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

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

 

Completing the square is the algorithm, I think.

https://en.wikipedia.org/wiki/Completing_the_square

 

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

 

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

 

LOL

 

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

or

 

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.

 

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

 

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.

 

"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).

 

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.

 

I don't believe so. I hadn't heard this term before, but ad quadratum seems to be related to architectural design or generating Platonic solids (see http://www.gatewaycoalition.org/files/millennium_sphere/products/AdQuadratum.pdf).

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.

 

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