Complete the square, yes, but what does it mean?

[mcastillo356]

TL;DR Summary

I've been searching in the net; I understand the whole quote I will write down; but the concept, the words "complete the square" keep me wondering if they really relate to a geometric and well known shape, or it is just an algebraic term.

Hi, PF

"Roots and Factors of Quadratic Polynomials
There is a well-known formula for finding the roots of a quadratic polynomial.

The Quadratic Formula
The two solutions of the quadratic equation

$Ax^2+Bx+C=0$,

where $A$, $B$, and $C$ are constants and $A\neq{0}$, are given by

$$x=\displaystyle\frac{-B\pm\sqrt{B^2-4AC}}{2A}$$

To see this, just divide the equation by $A$ and complete the square for the terms in $x$

$x^2+\displaystyle\frac{B}{A}x+\displaystyle\frac{C}{A}=0$

$x^2+\displaystyle\frac{2B}{2A}x+\displaystyle\frac{B^2}{4A^2}=\displaystyle\frac{B^2}{4A^2}-\displaystyle\frac{C}{A}$

$\Bigg(x+\displaystyle\frac{B}{2A}\Bigg)^2=\displaystyle\frac{B^2-4AC}{4A^2}$

$x+\displaystyle\frac{B}{2A}=\pm\displaystyle\frac{\sqrt{B^2-4AC}}{2A}$."

Attempt: The words "complete the square" for the terms in $x$ are both an algebraic and a geometric ideas in the proof of the two solutions for the quadratic equation:

(i)- There is an algebraic set out of a square when stating $\Bigg(x+\displaystyle\frac{B}{2A}\Bigg)^2$: this is an squared expression.
(ii)- The above expression, from a geometric point of view, is actually an square (height per equal width).

Am I right?

Greetings!

 

[Mark44]

I've been searching in the net; I understand the whole quote I will write down; but the concept, the words "complete the square" keep me wondering if they really relate to a geometric and well known shape, or it is just an algebraic term.
<snip>
$x+\displaystyle\frac{B}{2A}=\pm\displaystyle\frac{\sqrt{B^2-4AC}}{2A}$."

Attempt: The words "complete the square" for the terms in $x$ are both an algebraic and a geometric ideas in the proof of the two solutions for the quadratic equation:

(i)- There is an algebraic set out of a square when stating $\Bigg(x+\displaystyle\frac{B}{2A}\Bigg)^2$: this is an squared expression.
(ii)- The above expression, from a geometric point of view, is actually an square (height per equal width).

Am I right?

Yes, completing the square has both an algebraic meaning and a geometric one. Take a look at this wiki page, particularly the small animation at the right and near the top of the page - https://en.wikipedia.org/wiki/Completing_the_square

 

[fresh_42]

Completing the square is simply the step from $x^2+\dfrac{B}{A}x$ to $x^2+\dfrac{B}{A}x+\dfrac{B^2}{4A^2}=\left(x+\dfrac{B}{2A}\right)^2$ when a polynomial is extended (completed) to the square of another polynomial.

 

[FactChecker]

Yes, completing the square has both an algebraic meaning and a geometric one. Take a look at this wiki page, particularly the small animation at the right and near the top of the page - https://en.wikipedia.org/wiki/Completing_the_square

Wow! I wonder if anyone really thinks geometrically that way? I guess that some of the geniuses of mathematics can visualize that, just the way great chess players today can visualize the chess moves.

 

[WWGD]

Hope this is not obvious.Notice that while you add the expression $\frac {B^2}{4A^2}$, you must also ultimately subtract it as well, to avoid changing the form of your initial expression.

 

[Mark44]

Hope this is not obvious.Notice that while you add the expression $\frac {B^2}{4A^2}$, you must also ultimately subtract it as well, to avoid changing the form of your initial expression.

It's pretty obvious, IMO. If you're working with a single expression or just one side of an equation, if you add something to that expression or side, you have to also subtract it. In essence you are adding 0 to an expression, so aren't changing it.

 

[Mark44]

Wow! I wonder if anyone really thinks geometrically that way?

I think a lot of people do. It's been a long time since I've taught any algebra classes, but I seem to recall seeing the geometric approach being presented alongside the algebraic one. I've always been of the opinion that it's a good idea to combine symbolism with geometry wherever possible.

 

[mcastillo356]

Yes, completing the square has both an algebraic meaning and a geometric one. Take a look at this wiki page, particularly the small animation at the right and near the top of the page - https://en.wikipedia.org/wiki/Completing_the_square

Fine, I had already taken a look, but the animation didn't work. Right now it has.

Hope this is not obvious.Notice that while you add the expression $\frac {B^2}{4A^2}$, you must also ultimately subtract it as well, to avoid changing the form of your initial expression.

In my personal opinion, it is obvious for those with figurative, i.e., artistic skills; not my case. Architects should be able. In Spain it's required at the age of eighteen; well, I'm talking about 1983. An arquitect turned to teacher tried (unsuccessfully for almost all the classmates) to educate us: he gave us bidimensional four drawings, and we had to manage depicting the threedimensional underlying shape correctly.

 

[PeroK]

I'm a little bit sceptical of the value of the geometry in this case. This is an example of an algebraic manipulation where you have to plan to move towards a goal. The first important idea is that we can drop the $a$ for simplicity, then put it back in later. First, we solve$$x^2 + bx + c = 0$$The second idea is to look for $$x^2 + bx + c = (x + \alpha)^2 + \beta$$Because we can see how to solve the equation using the expression on the right-hand side. Next, we should see from the expansion of a square that $\alpha = \frac b 2$. Then we can solve for $\beta$. FInally, we get the full solution by replacing $b \rightarrow \frac b a$ and $c \rightarrow \frac c a$ in our solution for $a = 1$.

This way of thinking seems to me to give the student tools for tackling other problems. If there is a neat geometric interpretation, then all well and good, but good algebraic technique is a key skill that will be rewarded many times.

 

[Nugatory]

I'm a little bit sceptical of the value of the geometry in this case

Completing the square didn’t click for me until I thought of it as replacing $x$ with $x-\alpha$ to slide the parabola sideways until its extremum was on the y-axis.
(Although I didn’t phrase it that way in Algebra I).

 

[e_jane]

Wow! I wonder if anyone really thinks geometrically that way?

Before algebra was invented, "geometrically" was the only way to think about this stuff. Which makes it absolutely amazing to me that Archimedes almost invented calculus about two millennia before Isaac Newton did.

 

[mcastillo356]

Before algebra was invented, "geometrically" was the only way to think about this stuff. Which makes it absolutely amazing to me that Archimedes almost invented calculus about two millennia before Isaac Newton did.

"If I have seen further it is by standing on the shoulders of giants" (I. Newton)

 

[mcastillo356]

I'm a little bit sceptical of the value of the geometry in this case. This is an example of an algebraic manipulation where you have to plan to move towards a goal. The first important idea is that we can drop the $a$ for simplicity, then put it back in later. First, we solve$$x^2 + bx + c = 0$$The second idea is to look for $$x^2 + bx + c = (x + \alpha)^2 + \beta$$Because we can see how to solve the equation using the expression on the right-hand side. Next, we should see from the expansion of a square that $\alpha = \frac b 2$. Then we can solve for $\beta$. FInally, we get the full solution by replacing $b \rightarrow \frac b a$ and $c \rightarrow \frac c a$ in our solution for $a = 1$.

This way of thinking seems to me to give the student tools for tackling other problems. If there is a neat geometric interpretation, then all well and good, but good algebraic technique is a key skill that will be rewarded many times.

Geometry aside and avoiding controversy, I personally agree with the quote; moreover, I feel it is the quote I most thank and share. And also a good advice to keep in mind.

 

[mcastillo356]

Completing the square didn’t click for me until I thought of it as replacing $x$ with $x-\alpha$ to slide the parabola sideways until its extremum was on the y-axis.
(Although I didn’t phrase it that way in Algebra I).

HI, @Nugatory, would you explain this
quote?

 

[symbolipoint]

Strange how I never remember knowing or ever seeing the Geometric way of understanding this until at least decade after graduating. Then finding it and studying it felt great; easy to redo, too.

 

[mathwonk]

For me the geometric interpretation answers the question: "how did they think of that?", which the bare algebra seldom does, at least for me. I am one of those people who think visually, struggle with algebra, and always benefit from a geometric version of an argument. The geometry also allows me to guess further enhancements of an argument sometimes.

The geometric approach seems to have been more fundamental, since it was developed much earlier in human history than the algebraic one. Euclid uses geometry to express simple algebraic formulas, such as (a+b)^2 = a^2 + b^2 + 2ab, in Prop. 4, Book II of the Elements, without any algebraic notation, which indeed did not yet exist. He also solves essentially arbitrary quadratic equations purely by geometry, and uses the solutions to make geometric constructions, like that of a regular pentagon. Even earlier it seems the Babylonians gave geometric solutions equivalent to solving quadratic equations by methods that equate to completing the square when expressed in modern notation.

Much later, in his Two New Sciences, Galileo uses only geometry to compute the rate of fall of a body under influence of gravity, a quadratic expression, still entirely without benefit of algebraic notation, in about 1600. It was only around this time apparently that such problems began to find an abstract symbolic algebraic formulation, in the hands of Viete, then Descartes and others.

So there is much benefit to modern algebraic notation, in efficiency, but it is not second nature to some of us, and was not the way the solutions of these "quadratic" problems were originally discovered, indeed not by thousands of years.

I do however envy people who can think algebraically, as I think they are not limited so much in considering higher dimensions, for example, as I am.

 

[symbolipoint]

I thought of something at least a couple of times. The extended rectangular piece for the geometric derivation can be included at both sides instead of cutting it in half and then moving one of those halves. Still works.

 

[mcastillo356]

Hi, PF, I've found an exercise where completing the square is also a tool, but I have first to theoretically introduce (quoting the textbook) it again, similarly as in the first post:

Completing the Square

Quadratic expressions of the form $Ax^2+Bx+C$ are often found in integrands. These can be written as sums or differences of squares using the procedure of completing the square:

$$Ax^2+Bx+C=A\Bigg(x^2+\displaystyle\frac{B}{A}\,x+\displaystyle\frac{C}{A}\Bigg)$$

$$\qquad{=A\Bigg(x^2+\displaystyle\frac{B}{A}x+\displaystyle\frac{B^2}{4A^2}+\displaystyle\frac{C}{A}-\displaystyle\frac{B^2}{4A^2}\Bigg)}$$

$$\qquad{=A\Bigg(x+\displaystyle\frac{B}{2A}\Bigg)^2+\displaystyle\frac{4AC-B^2}{4A}}$$

It must be made the change $u=x+\displaystyle\frac{B}{2A}$

EXAMPLE 5 Evaluate $\displaystyle\int{\displaystyle\frac{1}{\sqrt{2x-x^2}}\,dx}$

Solution

$$\displaystyle\int{\displaystyle\frac{1}{\sqrt{2x-x^2}}\,dx}=
\displaystyle\int{\displaystyle\frac{dx}{\sqrt{1-(1-2x+2x^2)}}}$$
Let $u=x-1$,
$du=dx$
$$=\displaystyle\int{\displaystyle\frac{dx}{\sqrt{1-(x-1)^2}}}
=\displaystyle\int{\displaystyle\frac{du}{\sqrt{1-u^2}}}$$
$$\qquad{=\sin^{-1}u+C==\sin^{-1}{(x-1)}+C}$$

 

[Mark44]

Solution

$$\displaystyle\int{\displaystyle\frac{1}{\sqrt{2x-x^2}}\,dx}=
\displaystyle\int{\displaystyle\frac{dx}{\sqrt{1-(1-2x+2x^2)}}}$$

You have a typo in the line above. The coefficient of the $x^2$ term should be 1, not 2.

 

[quasar987]

To extrapolate a bit on e_jane's answer... I was watching recently the RI talk with John Stillwell () and he was saying that for the LONGEST TIME people were not working with symbols. These started being used as we do today only around the year 1600 (!) On the other hand, the square completion algorithm goes back at least to around 820 when the arab Al-Khwârismî (whose name was to become the word 'Algorithm') published the first algebra textbook. His book was called Kitābu 'l-mukhtaṣar fī ḥisābi 'l-jabr wa'l-muqābalah. It contains the word 'l-jabr which would become the word algebra. So anyway, this book contains the famous square completion algorithm for solving quadratic equations. But there are no symbols, remember? So the algorithm is all spelled out laboriously by invoking actual squares and rectangles.