# B What exactly is this concept called? -- Finding "?" in x^2+16x+? to get (x+8)^2

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1. Oct 8, 2018

### YoungPhysicist

x^2+16x+?
That ? Can only be 64 so the polynomial can transform into (x+8)^2 where the position where in this case “8” is must be an integer.

What exactly are these kind of concepts called?

2. Oct 8, 2018

### Staff: Mentor

We call it "quadratic supplement" and Wiki translates it to "completing the square".

3. Oct 8, 2018

4. Oct 8, 2018

### Staff: Mentor

Sounds like something you take when you need more vitamins vs being a carpenter and building a square.

5. Oct 8, 2018

### Staff: Mentor

Well, completing the square sounds like a paver at work.

6. Oct 8, 2018

LOL

7. Oct 8, 2018

### Staff: Mentor

Nutritional supplements aren't famous enough here - outside sports - that this association would come up. It would rather be

or

8. Oct 8, 2018

### YoungPhysicist

I don’t know if this is right,but after I searched “quadratic supplement”online,it seems like the process of turning the form of
ax^2+bx+c
to
a(x-h)^2+k

but in my case it kind of means the process of filling in a specific coefficient (either a or b or c)
in a way that the entire polynomial can be turn into (x+n)^2 where n must be an real number(sorry for the mispost up there)and the transformation process is based on
(a+b)^2 = a^2 +2ab+ b^2

Ps:It is like the specific propertie of a set of quadratic polynomials that its a(x-h)^2+k form has a = 1 and k = 0

9. Oct 8, 2018

### Staff: Mentor

The general idea is always to transform a quadratic polynomial $a_0x^2+a_1x+a_2$ into a form $c\cdot (x \pm a)^2 +b$ in order to isolate $(x \pm a)^2$ and solve it.

10. Oct 8, 2018

### YoungPhysicist

Get it.Thanks!!

11. Oct 9, 2018

### Staff: Mentor

In the US (and possibly UK?), we call this technique "completing the square."

12. Oct 9, 2018

### Staff: Mentor

Yes, that's what Wikipedia (my preferred dictionary for technical terms) gave me. My own translation "quadratic supplement" is not really optimal for "quadratische Ergänzung", which could also be translated by "quadratic completion" which is pretty close. I found the English name a bit funny, as square has multiple meanings in English (geometric figure, "don't be square", Times or Trafalgar square), whereas quadratic has not.

13. Oct 9, 2018

### Staff: Mentor

"Quadratic supplement" is not that far off from "completing the square," since the Latin for "square" is quadratus.
There are also a few other meanings for "square," such as "a square deal" (fair deal), or "to square accounts," (pay off a debt).

14. Oct 9, 2018

### pbuk

Yes it is "completing the square" in the UK. I have always assumed that the term comes from the geometric construction shown in the animation on the Wikipedia page: solving the problem involves finding the "missing piece" that literally completes the construction of a square. I don't normally use Wikipedia for mathematical terms, preferring Mathworld.

15. Oct 9, 2018

### Staff: Mentor

I spoke about translations. Wikipedia allows to switch from my native language directly to the corresponding English term, WolframAlpha does not, neither do ordinary dictionaries. It has also the advantage, that those pages do not correspond one on one, so you can sometimes find better descriptions, sources, formulas or graphics on other language sites. And for all who are really interested in accurate definitions and links, I strongly recommend nLab rather than WolframAlpha.

16. Oct 9, 2018

### Janosh89

I would love to know , as I have asked before, why the "missing piece" in #14 cannot be given a name. adquadratus ?
For
odd integers ,e.g. 223, -
$1112 +223 =1122$
would it then be alright to say 223 is the adquadratus ??

also in #13 don't forget the square meal ! [ from the wooden square plates on RN vessels of old , kept in a slotted rack so they they did not roll out in heaving sea states]

Last edited: Oct 10, 2018
17. Oct 10, 2018

### FactChecker

As a "fun" exercise, you can use the technique of completing the square to derive the quadratic formula.

18. Oct 10, 2018

### Staff: Mentor

I think you would be alone in the world to call it that. The "ad" in ad quadratum doesn't mean "to add" -- the Latin preposition means "toward" or "to".

Regarding your numbers, $111^2 + 223 = 111^2 + 222 + 1 = 111^2 + 2 \cdot 111 + 1 = (111 + 1)^2 = 112^2$. Geometrically, the two terms , $111^2$ and $222$, could be visualized as a square whose sides are 111 and two rectangles, each of length 111 and width 1. Completing the square means determining which square would fill in the small gap at the lower right in my drawing. In this case, a 1 X 1 square would do the job.

The areas of the 111 X 111 square, the two thin rectangles, and the small 1 X 1 square (not shown) add up to the combined figure, a 112 X 112 square.

19. Oct 10, 2018

### Staff: Mentor

Fun??? That's often the only way for me to check if I remembered the signs correctly.

20. Oct 10, 2018

### FactChecker

Ha! IMHO, one of the signs of a mathematician is the reliance on quick derivations rather than memorization (although, I understand that Gauss had a photographic memory -- just another thing that made him the ultimate genius.)

Last edited: Oct 10, 2018