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Completing the square

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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.
  • 13:42 - 13:45
    Anyway, I'll see you
    in the next video.
Title:
Completing the square
Description:

Khan Academy's video on completing the square.

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Video Language:
English
Duration:
13:45

English subtitles

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