Completing the square

Khan Academy's video on completing the square.

Welcome to the video on
completing the square.

What's completing the square?

Well, it's a way to solve
a quadratic equation.

And so before I--actually, let me just write
down a quadratic equation, and

then I will show you how
to complete the square.

And then we'll do another
example, and then maybe we'll talk

a little bit about why it's
called completing the square.

So let's say I have this
equation: x squared plus 16x

minus 57 is equal to 0.

So what are the tools in our
toolkit right now that we

could use to solve this?

Well, we could try
to factor it out.

We could say, what two numbers
add up to 16, and then when you

multiply them they're minus 57?

And you'd have to think
about it a little bit.

And you might get whole
numbers, but you're not even

sure if there are two whole
numbers that work

out like that.

This problem there are.

But, you know, sometimes the
solution is a decimal number

and you don't know it.

So the only time you can really
factor is if you're sure that

you could factor this into
kind of integer expressions.

You know, x plus some integer
or x minus some integer

times, you know, x plus
some other integer.

Or likewise.

The other option is to do
the quadratic equation.

And what we're going to see is
actually the quadratic equation

is just essentially a shortcut
to completing the square.

The quadratic equation is
actually proven using

completing the square.

So what is completing
the square?

So what do we do?

Well, before we move into this
video let's see what happens

if I square an expression.

Let me do it in this down here.

What is x plus a, squared?

Well that equals x squared
plus 2ax plus a squared.

Right?

So if you ever see something in
this form, you know that it's

x plus something squared.

So wouldn't it be neat if we
could manipulate this equation

so we can write it as x plus a
squared equals something,

and then we could just
take the square root?

And what we're going to do
is, actually, do just that.

And that is completing
the square.

So let me show you an example.

I think an example will
make it a little clearer.

Let me box this away.

This is what you
need to remember.

This is the whole rationale
behind competing the squares--

to get an equation into this
form, onto one side of the

equation, and just have a
number on the other side, so

you could take the square
root of both sides.

So let's see.

First of all, let's just check
to make sure this isn't

a perfect square.

If this were, this coefficient
would be equivalent to the 2a.

Right?

So a would be 8, and
then this would be 64.

This is clearly not 64, so
this right here is not

a squared expression.

So what can we do?

Well let me get rid of the
57 by adding 57 to both

sides of this equation.

So I would get x squared
plus 16x is equal to 57.

All I did is I added 57 to
both sides of this equation.

Now, what could I add here so
that this, the left-hand side

of this equation, becomes a
square of some expression

like x plus a?

If you just follow this pattern
down here, we have x squared

plus 2ax-- so you could view
this right here as 2ax.

Right?

That's 2ax.

And then we need to add
an a squared to it.

Right?

Plus a squared.

And then we would
have the form here.

But we know from basic algebra
that anything you do to one

side of an equation you
have to do to another.

So we added an a squared
here, so let's add an a

squared here as well.

And now you could essentially
rewrite this as a square

of some expression.

But before that we have to
figure out what a was?

Well how do we do that?

Well, what is a?

If this expression right
here is 2ax, what is a?

Well 2a is going to equal
16, so a is equal to 8.

And you could usually do
that just by inspection;

do it in your head.

But if you wanted to see it
done algebraically you could

actually write 2ax
is equal to 16x.

And then divide both sides
by 2x, and you get a is

equal to 16x over 2x.

And assuming that x doesn't
equal 0 this evaluates to 8.

So a is 8.

So if a is 8 we could rewrite
that expression-- I'll switch

colors arbitrarily-- as x
squared plus 16x

plus a squared.

Well, it's 64, because a is 8.

Is equal to 57 plus 64.

Right?

I went through a fairly tedious
explanation here, but all we've

really done to get from there
to there is we added 57 to both

sides of this equation to kind
of get it on the right-hand

side, and then we added 64 to
both sides of this equation.

And why did I add 64 to both
sides of this equation?

So that the left-hand side
expression takes this form.

Now that the left-hand side
expression takes this form

I can rewrite it as what?

x plus a, squared.

I can rewrite it in this form.

And we know that a is 8, so it
becomes x plus 8, squared,

is equal to-- and
what's 57 plus 64?

It's 121.

Now we have what looks like a
fairly straightforward-- it's

still a quadratic equation,
actually, because if you

were to expand this side
you'd get a quadratic.

But we can solve this without
using the quadratic equation

or without having to factor.

We can just take the square
root of both sides of this.

And if we take the square root
of both sides what do we get?

We get-- once again,
arbitrarily switching colors--

that x plus 8 is equal to, and
remember this, the plus or

minus square root of 121.

And what's the
square root of 121?

Well it's 11, right?

So then we come here.

Let me box this away.

This was just an aside.

So we get x plus 8 is equal
to plus or minus 11.

And so x is equal to-- subtract
8 from both sides-- minus

8 plus or minus 11.

And so x could equal-- so
minus 8 plus 11 is 3.

Right?

Let me make sure I
did that right.

x is equal to minus
8 plus or minus 11.

Yes.

That's right.

So x could be equal to 3.

And then if I took minus
8 minus 11, x could

also equal minus 19.

All right.

And let's see if
that makes sense.

So in theory this should be
able to be factored as x

minus 3 times x plus
19 is equal to 0.

Right?

Because these are the two
solutions of this equation.

And that works out, right?

Minus 3 times 19 is minus 57.

And minus 3 plus
19 is plus 16x.

We could have just immediately
factored it this way, but if

that wasn't obvious to us--
because, you know, at least

19 is kind of a strange
number-- we could do it

completing the square.

And so why is it called
completing the square?

Because you get it in this form
and then you have to add this

64 here to kind of complete the
square-- to turn this

left-hand expression into
a squared expression.

Let's do one more.

And I'll do less explanation
and more just chugging through

the problem, and that actually
might make it seem simpler.

But this is going to
be a hairier problem.

So let's say I have 6x squared
minus 7x minus 3 is equal to 0.

You could try to factor it,
but personally I don't

enjoy factoring things
when I have a coefficient.

And you can say, oh well why
don't we divide both sides

of this equation by 6?

But then you'd get a fraction
here and a fraction here.

And that's even worse to
factor just by inspection.

You could do the
quadratic equation.

And maybe I'll show you in a
future video, the quadratic

equation-- and I think I've
already done one where I proved

the quadratic equation.

But the quadratic
equation is essentially

completing the square.

It's kind of a shortcut.

It's just kind of
remembering the formula.

But let's complete the square
here, because that's what the

point of this video was.

So let's add the 3 to both
sides of that equation.

We could do-- well,
let's add the 3 first.

So you get 6 x squared
minus 7x is equal to 3.

I added 3 to both sides.

And some teachers will leave
the minus 3 here, and then try

to figure out what to add
to it and all of that.

But I like to get it out of the
way so that I can figure out

very clearly what number
I should put here.

But I also don't
like the 6 here.

It just complicates things.

I like to have it x plus a
squared, not some square root

coefficient on the x term.

So let's divide both sides of
this equation by 6, and you get

x squared minus 7/6 x is equal
to-- 3 divided by 6

is equal to 1/2.

And we could have made
that our first step.

We could have divided by 6
right at that first step.

Anyway, now let's try to
complete the square.

So we have x squared-- I'm just
going to open up some space--

minus 7/6 x plus something is
going to be equal to 1/2.

And so we have to add something
here so that this left-hand

expression becomes a
squared expression.

So how do we do that?

Well essentially we look at
this coefficient, and keep

in mind this is not just
7/6 it's minus 7/6.

You take 1/2 of it, and
then you square it.

Right?

Let me do it.

x plus a, squared, is
equal to x squared plus

2ax plus a squared.

Right?

This is what you have to
remember all the time.

That's all completing the
square is based off of.

So what did I say just now?

Well, this term is going
to be 1/2 of this

coefficient squared.

And how do we know that?

Because a is going to be 1/2 of
this coefficient if you just

do a little bit of
pattern matching.

So what's 1/2 of
this coefficient?

1/2 of minus 7/6 is minus 7/12.

So if you want you could
write a equals minus

7/12 for our example.

And I just multiplied
this by 1/2.

Right?

So what do I add to both sides?

I add a squared.

So what's 7/12 squared?

Well that's going to be 49/144.

If I did it to the left-hand
side I have to do it to

the right-hand side.

Plus 49/144.

And now how can I simplify
this left-hand side?

What's our next step?

Well we now know it
is a perfect square.

In fact, we know what a
is. a is minus 7/12.

And so we know that this
left-hand side of this equation

is x minus a-- or x plus a,
but a is a negative number.

So x plus a, and a is
negative, squared.

And if you want you can
multiply this out and confirm

that it truly equals this.

And that is going to be equal
to-- let's get a common

denominator, 144.

So 72 plus 49 equals 121.

121/144.

So we have x minus 7/12,
all of that squared

is equal to 121/144.

So what do we do now?

Well now we just take
the square root of both

sides of this equation.

And I'm trying to
free up some space.

Switch to green.

Let me partition this off.

And we get x minus 7/12 is
equal to the plus or minus

square root of that.

So plus or minus 11/12.

Right?

Square root of 121 is 11.

Square root of 144 is 12.

So then we could add 7/12 to
both sides of this equation,

and we get x is equal to
7/12 plus or minus 11/12.

Well that equals 7
plus or minus 11/12.

So what are the two options?

7 plus 11 is 18, over 12.

So x could equal 18/12, is 3/2.

Or, what's 7 minus 11?

That's minus 4/12.

So it's minus 1/3.

There you have it.

That is completing the square.

Hopefully you found that
reasonably insightful.

And if you want to prove the
quadratic equation, all you

have to do is instead of having
numbers here, write a x squared

plus bx plus c equals 0.

And then complete the square
using the a, b, and c's

instead of numbers.

And you will end up with
the quadratic equation

by this point.

And I think I did
that in a video.

Let me know if I didn't
and I'll do it for you.

Anyway, I'll see you
in the next video.