Talk:Quadratic formula

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This article was the subject of a Wiki Education Foundation-supported course assignment, between 7 September 2020 and 18 December 2020. Further details are available on the course page. Student editor(s): AH-2031.

Above undated message substituted from Template:Dashboard.wikiedu.org assignment by PrimeBOT (talk) 07:34, 17 January 2022 (UTC)

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:
 * $$\left ( \frac{-b}{2a},\frac{4ac-b^2}{4a} \right )$$

Focus:
 * $$\left ( \frac{-b}{2a},\frac{4ac-b^2+1}{4a} \right )$$

Directrix:
 * $$y=\frac{4ac-b^2-1}{4a}$$

...but I'm not insisting on it; just a suggestion. 2601:545:8201:6290:E445:9D52:D8D3:145F (talk) 23:29, 24 October 2020 (UTC)

Shreedharayacharya's formula
In India, quadratic formula is popularly known as Shreedharacharya's formula (or Sridharacharya's formula) as it was first given by an ancient Indian mathematician Shreedharacharya (or Sridharacharya) or Sridhara around 1025 A.D. Huzaifa abedeen (talk) 14:23, 4 November 2020 (UTC)

Citation needed
Does there really need to be a citation at the end of the subsection "Method 3"?  AltoStev Talk 12:43, 27 October 2021 (UTC)
 * Not really. But I did some cleaning up, removing reduncancies. There was nothing in the first part of subsection "Method 2" and in subsection "Method 3" that wasn't already there. - DVdm (talk) 13:33, 27 October 2021 (UTC)

Sign of discriminant never mentioned ?
I noticed that, oddly, the article never seems to mention that the formula $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$ is only valid if the discriminant $$b^2 - 4ac$$ is positive. Otherwise the right hand side of $$(2ax + b)^2 = b^2 - 4ac$$ has no real root and the notation $$\sqrt{b^2 - 4ac}$$ makes no sense (unless you explicitly define the notation $$\sqrt{x}$$ as designating one of the two complex roots of $$x$$, but to my knowledge this notation is usually considered forbidden for numbers other than positive reals). Isn't this an important omission? So I tried to fix it --193.52.194.235 (talk) 11:10, 21 February 2022 (UTC)


 * Complex numbers exist... Joao003 (talk) 12:33, 17 May 2023 (UTC)
 * And if it's 0, you can just use one solution.......... Joao003 (talk) 12:36, 17 May 2023 (UTC)
 * Yes, $√b2 − 4ac$, doesn't have an obvious principle value when $b2 − 4ac$ is not a non-negative real number. However $±√b2 − 4ac$, which is used in the quadratic formula, does meaningfully specify both square roots for all complex numbers.  (Well, it's a double root for $b2 − 4ac = 0$, but you know what I mean.)  — Q uantling (talk &#124; contribs) 12:57, 17 May 2023 (UTC)

Savage16 20
There is a user named savage 1620 who had recently vandalised wikipedia page we request him to be blocked immediately David dclork li (talk) 13:38, 28 April 2023 (UTC)

Oh my... I'm sorry. I was to delete my post but for some reason intruded into yours and am not sure if I removed any of your text. The Wiki's editing system is something else... (It's bad.) I apologize. --ErrorCorrectionOfficer (talk) 00:22, 4 May 2023 (UTC)

The Egyptian Berlin Papyrus
I undid your edit because it left us with the same sentence twice in a row. I suspect that your intent was otherwise; please make another edit accordingly. — Q uantling (talk &#124; contribs) 12:54, 18 May 2023 (UTC)
 * I think I figured out the intended edit, and have made it. If I have failed, please accept my apologies.  — Q uantling (talk &#124; contribs) 13:42, 18 May 2023 (UTC)
 * Nope, that's what I intended to do. Thanks! Graham 87 14:27, 18 May 2023 (UTC)

A dubious equivalent formulation
On the Quadratic formula section it says that the quadratic formula can be reduced to

$$x = \frac{-\frac{b}{2} \pm \sqrt{\left(\frac{b}{2}\right)^2-ac}}{a}$$It says that its dubious, which is quite strange, since it can be easily derived from the original quadratic formula:

$$\frac{-b \plusmn \sqrt{b^{2}-4ac}}{2a} = -\frac{b}{2a} \plusmn \frac{\sqrt{b^2 - 4ac}}{2a} = -\frac{1}{a} \frac{b}{2} \plusmn \frac{1}{a} \frac{\sqrt{b^2-4ac}}{2} $$

$$= -\frac{1}{a} \frac{b}{2} \plusmn \frac{1}{a}\frac{\sqrt{b^2-4ac}}{\sqrt{2^2}} = -\frac{1}{a} \frac{b}{2} \plusmn \frac{1}{a} \sqrt{\frac{b^2 - 4ac}{2^2}} $$

$$= \frac{-\frac{b}{2} \plusmn \sqrt{\frac{b^2}{2^2} - \frac{4ac}{2^2}}}{a} = \frac{-\frac{b}{2} \plusmn \sqrt{(\frac{b}{2})^2 - ac}}{a} $$

I suggest removing the [dubious - discuss] tag, since its easily provable that this formulation is equivalent

Sincerely, ALonelyPhoenix (talk) 19:12, 21 September 2023 (UTC)


 * I made some edits to remove the dubious claim that these formulas are more useful when using a calculator. Now it says that they "may" be more useful.  Also, I removed an intermediate result that wasn't useful, and generally shortened the section.  What do you think? — Q uantling (talk &#124; contribs) 19:26, 21 September 2023 (UTC)
 * Yeah, now its less ambiguous and more compact, considering the intermediate result could be directly obtained by one factorization and the [dubious] tag made it seem like those two weren't equal lol
 * Thanks! ALonelyPhoenix (talk) 23:53, 21 September 2023 (UTC)
 * I took this version out entirely (leaving just one variant involving $$\tfrac{b}{2a}$$), since it wasn't clear what the point was of having multiple trivially distinct variant formulas, and no source was given for any of them. –jacobolus (t) 20:37, 18 February 2024 (UTC)

High-level structure
Hi @Quantling, I reverted your move of the 'history' section up to the top, because I don't think it's the friendliest start for an audience of laypeople or middle-/high-school algebra students. (I also couldn't find any reliable sources with "Sridharacharya's formula" and even in a general web search it seems pretty rare; I replaced the bold with italic.)

But the current first section is also not very good to lead with. My hope is to expand the section, maybe retitled something less specific like "Meaning" or "Interpretation" or "Significance", and move that to the top. This section could elaborate about the meaning of the discriminant. It could maybe include "Special cases" as a sub-topic, which would try to demonstrate both symbolically and graphically what happens if a, b, and/or c = 0, what happens in the limit as these go infinite, etc..

The 'derivations' section was at some point in the past broken into two parts, with one ("completing the square") derivation on its own at the top, then other derivations later on in a separate section. Returning to that structure seems to me at least worth considering: Readers are likely to want to see some variant of the most common derivation near the top, but more examples can pretty safely be deferred in my opinion.

The history section can be much improved. It should be more specific/explicit about ancient Mesopotamian, Egyptian, Greek, Chinese, and Arabic/Persian methods, should talk about the way historically all of the quantities involved were considered to be non-negative, etc. It should elaborate about the specific contributions of Stevin, Descartes, and others, instead of just dropping names.

We should also ideally have some kind of section(s) about generalizations, applications, etc. down toward the bottom. And perhaps a section with some discussion about roots of quadratic equations with complex coefficients, maybe with image(s) showing a domain coloring plot of the complex plane.

I think the sub-section should probably be pulled out of derivations (where it is similar to the previous subsection) and made into a new section nearer the bottom titled just "Galois theory" or "Lagrange resolvents".

We also could use more figures (and a replacement of the current SVG figure, which has broken rendering). I might make some figures using Desmos (probably just leave them as highish resolution png images), and link through from the image description pages for interactive versions.

–jacobolus (t) 19:04, 19 March 2024 (UTC)


 * I like the history section as first after the lede. Strangely, it is because it is not like the other sections and thus both, it makes the needle easier to find for those who are looking for it in the haystack, and it is easy to scroll past because it is smaller than the other sections combined, for those who are not looking for it.  I admit that that isn't the strongest logic ... but perhaps we don't have to reinvent the wheel here, and there is some general Wikipedia advice on this topic?  Also, over the years there have been many editors wanting to add Sridharacharya's formula or Sridharacharya's method (often in bold, though always reverted or otherwise diminished) so my gut tells me that it is real, and that there is value to having it somewhere near (or possibly in) the lede.
 * I agree that generalizations should be at/near the end. And proofs could be in a single section.  — Q uantling (talk &#124; contribs) 20:19, 19 March 2024 (UTC)
 * I don't mind mentioning Sridharacharya's formula or similar, and I don't doubt that someone calls it this, but from the available sources I could find in a web search, I don't think the bold is justified: this doesn't seem like a widely adopted name, and I frankly don't think the attribution is justified considering how many different people through history have made various contributions to this topic. (Can you find any reliable sources mentioning this name specifically?)
 * I similarly took bold out of théorème d'Al-Kashi in (and took a mention thereof out of the lead section of that article), while also dramatically expanding the text so it's clear what Al-Kashi's version actually said and what it's relation is to the "modern" version, which I think is more useful than just naming things for some particular chosen person who made relevant contributions.
 * The section here should similarly be significantly longer (3x the current length or more) to accurately and somewhat completely cover the topic, and I think it's substantially distracting to have as the first thing mentioned for an imagined struggling 14-year-old arriving here to figure out about the quadratic formula for their algebra class.
 * And proofs could be in a single section. – That is, you don't like my proposal that we consider putting one derivation near the front and deferring alternative derivations to further down the page? –jacobolus (t) 00:34, 20 March 2024 (UTC)
 * ... putting one derivation near the front and deferring alternative derivations to further down the page. That works for me.  Apologies for the confusion. — Q uantling (talk &#124; contribs) 17:19, 20 March 2024 (UTC)
 * Okay, I did such a split. Does that read okay? I think we should try to make a graphical depiction of completing the square section somewhat along the lines of File:Completing_the_square.svg, but using the expressions for a generic quadratic equation. I find the video File:Completing the square.ogv somewhat confusing though. –jacobolus (t) 03:45, 23 March 2024 (UTC)
 * Among the first few hits from Google, I see Sridharacharya Formula, Sridharacharya Formula - Definition, Derivation, Proof with Example, Shreedhara Acharya's formula, Why is the quadratic equation also called the Shri Dharacharya formula?. Certainly not as many hits as "quadratic formula" would give, but perhaps more than any other " formula" search would give for this formula.  Is there a way to find out how many users have searched the English Wikipedia for "Śrīdharācāryya formula", "Śrīdharācāryya method", or any of the spelling variations? — Q uantling (talk &#124; contribs) 17:32, 20 March 2024 (UTC)
 * Yeah, but these hits are a quora answer, a wiki largely duplicative of the material at Wikipedia, and two different exam coaching provider's websites who probably sourced their material from a web search. None of these are "reliable sources" by Wikipedia standards. I don't disagree that someone somewhere calls it this, but it doesn't seem like a very widespread name (at least, not in the scholarly literature). The reliable sources I have seen discussing "the Hindu method" or "Śrīdhara's rule" (e.g. Gupta 1966, Renfro 2007) are referring to the approach to completing the square by first multiplying by 4 times the quadratic coefficient to avoid fractions, not to the "quadratic formula" as such. (Though if you start with generic coefficients and then follow Śrīdhara's method of completing the square, or any other generic method of solving a quadratic equation, what you end up with is of course something equivalent to the quadratic formula.) –jacobolus (t) 17:46, 20 March 2024 (UTC)
 * I changed the heading of the relevant derivation to . –jacobolus (t) 05:01, 23 March 2024 (UTC)

Discriminant with a bullet list
@Number 3434, I reverted your change to make the paragraph in the lead about the discriminant into a bullet list. I think this is too visually heavy and distracting at that spot, but it might be worth adding a separate section discussing it in greater detail (though I still would recommend against using a bullet list where paragraphs will do). There used to be a bullet list in the version of this article from a few months ago, special:permalink/1208763608, but I found the lead to be too long; remember, some readers just want a quick overview. You might also want to try expanding the part about quadratics at Discriminant or the part about disciminants at Quadratic equation. –jacobolus (t) 16:02, 30 April 2024 (UTC)