Talk:Completing the square

Inverse Quadratic
It looks like completing the square can be used to find the inverse:

f(x) = a(x-h)^2 + k

rearranging: f-1(x) = h+-sqrt((f(x)-k)/a)   :i.e. the inverse. For some reason finding a standard form for the inverse on the internet is rather harder than finding roots.

Can we add this to this page?

159.245.32.2 (talk) 18:49, 16 May 2011 (UTC)Martin

Please verify the complete the square formula
Please verify the complete the square formula. I changed the page (before I had an account) by altering the equation to what I believe is the correct form, on 9/25/2006. Specifically, I changed the part of the equation that read: "4a^2" to "4a". I made this change while doing some calc II homework, when I realized that I was getting incorrect results with the equation that was on the page prior to my change on 9/25/2006.

After looking a bit further, I realized that the equation had previously been entered correctly, but some user (Huadpe) "corrected" the equation later to the "wrong" value: 4a^2. Although I think the equation is correct now, I'd appreciate it if a more veteran user/moderator of wikipedia with solid math knowledge could confirm the accuracy of the equation on the page, and somewhere make a public note (or cite) to that effect.

I'm a very experienced user of google and also a computer science major, yet I had difficulty finding a simple "plug-and-play" equation that would algebraically provide a completed square given the coefficients a, b, and c from ax^2 + bx + c. After much searching, I found that Wikipedia provided this equation, but it happened to have this minor error. I fear that other math students (likely at lower levels of math, or perhaps some that need refreshers like me) are likely to come to this page as their primary source in the future, and the equation will be modified to the incorrect version again. I'm not sure how wikipedia prevents this, but please put this on your "watch list".

Thanks all! Austin

Austinflorida 09:23, 21 October 2006 (UTC)

I decided to add an extra step to the equation to clarify the step where some people may be inclined to make an incorrect "correction." Without the added step, it seems that it might be easier for a user, even after a few glances, to inadvertently believe that a correction is necessary, when in fact the steps ARE algebraically correct (verified with TI Derive 6).

Austinflorida 09:50, 21 October 2006 (UTC)

Questions / Possible Additions to the Article
Does completing the square also apply to multivariate equations? The procedure is the same, say you have: $$x^{2} - 4xy + 5y^{2} = (x-2y)^{2} + y^{2},$$ This corresponds to completing the square with x, holding y constant - nothing too extraordinary. However, you can extend this process to three variables. Take for example $$ x^{2} + 4xy + 8y^{2} + 20yz + 6z^{2} -4xz = (x + 2y - 2z)^{2} + (2(y+3z))^{2} - 34z^{2} ,$$ (if my algebra is correct!) This is also an application of completing the square, allowing us to write a quadratic polyomial of three variables in which all terms have degree two, as the sum of three squares.

(My inspiration was a multivariate calculus problem: Create a tranformation to map the ellipsoid $$x^{2} + 4xy + 8y^{2} +4yz + 6z^{2} - 2xz = 9,$$ onto the unit sphere.)

--D Mac 04:39, 17 November 2005 (UTC)

These discussions cut to the heart of quadric surfaces.

Consider equations of the form a*x^2 + b*y^2 + c*x*y + d*x + e*y + f = 0, where a,b,c,d,e,f are constants. In other words, we are talking about a polynomial of several variables where the maximum degree is 2 (x^2, xy, etc). Equations of this form are combinations of scaling, rotation, and translation of an ellipse/hyperbola/parabola. This can be seen more rigorously with linear algebra. —Preceding unsigned comment added by 98.203.237.75 (talk) 04:16, 12 January 2008 (UTC)

Completing the square works no matter what the coefficients of the polynomial, as long as they don't contain the variable that is being 'completed'. I have modified the article to reflect this.

Michaelbusch 04:00, 24 August 2006 (UTC)

Vandalism
Last evening, an anon user blanked this page. I reverted it and warned the user. Please pay special attention to this page on your watch list to spot any future vandalism. 48v 00:10, 5 September 2006 (UTC)

Reorganization
I just did a significant reorganization of this article. I changed very little content, but added an Overview section heading for everything that was in the intro previously but for its first paragraph, moved the canonical quadratic equation to its own example, and subsectioned out and reordered the several examples (from most specific to most general). I tweaked a lot of math markup to look nicer, and changed the discourse on just manipulating a x^2 + b x + c to actually finding its roots, as this is arguably the most significant application of the topic (which I also noted).

I was considering removing some examples for space, perhaps what are now the second and fourth. But I decided to let this version stand and see what comments came first. Baccyak4H 16:38, 27 October 2006 (UTC)

Cleanup
I've done a major cleanup, taking advantage of the new and improved texvc, which can now handle the "align" environment. Along the way I polished the prose and pruned the examples. I did not touch the intro, though it needs work. --KSmrqT 08:52, 26 November 2006 (UTC)

Upload Image to Geometric Perspective
Could someone with a user account please upload the following image and place it in the Geometric Perspective section. Thanks. I believe this will add some intuitive perspective to the algebraic manipulations. http://1073741824.org/square_thumb.png —Preceding unsigned comment added by 98.203.237.75 (talk) 10:03, 8 January 2008 (UTC)


 * I could not find the image, not even with archive.org. But also I have thought of providing a geometrical representation (with decompositions).


 * If there are no objection or proposals for modification, then I'd insert my image version into the main article.


 * Physis (talk) 08:21, 23 January 2010 (UTC)


 * Sorry, I have just noticed, that there exists already an image in the article, better than mine. Physis (talk) 08:58, 23 January 2010 (UTC)

Where is that equation?
Where is that equation to complete squares? — Preceding unsigned comment added by Pasanbhathiya2 (talk • contribs) 17:30, 10 December 2010 (UTC)

I just added it. Just check you, math specialist,s if that equation is correct. Thanks!! Happy Editing! P.S. I don't know how to add it in this wiki text. Would you somebody please do that,instead of my image? After you do that remove the image and it's ok! Just wanted to help and found no other way rather than adding an image and I didn't understand how you use wiki texts! Thanks once again!! Pasanbhathiya2 (talk) 17:43, 10 December 2010 (UTC)


 * I have removed the equation. This is about the polynomial ax2+bx+c. It is not about the equation ax2+bx+c=0. So the values of x which you added are not really relevant in this context. See my edit summary and a note on your talk page. DVdm (talk) 20:26, 10 December 2010 (UTC)

Inconsistencies
I don't know whether this page has been recently vandalised but it does not make a great deal of sense. In the opening paragraph there is no reason to have a negative sign in the a(x - h) + k. Indeed this goes against the idea of completing the square as stated in the article, the main idea of which is to take a square component plus a rectangular component and make a larger square by breaking up the rectangle. You then have a little bit which needs to be filled in (the "completing the square"). Putting a negative sign in a(x - h) + k does the opposite. It makes a smaller square from a larger square with a section left over. Thus look at the geometrical picture at the bottom: it is transforming x^2 + bx into the larger square — and is throwing in it's own confusion by treating this as having to do with a quadratic equation, rather than being purely a quadratic polynomial transformation. So there seem to be multiple problems here of a pedagogic nature. Some of the confusion would be lessened if it were made clear that the negative sign on the h is purely to make the graphing more intuitive, since it corresponds in a shift of the parabola to the right. (And that this graphing has no real connection with the geometrical meaning of "completing the square". Eluard (talk) 02:29, 21 June 2011 (UTC)


 * It doesn't say a(x &minus; h) + k; it says a(x &minus; h)2 + k.
 * The reason for the minus sign is that that makes h the value of x for which the square vanishes, and thus in later problems it is the x-coordinate of the vertex of the parabola.
 * There is nothing essential about the "complete" square being larger than the square to which the rectangle is added. Moreover, you should not assume &minus;h is negative.  If h is negative, then &minus;h is positive.
 * But perhaps it could be explained more clearly in some respects. Michael Hardy (talk) 07:22, 21 June 2011 (UTC)
 * a(x &minus; h)2 + k is the vertex form of a quadratic function. Aboctok (talk) 13:04, 17 August 2018 (UTC)
 * But perhaps it could be explained more clearly in some respects. Michael Hardy (talk) 07:22, 21 June 2011 (UTC)
 * a(x &minus; h)2 + k is the vertex form of a quadratic function. Aboctok (talk) 13:04, 17 August 2018 (UTC)
 * a(x &minus; h)2 + k is the vertex form of a quadratic function. Aboctok (talk) 13:04, 17 August 2018 (UTC)
 * a(x &minus; h)2 + k is the vertex form of a quadratic function. Aboctok (talk) 13:04, 17 August 2018 (UTC)

The matrix case
The bit that begins 'The matrix case looks very similar:' tempts the question what matrix case? Matrices have not been previously mentioned. That, and other things, makes the article look as if many different people have just thrown in odd bits here and there. Note that someone who really needs to know how to complete the square is probably rather new to mathematics, and clarity is essential for such a person. — Preceding unsigned comment added by 81.131.57.37 (talk) 15:29, 21 October 2012 (UTC)

Also, the conditions for the matrix case are not clear: does A have to be symmetric? and it doesn't say that all terms should be scalar in the equation.... Could someone check these for me and edit? Thank you!--129.215.24.65 (talk) 17:43, 5 March 2013 (UTC)

Checked and added the non-symmetric case. 129.132.146.210 (talk) 14:18, 8 April 2013 (UTC)

This article appears to contradict another article
See the discussion here. Jarble (talk) 02:56, 6 January 2015 (UTC)

A generalized statment
There is a generalization of completing the squares for higher-degree polynomials as follows:

Let $$K$$ be a field of characteristic $$\ne 2$$, $$n \in \mathbb N$$ and $$p \in K[x]$$ a polynomial of degree $$2n$$. It can be shown by induction that there exist polynomials $$h \in K[x]$$ of degree $$n$$ and $$f \in K[x]$$ of degree $$<n$$ such that $$p$$ can be written as $$p=h^2+f$$.

This can be proved pretty easily (by induction). I had added it but it was put down by the self-proclaimed commandant D.Lazard. An encyclopedia is about knowledge, expanding knowledge and contribution. Mathematics is about generalization, abstraction and finding patterns. While plenty of other mathematical articles on Wikipedia talk about generalizations, why shouldn't this little one here be allowed? — Preceding unsigned comment added by 89.154.51.249 (talk) 12:52, 28 January 2021 (UTC)


 * Please put new talk page messages at the bottom of talk pages and sign your messages with four tildes ( ~ ) — See Help:Using talk pages. Thanks.
 * Wikipedia is about wp:reliable sources. And personal attacks are not allowed. - DVdm (talk) 13:00, 28 January 2021 (UTC)

It's not an attack, it's the truth. In fact, he was the one who attacked my contribution by deleting it. I don't allow the suppression of knowledge! Here: http://people.math.harvard.edu/~elkies/many_pts.pdf Page 4, Lemma M. Set r=2 and you get my generalization. Any more questions? Do you always need a scientific paper or can't you just think for yourself? Elkies says: "This lemma is, of course, well-known and elementary". — Preceding unsigned comment added by 89.154.51.249 (talk) 13:08, 28 January 2021 (UTC)


 * Just a comment from an uninvolved editor- 89.154.51.249 you are coming off as combative, you need to read WP:RS and a few other policies before you start saying how things should be- there is a reason why they are the way they are- and if you don't like that- you need to work on changing policy, not just decide to ignore it. And calling someone names just because they disagreed with your edits- is rude. And its not likely to encourage people to see your point of view. Nightenbelle (talk) 13:13, 28 January 2021 (UTC)
 * The beauty of mathematics is that you can verify mathematical statements directly by logical reason instead of needing a so-called reliable source, especially if it is as elementary as the one I made, which can be proved by an above-average first-semester mathematics student. (I can speak for myself because I edited/improved the first version after I had overseen a detail.) Even after I have given a professional resource (which, as I said, is basically not necessary to verify a mathematical statement), D.Lazard continues blocking. I don't understand because this mathematical statement is true and a generalized statement (which is about completing p to the square h^2 by f, it's just more terms/monomials than one to complete) and many other mathematical articles have generalized statements. Either he should let it be here in this article or find a more suitable article to put it in or create a new article such that people can see that the case n=1 is exactly the main topic here. But I think the best way for now is letting the statement in this article and with time, if other articles or more information appears, it can be moved or copied to somewhere else. Wikipedia supervisors should be careful what they block. If they block in accordance with their dogmas and not by a legitimate reason (for example, if my statement had been wrong or was in fact not a generalization of the main topic), then this encyclopedia fails its original purpose. Instead of showing gratitude, this is just a way to discourage people who only mean well and want to contribute to the knowledge of mankind by taking initiative and doing other people's work. Actually, we could all benefit from each other. — Preceding unsigned comment added by 89.154.51.249 (talk) 09:57, 29 January 2021 (UTC)

Wikipedia is not a blog. So, you have to follow Wikipedia policies for editing Wikipedia pages, and adding content. Here, you are WP:edit warring, instead of following the Wikipedia process WP:BRD. This may lead you to be blocked for editing.

In the present case, nobody disputes that the result is known, that it may be found in reliable sources (although the source you refer to is not what is considered here as a reliable source, even if its author is well known). I consider also that it deserve to be mentioned somewhere in WP, although it is not clear where it is worth to add it. In any case, its place is not in this article, as it is not a direct generalization of the subject of this article, and its proof is much too technical for the audience of this very elementary article.

This is only my opinion, but per Wikipedia policies, a consensus involving several editors is needed before adding again this content. So please stop edit warring. D.Lazard (talk) 14:10, 28 January 2021 (UTC)


 * "as it is not a direct generalization of the subject of this article"
 * How does a direct generalization then look like? Set n=1 and you get exactly the subject of the article. In general, the statement is that it is sufficient to complete (only) the terms with degree < n of a polynomial of degree 2n (the addition/deviation is called f here) to get a square of a polynomial of degree n (called h^2 here).


 * "and its proof is much too technical for the audience of this very elementary article"
 * I never suggested to publish a proof with it, but only mentioned that it can be done by induction. There is also the option the add a proof in a box which can be hidden or shown optionally. If you want, I can do that. In the end, none of us knows the audience. But the fact is that by blocking my contribution, you refuse access to abstraction and self-study to interested people who so far have only a fairly elementary level of knowledge, but who would be inspired by generalizations. For example, a talented 14-year-old teenager interested in mathematics would have the chance for further education by my contribution. On the other side, those who might be overwhelmed, would not be forced to read such a section, as is the case with many other mathematical articles. Wikipedia combines school and university mathematics. The articles "parabola", "derivatives", "divisor", "greatest common divisor" etc. have each their own generalization(s).


 * "This is only my opinion"
 * Only your opinion, but you are granting yourself the right to block my contribution. Even after I have given a source, you are blocking by saying "it is much too technical". Let the readers decide! If they like, they will read. If not, they can skip, as always. — Preceding unsigned comment added by 89.154.51.249 (talk) 15:53, 28 January 2021 (UTC)

Completing the Square geometry
Can we please not use the variable "a" for the area. This will confuse any students stupid enough to expect to learn anything from Wikipedia maths article because "a" is normally the coefficient of the x^2 term. "k", or "-c" would be fine. I know that this will not confuse expert mathematicians, but experts already know this stuff. Tuntable (talk) 00:27, 8 February 2021 (UTC)


 * I second this, but the graphic would have to be updated accordingly, so I cannot make the change myself as I am not familiar with how to deal with images on wikipedia Joel Brennan (talk) 11:27, 13 May 2021 (UTC)

Wiki Education assignment: 4A Wikipedia Assignment
— Assignment last updated by Ahlluhn (talk) 00:57, 31 May 2024 (UTC)