Talk:Completing the square

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 Field: Basics

Inverse Quadratic[edit]

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

Please verify the complete the square formula[edit]

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[edit]

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 (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)


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)


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)


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[edit]

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. —Preceding unsigned comment added by (talk) 10:03, 8 January 2008 (UTC)

Completing the sqare — an algebraic method represented here in geometrical concepts (decomposition). Using layers, the main step can be represented in a single figure, and the motivation behind this step also has a pure geometrical meaning.
I could not find the image, not even with 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?[edit]

Where is that equation to complete squares? — Preceding unsigned comment added by Pasanbhathiya2 (talkcontribs) 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)


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(xh) + k; it says a(xh)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 −h is negative. If h is negative, then −h is positive.
But perhaps it could be explained more clearly in some respects. Michael Hardy (talk) 07:22, 21 June 2011 (UTC)

The matrix case[edit]

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 (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!-- (talk) 17:43, 5 March 2013 (UTC)

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