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

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In summary, completing the square is a technique used to transform a quadratic polynomial into a form where the squared term is isolated and can be easily solved. It is often referred to as "quadratic supplement" or "completing the square" and is based on the property (a+b)^2 = a^2 + 2ab + b^2. In the US and UK, it is commonly known as "completing the square" and is related to architectural design and the construction of Platonic solids.
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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?
 
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  • #2
We call it "quadratic supplement" and Wiki translates it to "completing the square".
 
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  • #4
fresh_42 said:
We call it "quadratic supplement" and Wiki translates it to "completing the square".

Sounds like something you take when you need more vitamins vs being a carpenter and building a square.
 
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jedishrfu said:
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.
 
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  • #6
LOL :smile:
 
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jedishrfu said:
Sounds like something you take when you need more vitamins
Nutritional supplements aren't famous enough here - outside sports - that this association would come up. It would rather be
upload_2018-10-8_17-29-51.png

or
upload_2018-10-8_17-31-15.png
 

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  • #8
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
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.
 
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  • #10
Get it.Thanks!
 
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fresh_42 said:
Well, completing the square sounds like a paver at work.
In the US (and possibly UK?), we call this technique "completing the square."
 
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  • #12
Mark44 said:
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.
 
  • #13
"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).
 
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  • #14
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
pbuk said:
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.
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
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]
 
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  • #17
As a "fun" exercise, you can use the technique of completing the square to derive the quadratic formula.
 
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  • #18
Janosh89 said:
I would love to know , as I have asked before, why the "missing piece" in #14 cannot be given a name. adquadratus ?
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).
Janosh89 said:
For odd integers ,e.g. 223, -
##1112 +223 =1122##
would it then be alright to say 223 is the adquadratus ??
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.

ComplSqr.png

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.
Janosh89 said:
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]
 

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FactChecker said:
As a "fun" exercise, you can use the technique of completing the square to derive the quadratic formula.
Fun? That's often the only way for me to check if I remembered the signs correctly. :wink:
 
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  • #20
fresh_42 said:
Fun? That's often the only way for me to check if I remembered the signs correctly. :wink:
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.)
 
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FactChecker said:
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 way that he was the ultimate genius.)
This would explain why my library is far better than my memory <sigh>.
 
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  • #22
Found out / discovered that addendum is the gerund of the Latin verb addere . [to add together]
Painful memories of chalk dust flooding back. ..
 
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1. What is the concept of finding "?" in x^2+16x+? to get (x+8)^2?

The concept of finding "?" in x^2+16x+? to get (x+8)^2 is known as completing the square. It involves manipulating a quadratic equation in order to write it in its perfect square form, (x + a)^2, where "a" is the missing number that needs to be found.

2. Why is it important to know how to find "?" in x^2+16x+? to get (x+8)^2?

Completing the square is a crucial technique in solving and graphing quadratic equations. It allows us to easily identify the vertex of a parabola, which is a key point in understanding the behavior of a quadratic function. It is also used in various real-life applications such as calculating maximum or minimum values in optimization problems.

3. How do you find the missing number in x^2+16x+? to get (x+8)^2?

The missing number in x^2+16x+? can be found by following these steps:

  1. Factor out the coefficient of x^2 from the equation, giving you x^2+16x+a.
  2. Divide the coefficient of x (16) by 2 and square the result, giving you (16/2)^2 = 64.
  3. Add and subtract the value you found in the previous step inside the parentheses, giving you x^2+16x+64+a-64.
  4. Group the first three terms and the last two terms together, and factor the first group, giving you (x^2+16x+64)+a-64.
  5. Recognize that the first group is a perfect square, (x+8)^2, and rewrite the equation as (x+8)^2+a-64.
  6. Finally, set the equation equal to the original given equation x^2+16x+? and solve for a by equating the coefficients of the two equations. In this case, a-64=? and a=64.

4. Can completing the square be used for all quadratic equations?

Yes, completing the square can be used for all quadratic equations. However, it may not always be the most efficient or practical method for solving a particular quadratic equation. In some cases, factoring or using the quadratic formula may be more appropriate.

5. Are there any shortcuts or tricks for completing the square?

There are a few shortcuts or tricks that can make completing the square easier, such as recognizing common perfect square expressions and using patterns to help with factoring. However, the process of completing the square ultimately involves following a specific set of steps and there is no shortcut for that. Practice and familiarity with the method can help make it easier and quicker to complete the square.

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