a > 0)
x2 == 0 if
c == 0 && a != 0 && b <= 0

Combined signs (assuming d >= 0)
x1 > 0 && x2 > 0 if
(a < 0 && b > 0 && c < 0) || (a > 0 && b < 0 && c > 0)
x1 < 0 && x2 < 0 if
(a < 0 && b < 0 && c < 0) || (a > 0 && b > 0 && c > 0)
x1 > 0 && x2 < 0 if
a > 0 && c < 0
x1 < 0 && x2 > 0 if
a < 0 && c > 0
x1 == 0 && x2 == 0 if
b == 0 && c == 0 && a != 0

--93.220.53.23 (talk) 19:21, 18 April 2014 (UTC)

Pretty Pictures

Uh... did you say, "stubbish" or "rubbish"?! :) ... Well, "rubbish" is a bit harsh, but I'm shocked, SHOCKED, I tell you!, that the only graphical mathematical representation is not a representation of the parabola crossing the x-axis!! That's what it's all about BABY! Don't make me do it! I will look like crap! :) 192.0.204.219 (talk) 14:45, 26 November 2014 (UTC)

random question

"determinant" or "discriminant"? Search within Wikipedia seems to indicate that the latter is correct. 192.0.204.219 (talk) 14:45, 26 November 2014 (UTC)

Section on importance

As the article is being prepped for GA status I want to draw your attention to the section entitled Importance of this solution. I find the writing in this section to be very sophomoric. Furthermore, it doesn't seem to say much beyond – This is important, so you'd better memorize it! I would contemplate removing it, but I wonder whether or not it can be salvaged. The formula is one of, if not the most, significant result(s) in elementary algebra and many authors do recommend that it be memorized ... surely we should be able to explain why this is so to a general audience. There is also a certain sensitivity to this issue, as the formula is often brought up as an example of the useless mathematics being force fed to the general populace by some members of our society who would claim that they never had to use it after their high school math class. Any thoughts or suggestions about what to do with this section? Bill Cherowitzo (talk) 05:12, 28 March 2015 (UTC)

I think this section could be removed, as you discuss. The similar Pythagorean theorem is a GA with no such section. However, maybe it can be replaced by a small section called Applicationsor something similar, which could discuss some of the other equations that can be put in quadratic form (perhaps the classic example of finding when a thrown object hits the ground). The applications section could contain a reference to those who say the quadratic formula is not particularly useful, because it is a notable claim.Brirush (talk) 11:30, 28 March 2015 (UTC)

Well, I tried a rewrite of the section into a "Memorization" section giving points for and against memorizing the formula. Let me know what you think. I didn't do the applications, as it didn't seem to be worthwhile.Brirush (talk) 20:22, 29 March 2015 (UTC)

A good start, but it does need some fleshing out. I'll try to help out if I can find good sources. I'm not sure that applications are needed, but I'd like to beef up the importance aspect ... beyond just having some folks say that it is important. Bill Cherowitzo (talk) 02:37, 30 March 2015 (UTC)

Renominating as Good Article

I have made a number of small changes which I believe address the main outstanding issues raised with this article when it was initially nominated for good article status. Many of the 'we's have been left as is, since this kind of voice is common in mathematical literature (this caveat is mentioned explicitly in WP:TONE), and now that the references to 'the student' have been removed, the voice seems suitable. As such I believe it is time to renominate this article for good article status. Is that a good idea? --♫CheChe♫ talk 15:53, 19 October 2016 (UTC)

Error in the general equation.

The article states that the general form of the quadratic equation is "ax2 + byy + c = 0" ... the "2" should be a superscript, but I don't feel like figuring it out.

Shouldn't the 2nd term be "bx"? — Preceding unsigned comment added by Catbirdfan (talk • contribs) 17:47, 5 March 2018 (UTC)

This was put in by a disruptive editor. It is now fixed.--Bill Cherowitzo (talk) 18:18, 5 March 2018 (UTC)

Please add something =

About when the part under the square root is negative. If it's not allowed, or not possible for some reason, please say that. I know there are rules about this but I can't remember them so it seems an appropriate thing to add. Thanks! — Preceding unsigned comment added by BruceEckel (talk • contribs) 22:28, 27 April 2018 (UTC)

Also, many of the equations have a "." after them. I would have taken this off but I'm not sure if that's a Wikipedia style. — Preceding unsigned comment added by BruceEckel (talk • contribs) 20:44, 28 April 2018 (UTC)

Visualizing the page on the wikipedia app

Take for example (though there are many occurrences) the text "From this we can see that the sum of the roots of the standard quadratic equation is given by" which is followed by "-b divided by a". Using the wikipedia app, on a Huawei Android, the symbols for "-b divided by a" come out very badly. It looks more like "-slash divided by a" with a "b" floating somewhere just above. I don't know if this is a problem with the page or with the app. Masonmilan (talk) 12:24, 19 November 2018 (UTC)

Diagram Request

I am removing the request for a Mathematical Diagram placed in 2015 because it seems as though an appropriate diagram has been provided since then. Simply Another Editor (talk) 04:29, 18 December 2018 (UTC)

Untitled

Seeing as:

this article is fairly stubbish
so is Quadratic equation
I'm planning on writing up other methods of solving quadratics, namely factorization and completing the square

I'm planning on merging content from here into Quadratic equation and leaving a redirect. Does anyone have any objections? I'll leave this a week (note that a Wikipedia-Week, for me at least is more like a month...) -- Tarquin 16:37, 9 September 2002 (UTC)

Sounds good to me. AxelBoldt 02:35, 10 September 2002 (UTC)

Seems like there are a lot of redundant derivations of the quadratic formula here and it ignores the more basic insight. That is, the first three derivations are essentially the same. The underlying mathematical realization is that x^2-q is easier to solve for roots than ax^2 + bx + c. Completing the square, translating the graph by the coordinate of the vertex, and the substitution method are essentially the same method other than the realization that completing the square and substitution are algebraic and you have to motivate the knowledge of the vertex outside of the theory, i.e. through calculus. Then you go on to explain the calculus based derivation and that along with he second method is completely redundant.

This is a really nice article if you are studying the algorithms for deriving the quadratic formula but a terrible one in terms of mathematical insight. With the exception of the resolvants part. Zoratao (talk) 16:22, 17 January 2019 (UTC)

precede plus-or-minus with separate expressions?

Opinions would be appreciated on the idea of presenting the formula in two parts in the lead, and only afterward saying that the plus-or-minus symbol is a common abbreviation.

Here's why I ask: When classes of sophomores in a fairly serious high school are asked to program the formula as an introductory exercise in a computer-programming language, it is routine that many people will wonder "How do I program the plus-or-minus operator?" They have all been exposed to the formula for a couple years, but they are unclear that the "operator" is used as a convenient way to write two different expressions, one with an addition and the other with a subtraction. People at that level of confusion seem like exactly the audience that could benefit from this article.

Someone has already kindly typed the separate expressions and explained the meaning of plus-or-minus, in "Method 1" for deriving the formula. But I'm guessing that a confused person is unlikely to delve into the derivations. So I'm mildly recommending moving the explanation into the lead.

On the other hand, some editors may regard the plus-or-minus form as hallowed by tradition. So I'd hate for them to choke on their morning beverage if they woke up to find that it had been relegated to second place by an overly-bold newbie like me.

Thanks for any help. --DavidHolmes0 (talk) 16:18, 19 April 2019 (UTC)

Not likely to choke on my morning coffee, but it seems to me that you would be tossing away a very teachable moment with this suggestion. The ± symbol is used extensively in scientific expressions. If students don't encounter it here (in the quadratic formula) they will come across it later on when the context might not be as transparent.--Bill Cherowitzo (talk) 03:50, 20 April 2019 (UTC)

Po-Shen Loh

Where in the article could we add the method discovered by Po-Shen Loh? This new method has been covered in Technology Review and Popular Mechanics. The first steps seem to be identical to the ones mentioned in § By Lagrange resolvents --Distelfinck (talk) 17:55, 10 December 2019 (UTC)

I would say it's probably not worth it to add. –Deacon Vorbis (carbon • videos) 18:30, 10 December 2019 (UTC)

You don't think the method is simpler and more intuitive? --Distelfinck (talk) 19:55, 10 December 2019 (UTC)

Not especially, nor does it really seem all that novel compared to other methods. But even if I did, that doesn't mean we should necessarily include it. –Deacon Vorbis (carbon • videos) 20:44, 10 December 2019 (UTC)

Thanks to your link I found this: The new method had already been added to the article, and was removed due to lack of secondary sources. --Distelfinck (talk) 21:44, 10 December 2019 (UTC)

There is a secondary source now. Does this mean we can now include this method? I personally like it and it appears to be genuinely different - I mean the idea of using the average of the roots to actually derive their values. I also think that it may even be the easiest proof of them all. (Admittedly, completing the square seems like a quite simple technique to me, but it is nice that you don't actually have to do this to prove the quadratic formula, and I was not aware of this.) KarlFrei (talk) 08:17, 17 December 2019 (UTC)

This is nothing new, I should add. It's merely using Vieta's formulas, and in the general case with a non-monic polynomial, the derived formula is absolutely no improvement whatsoever over the usual method. I sense that the general method continues to be taught today for that reason.--Jasper Deng (talk) 13:39, 17 December 2019 (UTC)

I realize that Vieta's formulas are not new, and that the final result is the same (as to how could it not be). Nevertheless, the step of expressing the roots in terms of their average and calculating them immediately from that is unusual. It is a straightforward and natural way of solving the problems of type "product is this, sum is this, what are the values?" which students typically do by trial and error. (As to it "still being taught today", the paper has only been out for two months! For changes in education we probably need to wait more like two decades :-) ). KarlFrei (talk) 09:20, 18 December 2019 (UTC)

That doesn't look like a reliabe source in the Wikipedia sense. - DVdm (talk) 11:18, 17 December 2019 (UTC)

Hardly. To find out what reliable sources are for Wikipedia, see wp:reliable sources. - DVdm (talk) 13:31, 17 December 2019 (UTC)

How do you feel about the New York Times? KarlFrei (talk) 13:03, 6 February 2020 (UTC)

I don't think that the NYT qualifies as a standard textbook or as a relevant professional journal. If this is really important, textbooks will pick it up sooner ot later. Give it five years or so... - DVdm (talk) 13:10, 6 February 2020 (UTC)

I find it sad that you are putting such a high bar before inclusion of this material in Wikipedia. It is a nice proof and clearly correct (as everybody can check for themselves), so I find your insistence on having exactly the right kind of reliable source before this can be included rather strange. The result was apparently known in some form for over 2,000 years. You seem to feel that it somehow harms this article if this proof were added to it. I do not understand this at all. KarlFrei (talk) 10:49, 25 February 2020 (UTC)

Correcting and adding information

I corrected the information about types of roots based on value of discriminant and added information about sum and products of the roots with citations:^[1]

^[2] ] --AH-2031 (talk) 16:25, 2 October 2020 (UTC)

References

^ Schuloff, A. (2007). COMPLEX CONJUGATES. The Mathematics Teacher, 100(6), 391-391. Retrieved October 2, 2020, from http://www.jstor.org/stable/27972264
^ Hong, D., & Mi Choi, K. (2014). A comparison of Korean and American secondary school textbooks: The case of quadratic equations. Educational Studies in Mathematics, 85(2), 241-263. Retrieved October 2, 2020, from http://www.jstor.org/stable/43589818

Could mention simpler expressions for vertex, focus, and directrix

The article could also mention somewhat simpler expressions for the y-coordinates of the vertex and the focus, and for the equation for the directrix, avoiding showing quantities in parentheses, and the negation thereof, based on the negation of the discriminant.

Vertex: