Talk:Quadratic equation

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)

I think it should be "equations with real or complex coefficients" as opposed to "equations with real and complex coefficients", unlike what user KestrelOshin (talk · contribs) wants to say. "Equations with real and equations with complex coefficients" would be ok, but here we talk about the individual coefficients, which are real or complex. I have restored the standing wp:consensus version and warned the user about wp:edit warring. - DVdm (talk) 06:33, 31 May 2018 (UTC)

The second sentence in this passage seems to contradict the first, and sounds unbelievable.

In 1637 René Descartes published La Géométrie containing the quadratic formula in the form we know today. The first appearance of the general solution in the modern mathematical literature appeared in an 1896 paper by Henry Heaton.

What exactly is Heaton supposed to have published first? Modern algebraic notation was in use throughout the 19th century, and Lagrange had a long treatise on solving equations circa 1800. Euler's textbook on algebra was around 1770 and established much of the notation currently used. What was there left to do in 1896? It sounds preposterous even if we are talking only about notation rather than development of formulas and methods. Sesquivalent (talk) 06:57, 11 June 2020 (UTC)

update: Euler's "Elements of Algebra" has the modern formula. He writes the quadratic equation to be solved as ax^2 = bx+c so that some of the signs are different but it is otherwise the same thing. Sesquivalent (talk) 20:41, 11 June 2020 (UTC)

I have removed this sentence, and the similar sentence in Henry Heaton. This results from a misunderstanding of Heaton's paper: He presented a new method for deriving the quadratic equation, and claimed that it was new. Someone understood wrongly that it was the formula, not the derivation method that was claimed new. D.Lazard (talk) 08:45, 11 June 2020 (UTC)

Merci. I found Heaton's paper and it is interesting but obscure. He squares to obtain a degree 4 equation, which introduces false roots, but the effect of his calculations is to force the false roots of a related equation to be zero, and he recovers the true roots as the non-zero ones. Sesquivalent (talk) 20:02, 11 June 2020 (UTC)

the whole section uses the word formulas instead of formulae.

Isn't formulas grammatically wrong? SSKS1 (talk) 11:54, 13 August 2021 (UTC)

Both formulas and formulae are correct (see MOS:MATH#TONE). As far as I know, formulas is more common in mathematical English. D.Lazard (talk) 12:52, 13 August 2021 (UTC)

There is a drawback in using the symbols ${\displaystyle x_{1}{\text{ and }}x_{2}}$ for roots of an equation as they may be confused with the variables. Instead Greek letters like α and β or ${\displaystyle r_{1}{\text{ and }}r_{2}}$ should be used for roots or zeroes. Also, there were no such references advocated such use. Furthermore, my edits that gave the reason for the results of the Vieta's formulas were reverted swiftly just under the pretext of "so-called" controversial edits. Nishānt Omm (talk) 06:11, 6 May 2022 (UTC)

I think that the Greek letters like α and β are far more easily confused with a and b than x₁ and x₂, and unlike the latter, are almost never used in the literature. Indexed variables r₁ and r₂ are used for an equation with an unknown variable r. - DVdm (talk) 08:07, 6 May 2022 (UTC)

(edit conflict) Firstly, an edit is controversial as soon as some experimented editors disagree with it, and giving your reasons in an edit summary cannot imply that other editors must agree with them, and, indeed, my revert means that I disagree with them.

This article is intended for beginners in mathematics who are probably not accustomed with Greek letters. So, using them is not a good idea. A confusion with the variables (which ones? x is not a variable, it is a unknown) seems unlikely, as ${\displaystyle x_{1}{\text{ and }}x_{2}}$ are defined as "the roots of a quadratic polynomial" in the end of the sentence containing the formula. Moreover, the remainder of the article uses x for both roots (thanks to ± notation, see the first image in the lead). So ${\displaystyle x_{1}{\text{ and }}x_{2}}$ is definitively the less astonishing notation. D.Lazard (talk) 08:31, 6 May 2022 (UTC)

I agree with the others that ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ to represent the roots of a quadratic is unusual. In fact, they are typically reserved for denoting angles. ${\displaystyle x_{1}}$ and ${\displaystyle x_{2}}$ work well enough, though, ${\displaystyle r_{1}}$ and ${\displaystyle r_{2}}$ may be preferable to emphasize that they are roots and to be consistent with the main article on Vieta's formulas.

At the moment, the wording of the section relies on a bit of unnecessary inference to realize that ${\displaystyle x_{!}}$ and ${\displaystyle x_{2}}$ are the roots of the quadratic, so I'll make an edit now to clarify that. As for notation, I don't think it matters much whether one uses ${\displaystyle x_{n}}$ or ${\displaystyle r_{n}}$, so I'll leave it as the former for now. ΣΨ (talk) 09:40, 6 May 2022 (UTC)

I also agree that α and β can be confused with a and b. But, Can you explain your claim "Indexed variables r1 and r2 are used for an equation with an unknown variable r."? How can you say that? SeeVieta's formulas § Example. Nishānt Omm (talk) 09:52, 6 May 2022 (UTC)

DVdm just means that when we have an unknown variable, then we often denote its possible values by indexing them (i.e. labelling them with positive integers). DVdm just arbitrarily chose ${\displaystyle r}$ as the variable here instead of ${\displaystyle x}$ which is a bit weird since ${\displaystyle r}$ is rarely used as a variable in polynomials. In the example you linked, something else is going on. The author chose to use ${\displaystyle r_{1}}$ and ${\displaystyle r_{2}}$ instead of ${\displaystyle x_{1}}$ and ${\displaystyle x_{2}}$ to make it more clear that the these numbers are roots dependent on predetermined coefficients rather than an independent variable like ${\displaystyle x}$. As I stated in my previous reply, it doesn't really matter which we choose as long as we're clear with our definitions. Both have their advantages depending on the context; for what it's worth, I believe that ${\displaystyle r_{n}}$ is more fitting here. ΣΨ (talk) 10:21, 6 May 2022 (UTC)

As x (with or without subscript) is used for the roots in the remainder of this article, this must not be changed in a single section, for evident reasons of coherency and possible resulting confusion for some readers. D.Lazard (talk) 10:31, 6 May 2022 (UTC)

@ΣΨ: Just a quibble for the record, I did not arbitrarily choose ${\displaystyle r}$ as the variable here instead of ${\displaystyle x}$. User Nishānt Omm did that in his opening statement. I replied to that. Cheers. - DVdm (talk) 16:37, 6 May 2022 (UTC)

I'd like to suggest a slight improvement to the paragraph near the beginning of the article which includes this verbiage:

"The quadratic formula [insert quadratic formula here] expresses the solutions in terms of a, b, and c. Completing the square is one of several ways for getting it."

Is there a way to express "getting it" that is more aligned with an encyclopedic tone? Perhaps "solving for x"? – Kekki1978 ^{(talk ✉ | contribs ✎)} 14:29, 26 December 2022 (UTC)

"Getting" is perfectly adapted here, and I do not understand why you think it must be changed. "Solving" is improper, as a formula cannot be solved. "Proving" is correct, but not really adapted, since completing the square is mainly a method for (re)discovering the formula. "Discovering" is confusing as the formula has been discoveed centuries ago. "Deriving" is commonly used in Wikipedia, but I believe that this is a pedagogical jargon for "proving". So "getting" seems the more accurate word. D.Lazard (talk) 14:47, 26 December 2022 (UTC)

I believe that the antecedent for the pronoun "it" is in need of clarity for the reader. Getting what, specifically? – Kekki1978 ^{(talk ✉ | contribs ✎)} 15:27, 26 December 2022 (UTC) To further explain, this guidance from the Undergraduate Writing Center at the University of California at Los Angeles says to avoid writing so that pronouns refer to an implied idea. There are numerous other reputable sources which advise staying away from using missing, ambiguous, or faraway antecedents. I can cite them if needed. To abide by such guidance, "getting 'it'" can be written better so that the antecedent of "it" is clear, and when a statement can be written more clearly, it should be written more clearly. What about stating something like, "Completing the square is one of several ways of getting an equivalent equation which more clearly indicates the value(s) of x"? Regardless of the wording, I believe the antecedent issue warrants an edit. – Kekki1978 ^{(talk ✉ | contribs ✎)} 16:20, 26 December 2022 (UTC)

As usual, "it" refers to the preceding singular noun phrase and to the subjet of the preceding sentence, namely "the quadratic formula". There is no implicit idea here. "It" could be replaced by "the quadratic formula", but such a repetition of this phrase seems not useful (pronouns have been invented for avoiding such repetitions). D.Lazard (talk) 17:54, 26 December 2022 (UTC)

D.Lazard, based on your responses, it seems fair to assume that you wrote the sentence. Are you trying to communicate that completing the square is one way to derive the quadratic formula from a quadratic polynomial? – Kekki1978 ^{(talk ✉ | contribs ✎)} 22:57, 26 December 2022 (UTC)

Why not simple Completing the square is one of several ways for deriving the formula". --Salix alba (talk): 23:34, 26 December 2022 (UTC)

OK. Change done. D.Lazard (talk) 11:59, 27 December 2022 (UTC)

Yep, sounds better. - DVdm (talk) 12:13, 27 December 2022 (UTC)

"Deriving" is not simply "pedagogical jargon for 'proving'". Deriving a result is following a process which finds the result, i.e. produces that result as an outcome. Proving a result is providing a logical reason why the result must be valid; it may produce that result, or it may merely provide justification for the result, which must be provided by other means. An example of a case where the distinction is relevant is proving a result by induction: you need to start out already knowing what result you are to prove, and so the induction doesn't derive it, but it does prove it. In this sense completing the square is indeed a method of deriving the formula, not merely proving it. JBW (talk) 12:25, 27 December 2022 (UTC)

Strangely, I didn't immediately think of an example which is more relevant here. Substituting the formula into the quadratic equation and simplifying is a method of proving the validity of the formula, but it does not derive it. JBW (talk) 12:33, 27 December 2022 (UTC)

It is exactly what I had in mind when saying that "proving" is not well adapted here. When talking of jargon, it was for comparing "getting" with "deriving". My opinion is that the meanings are the same in this case, but that "deriving" is more jargonny, as referring to a technical concept (succession of computational steps) that is not formally defined. D.Lazard (talk) 14:26, 27 December 2022 (UTC)

I am late to this discussion, but if the informality of "getting it" is a motivation, we could simply change that to "obtaining it". But if "deriving" is better than "obtaining" then ... never mind. —Quantling (talk | contribs) 19:42, 3 January 2023 (UTC)

LaTeX not displaying properly... please fix. Not super comfortable with it so can't do it myself anyways. watermelon66 (talk) 00:20, 3 February 2024 (UTC)

I get a bunch of red warning text instead of LaTeX sometimes, especially when previewing a page that I am editing. If your situation is the same as mine: I believe it is a bug in Wikipedia, in that reloading the page, or adding an innocuous space to the page I am editing (even outside the math markup) usually fixes the problem. —Quantling (talk | contribs) 00:28, 3 February 2024 (UTC)

I am not editing the page, simply viewing it. I was brushing up on the topic when I figured I couldn't understand a thing. Luckily, I have some Latex knowledge (in another format, but still) so I could read it. I don't think the same can be said for others though. watermelon66 (talk) 00:30, 3 February 2024 (UTC)

Some examples:

watermelon66 (talk) 17:15, 3 February 2024 (UTC)

Apparently, the error comes from the tool you use for reading the Wikipedia article, since your screen copies display many $ that are not in the Wikipedia article nor in its source. Please use a standard browser or the Wikipeida application for smartphones. D.Lazard (talk) 17:36, 3 February 2024 (UTC)

Hi.

I am using Microsoft Edge on a Windows 11 PC. That should qualify as a standard browser and not a smartphone... watermelon66 (talk) 17:38, 3 February 2024 (UTC)

When I go to visual editing, it looks fine. But when I read the actual article, it doesn't display right. watermelon66 (talk) 17:44, 3 February 2024 (UTC)

When a and c are positive quantities but b is a negative quantity such that b² > 4ac, inequalities include:

• 0 < a < ⁠b²/4c⁠ where b < 0 but c > 0.
• b < –2√ac where a > 0 and c > 0.
• 0 < c < ⁠b²/4a⁠ where a > 0 but b < 0.

2603:7000:B500:5D4:FCF3:6580:ECA7:89DF (talk) 23:00, 6 August 2024 (UTC)

The context is wrong in saying "solutions X= X1 and X= X2". Instead it should be "X= X1 or X= X2". Please refer to some qualified text books for reference.

Eh?? If there are two solutions, they both exist, and in this context they are not alternatives. "And" is therefore correct. In the context where there are two possible answers to a problem, then you can use "or".

Example: "There are two solutions to the equation: x=7 and x=−4. Thus the answer could be either 7 or −4. However as the question states that x represents the number of bananas in the bunch, and you cannot have negative bananas, the correct answer is x=7."---Ehrenkater (talk) 21:50, 20 September 2024 (UTC)