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…
(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.
(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.”


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
(x+3)^2 = 36. So
x+3 = plus-or-minus root-36;
x = -3 plus-or-minus 6;
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 {\Bbb C}
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
Ax^2+Bx+C =0 \Leftrightarrow x =  {{-B \pm \sqrt{B^2 - 4AC}}\over{2A}}\,,
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.



  1. 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.

  2. 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
    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*

    “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.

    where’s *ben*, dammit.
    bloggers like that are *rare*.

    his everlasting mark
    (while it lasts)

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

  4. 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).

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