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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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And if you want you can
multiply this out and confirm
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that it truly equals this.
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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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121/144.
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So we have x minus 7/12,
all of that squared
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is equal to 121/144.
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So what do we do now?
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Well now we just take
the square root of both
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sides of this equation.
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And I'm trying to
free up some space.
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Switch to green.
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Let me partition this off.
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And we get x minus 7/12 is
equal to the plus or minus
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square root of that.
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So plus or minus 11/12.
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Right?
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Square root of 121 is 11.
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Square root of 144 is 12.
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So then we could add 7/12 to
both sides of this equation,
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and we get x is equal to
7/12 plus or minus 11/12.
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Well that equals 7
plus or minus 11/12.
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So what are the two options?
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7 plus 11 is 18, over 12.
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So x could equal 18/12, is 3/2.
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Or, what's 7 minus 11?
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That's minus 4/12.
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So it's minus 1/3.
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There you have it.
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That is completing the square.
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Hopefully you found that
reasonably insightful.
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And if you want to prove the
quadratic equation, all you
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have to do is instead of having
numbers here, write a x squared
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plus bx plus c equals 0.
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And then complete the square
using the a, b, and c's
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instead of numbers.
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And you will end up with
the quadratic equation
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by this point.
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And I think I did
that in a video.
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Let me know if I didn't
and I'll do it for you.
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Anyway, I'll see you
in the next video.