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Welcome to the video on
completing the square.

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What's completing the square?

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

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

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then I will show you how
to complete the square.

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

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

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

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minus 57 is equal to 0.

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

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could use to solve this?

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Well, we could try
to factor it out.

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

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multiply them they're minus 57?

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

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

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sure if there are two whole
numbers that work

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out like that.

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This problem there are.

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

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and you don't know it.

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

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you could factor this into
kind of integer expressions.

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

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times, you know, x plus
some other integer.

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Or likewise.

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The other option is to do
the quadratic equation.

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

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is just essentially a shortcut
to completing the square.

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The quadratic equation is
actually proven using

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completing the square.

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So what is completing
the square?

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So what do we do?

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

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if I square an expression.

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Let me do it in this down here.

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What is x plus a, squared?

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

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Right?

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

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x plus something squared.

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

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

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and then we could just
take the square root?

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

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And that is completing
the square.

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So let me show you an example.

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I think an example will
make it a little clearer.

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Let me box this away.

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This is what you
need to remember.

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This is the whole rationale
behind competing the squares--

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

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

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you could take the square
root of both sides.

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So let's see.

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

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a perfect square.

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

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Right?

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

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

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a squared expression.

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So what can we do?

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

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sides of this equation.

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

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

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

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of this equation, becomes a
square of some expression

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like x plus a?

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

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

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Right?

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That's 2ax.

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

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Right?

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Plus a squared.

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And then we would
have the form here.

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

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side of an equation you
have to do to another.

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

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squared here as well.

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And now you could essentially
rewrite this as a square

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of some expression.

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

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Well how do we do that?

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Well, what is a?

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

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

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And you could usually do
that just by inspection;

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do it in your head.

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But if you wanted to see it
done algebraically you could

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actually write 2ax
is equal to 16x.

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

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equal to 16x over 2x.

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

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So a is 8.

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

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colors arbitrarily-- as x
squared plus 16x

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plus a squared.

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

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Is equal to 57 plus 64.

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Right?

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

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

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

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

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

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

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Now that the left-hand side
expression takes this form

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I can rewrite it as what?

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x plus a, squared.

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I can rewrite it in this form.

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

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

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It's 121.

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

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still a quadratic equation,
actually, because if you

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

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But we can solve this without
using the quadratic equation

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or without having to factor.

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

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

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

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

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minus square root of 121.

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And what's the
square root of 121?

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Well it's 11, right?

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So then we come here.

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Let me box this away.

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This was just an aside.

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

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

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8 plus or minus 11.

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

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Right?

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Let me make sure I
did that right.

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x is equal to minus
8 plus or minus 11.

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Yes.

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That's right.

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So x could be equal to 3.

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

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also equal minus 19.

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All right.

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And let's see if
that makes sense.

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

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minus 3 times x plus
19 is equal to 0.

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Right?

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Because these are the two
solutions of this equation.

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And that works out, right?

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Minus 3 times 19 is minus 57.

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And minus 3 plus
19 is plus 16x.

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

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

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

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completing the square.

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And so why is it called
completing the square?

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

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

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left-hand expression into
a squared expression.

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Let's do one more.

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

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

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But this is going to
be a hairier problem.

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

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

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enjoy factoring things
when I have a coefficient.

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

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of this equation by 6?

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

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

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You could do the
quadratic equation.

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

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

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the quadratic equation.

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But the quadratic
equation is essentially

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completing the square.

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It's kind of a shortcut.

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It's just kind of
remembering the formula.

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

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point of this video was.

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

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

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

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I added 3 to both sides.

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

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

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

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very clearly what number
I should put here.

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But I also don't
like the 6 here.

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It just complicates things.

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

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coefficient on the x term.

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

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

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is equal to 1/2.

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And we could have made
that our first step.

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

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

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

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

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

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expression becomes a
squared expression.

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So how do we do that?

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Well essentially we look at
this coefficient, and keep

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

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

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Right?

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Let me do it.

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

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2ax plus a squared.

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Right?

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This is what you have to
remember all the time.

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

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So what did I say just now?

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

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coefficient squared.

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And how do we know that?

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

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do a little bit of
pattern matching.

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So what's 1/2 of
this coefficient?

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

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So if you want you could
write a equals minus

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7/12 for our example.

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And I just multiplied
this by 1/2.

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Right?

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So what do I add to both sides?

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I add a squared.

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So what's 7/12 squared?

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Well that's going to be 49/144.

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

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the right-hand side.

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Plus 49/144.

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

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What's our next step?

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Well we now know it
is a perfect square.

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

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

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

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

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00:11:47,980 --> 00:11:50,350
And if you want you can
multiply this out and confirm

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that it truly equals this.

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00:11:53,130 --> 00:11:55,920
And that is going to be equal
to-- let's get a common

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denominator, 144.

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So 72 plus 49 equals 121.

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00:12:04,070 --> 00:12:06,300
121/144.

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00:12:06,300 --> 00:12:09,210
So we have x minus 7/12,
all of that squared

248
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is equal to 121/144.

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00:12:13,180 --> 00:12:14,300
So what do we do now?

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00:12:14,300 --> 00:12:15,570
Well now we just take
the square root of both

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00:12:15,570 --> 00:12:17,700
sides of this equation.

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00:12:17,700 --> 00:12:20,140
And I'm trying to
free up some space.

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00:12:20,140 --> 00:12:22,215
Switch to green.

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00:12:22,215 --> 00:12:25,320
Let me partition this off.

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00:12:25,320 --> 00:12:33,310
And we get x minus 7/12 is
equal to the plus or minus

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00:12:33,310 --> 00:12:33,940
square root of that.

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00:12:33,940 --> 00:12:38,120
So plus or minus 11/12.

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00:12:38,120 --> 00:12:38,390
Right?

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00:12:38,390 --> 00:12:39,660
Square root of 121 is 11.

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00:12:39,660 --> 00:12:42,420
Square root of 144 is 12.

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00:12:42,420 --> 00:12:44,480
So then we could add 7/12 to
both sides of this equation,

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00:12:44,480 --> 00:12:53,100
and we get x is equal to
7/12 plus or minus 11/12.

263
00:12:53,100 --> 00:12:58,660
Well that equals 7
plus or minus 11/12.

264
00:12:58,660 --> 00:13:00,050
So what are the two options?

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00:13:00,050 --> 00:13:03,930
7 plus 11 is 18, over 12.

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00:13:03,930 --> 00:13:08,210
So x could equal 18/12, is 3/2.

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00:13:08,210 --> 00:13:11,010
Or, what's 7 minus 11?

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00:13:11,010 --> 00:13:12,760
That's minus 4/12.

269
00:13:12,760 --> 00:13:15,370
So it's minus 1/3.

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00:13:15,370 --> 00:13:16,630
There you have it.

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00:13:16,630 --> 00:13:17,940
That is completing the square.

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00:13:17,940 --> 00:13:20,220
Hopefully you found that
reasonably insightful.

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00:13:20,220 --> 00:13:23,340
And if you want to prove the
quadratic equation, all you

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00:13:23,340 --> 00:13:27,320
have to do is instead of having
numbers here, write a x squared

275
00:13:27,320 --> 00:13:29,820
plus bx plus c equals 0.

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00:13:29,820 --> 00:13:34,130
And then complete the square
using the a, b, and c's

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00:13:34,130 --> 00:13:35,060
instead of numbers.

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00:13:35,060 --> 00:13:37,180
And you will end up with
the quadratic equation

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00:13:37,180 --> 00:13:38,110
by this point.

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00:13:38,110 --> 00:13:39,510
And I think I did
that in a video.

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00:13:39,510 --> 00:13:41,600
Let me know if I didn't
and I'll do it for you.

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00:13:41,600 --> 00:13:44,540
Anyway, I'll see you
in the next video.