### Complete The Square

Exam II. (Math 104.)

Polynomials. Quadratics in particular.

Bad news.

The campus-wide median score slipped

considerably and so did ours.

A lot of damage was done on the page

having problems of the forms

(a.) Complete the Square…

and

(b.) Use the Quadratic Formula…

(… to solve a given quadratic equation.).

I’ve written a little about QF in the past;

one of my most popular posts. And I like

getting attention as much as the next blogger.

Here are some remarks on “completing the square”.

So. Imagine me… imagine *us*…

scribbling on a blackboard and talking.

Let’s look at a quadratic equation

whose answer we *already know*.

Suppose that [ (x = 3) OR (x = -9)].

Then (“obviously”… check it!)

(x-3)(x+9) = 0.

“FOIL”

(or, finding this distasteful, expand

the product-of-binomials on the Left

Hand Side by some “other” algorithm):

x^2 + 6x – 27 = 0.

Now. Suppose it were some bloody

Exam Problem. “Complete the Square.”

Digress.

The trick is to find

a “perfect square trinomial”

*having the same variable terms*

as *our* trinomial (x^2 + 6x – 27).

We want

(x+?)^2 = x^2 + 6x + ___.

And the thing *here* is that

(x+?)^2 = x^2 + (2*?)*x + ?^2.

It follows that 2*? = 6, and so ? = 3.

Obviously, then, ?^2 = 9.

We’ve established that

(x+3)^2 = x^2 + 6x + 9

has the same variable terms

as our original polynomial.

We’re ready to go. (End digression.)

Start with x^2 + 6x – 27 = 0.

Add on both sides (to “isolate”

the variable terms):

x^2 + 6x = 27.

Here comes the magic…

the step that gives “Complete the Square”

its name. Add on both sides *again*:

x^2 + 6x + 9 = 27 + 9.

(the point here was to get the

“perfect square trinomial”

we computed in the digression;

the LHS has been transformed

into a easily-manipulated form

[abstraction to the rescue!].)

Now just use the “perfect square” property:

x^2 + 6x + 9 = 27 + 9

becomes

(x+3)^2 = 36. So

x+3 = plus-or-minus root-36;

x = -3 plus-or-minus 6;

finally

x = 3 or x = -9

(as we already knew from the

factored form we began with).

With the example in hand, the best

thing is to go off and *do* a bunch

of similar examples. And *then*

consider the “abstract” version

used in deriving the Quadratic Formula.

Never; the less.

Let A, B, and C be Complex Numbers.

(OK… let ’em be elements of your

favorite Field. Rationals, Reals,

and Complexes all work; so do many

other Fields not considered in

School Mathemathics (alas).

The point of using …

the Complex Number field…

is that the “answers” will always “exist”.

Famously x^2 + 1 = 0 has no (so-called)

“real number” answers; this kind of thing

doesn’t happen in “algebraically closed”

fields like the Complex Numbers.)

Suppose further that A\not=0.

The Following Are Equivalent.

Ax^2 + Bx + C = 0

x^2 + (B/A)x = -C/A

x^2 + (B/A)x + (B/[2A])^2 = -C/A + B^2/[4A^2]

(x+ B/[2A])^2 = [-4CA]/[4A^2] + B^2/[4A^2]

x + B/[2A] = \pm \sqrt {[B^2 – 4AC]/[4A^2]

x = [-B \pm \sqrt {B^2 – 4AC}]/[2A].

In other words, when A is nonzero one has

*i.e.*, QF: we have derived

the Quadratic Formula.

(Remark: one need *not* be able to reproduce

this calculation to “prove” QF. Suppose you’ve

*memorized* it at some point but now you’re not

so sure. To check that *your* formula really is

the true “quadratic” formula, just choose either

the “plus” or the “minus” in the appropriate

place in the code [where i’ve abbreviated \pm

where i haven’t just written-it-out].

Then just “plug in” the whole mess on

Ax^2 + Bx + C and turn the crank until

out pops zero [if your formula and calculations

are correct]. It’s easier to *check*

that the formula works than to *derive* it.

It’s a good exercise, too, in my opinion…

but I don’t think I’ve ever assigned it

[or seen it assigned in a textbook].

)

Here’s a verbal recap.

We began by “dividing away the leading coefficient”.

This is where we “use” the condition A\not=0.

(Of course, it also makes sense to say that we “used”

this condition already in calling Ax^2 + Bx + C = 0

a quadratic equation in the first place…

if A were 0, one would have the *linear* equation

Bx+C=0. (We can handle these *without* memorizing

the “Linear Formula” [the equation is equivalent…

when B\not=0… to x = -C/B]. Instead one learns

“steps” [subtract C from both sides; divide by B].

For that matter, while I’m thinking about it,

neither does one expect students to learn that

“y = Mx + B” equivalences-to “x = (y-B)/M”

[when, dammit, M\not=0… I can’t help myself…];

rather, again, one simply carries out certain “steps”

to *dervive* this result whenever it’s needed.

Formulas-versus-procedures is a major battleground

in the Math Ed Wars so I think about this stuff.

A great deal sometimes.

)

Having “divided through by A” we get

(the so-called “monic” polynomial…

*having lead coefficient 1*

is a big enough deal to deserve

its own name…) x^2 + (B/A)x + C/A = 0;

subtracting the constant on sides gives

x^2 + (B/A)x = -C/A.

The number (B/A) is in the position of

the “6” in our first example.

Just as it turned out in the example,

where we add 9 to both sides because

9 is (6/2)^2, in *every* instance

of “complete the square” we add

the square of half of the constant-coefficient

(of the monic form) to both sides of the equation.

x^2 + (B/A)x + (B/[2A])^2 = -C/A + B^2/[4A^2]

Factor on the left; take square roots

on both sides (slightly tricky;

don’t forget the \pm!); clean up.

Coffee!

February 24, 2011 at 11:40 pm

Nice explanation of the standard way to complete the square. James Tanton has a video or two explaining an alternate way. (And then deriving the QF.) I taught it his way last semester, and it seemed to stick better.

February 26, 2011 at 5:54 pm

i looked at enough of one of tanton’s videos

to see that he began his derivation by (TFAE):

Ax^2 + Bx + C = 0

(AX)^2 + ABx + AC = 0

(2AX)^2 + 4ABx + 4AC = 0.

then i quit.

i can honestly say that this appears to be

outstanding work by professor tanton.

(also i like the sound of his voice.)

but i’m repelled by “new media” pretty reliably

and i turned it off at about 100\pm20 seconds.

like (talk) radio and television, it’s too slow.

(mostly; sometimes they’re too fast… the point

is that in *reading*, we each set our own pace.)

probably this format… where we see only the writing…

is much *better* than a full-blown video of a lecture.

if i had the tools at hand, i’d probably try

this method of presentation as an exercise.

(and it’s hard to imagine a *better* gig

if one could only make it *pay*…)

now. *obviously* i don’t intend to critique

the video. again: i haven’t even *seen* it.

but. no… wait.

here’s some more stuff in its praise.

*the “multiply both sides by 4A”

approach looks, to these jaded eyes,

like a *darn good idea*.

multiplication is easier

to think about than division.

(i’ve begun to believe that this

is one of the *main themes* of

history-and-philosophy-of-maths…

to say nothing of math-ed…

much too much to go into here

[“fractions are always hard”…

and then there are “quotient spaces”…].)

meanwhile. heck. ain’t *i*

the very guy that’s always

taking every available opportunity

to proclaim that “our medium

is handwriting”? umm. yeah.

i’m that guy.

our medium *is* handwriting. (see?)

and dammit if videos like tanton’s

(and vi hart’s) don’t *prove* it!

but. (end “praise”.)

what am *i* gonna do with it?

our medium is *not*

videos-about-handwriting!

“access to tools” was the slogan of,

at least, a sub-sub-culture:

the Whole Earth Catalogue

blew many a mind i think.

(anyway it sure blew mine

[and co-ev quarterly was also

pretty great. stewart brand

and some of the rest of the crew

became high-tech enthusiasts

somewhere along the line;

i lost track.].)

so. paper and pencils i’ve got. erasers.

in fact, and this might be *very* important:

i’ve got enough that i’ll gladly *give ’em away*

if it helps keep the discussion going.

i give away my zines too of course

(i use high-tech tools to make ’em

of course… but the by-hand objects

i’m reproducing are very *low*-tech

[make a virtue of necessity]).

computers and telephones and such?

video cameras? “smartboards”?

of course not.

and nobody appears to be

giving ’em away real freely, either

… not around here, anyway…

not while they still actually *work*.

if and when you *get* ’em?

can’t *maintain* ’em.

teach a man to fish.

then kill him and *take* his fish.

rape his wife and sell his kids

into slavery. western culture, baby.

we’re all indians now.

hmmm. the drugs are kicking in.

if i were hunter thompson,

i’d probably throw away

everything i’ve written

in this post *so far*

and begin the *real* post.

but hunter’s gone.

and *i*… have an uneasy feeling.

(besides wanting to throw up, i mean.

we can more or less take that for granted.)

@sue v.

all by way of backhanded apology

for taking so little interest in

this stuff you’re so enthusiastic about.

sure and i’m glad that you brought this

stuff to my attention here…

but… “get it in writing!”.

archimedes drew in the sand.

PS:

where’s *ben*, dammit.

bloggers like that are *rare*.

OT

his everlasting mark

(while it lasts)

V.

February 26, 2011 at 11:29 pm

You want writing? No problem. I blogged about the serendipity of learning this the day before I was planning to teach completing the square.

March 4, 2011 at 1:38 pm

a fine post (and outstanding comment thread);

thanks for the reminder. if memory serves,

i’ll’ve seen the post but missed the rest of

the thread (till now).

i guess it’s just “teach complete-the-square” *time*…

comes around several times a year (thank The Force

i’m an algebra teacher).

March 4, 2011 at 11:29 pm

http://xkcd.com/857/

“archimedes” in xkcd.

June 30, 2013 at 8:39 am

http://function-of-time.blogspot.com/2011/11/completing-square.html

the “genius method” by rockstar-in-denial kate nowak

http://function-of-time.blogspot.com/2013/06/rock-stars-and-gift-culture.html