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www.mathcentre.ac.uk/.../Completing%20the%20Square_maxima_minima.mp4

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    Completing the square is a
    process that we make use of in a
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    number of ways. First, we can
    make use of it to find maximum
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    and minimum values of quadratic
    functions, second we can make
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    use of it to simplify or change
    algebraic expressions in order
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    to be able to calculate the
    value that they have. Third, we
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    can use it for solving quadratic
    equations. In this particular
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    video, we're going to have a
    look at it for finding max- and
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    min-imum values of functions,
    quadratic functions.
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    Let's begin by looking at a very specific
    example.
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    Supposing we've got x squared,
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    plus 5x,
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    minus 2. Now.
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    x squared, it's positive, so one
    of the things that we do know is
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    that if we were to sketch the
    graph of this function.
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    It would look something perhaps
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    like that. Question is where's
    this point down here?
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    Where's the minimum value of
    this function? What value of x
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    does it have? Does it actually
    come below the x-axis as I've
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    drawn it, or does it come up
    here somewhere? At what value of X
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    does that minimum value occur?
    We could use calculus if we knew
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    calculus, but sometimes we don't
    know calculus. We might not have
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    reached it yet.
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    At other occasions it might be
    rather like using a sledgehammer
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    to crack or not, so let's have a
    look at how we can deal with
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    this kind of function.
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    What we're going to do is
    a process known as
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    "completing ... the ... square"
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    OK, "completing the square",
    what does that mean?
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    Well, let's have a look at something
    that is a "complete square".
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    That is, an exact square.
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    So that's a complete and exact
    square. If we multiply out the
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    brackets, x plus a times by x
    plus a, what we end up with
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    is x squared...
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    that's x times by x...
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    a times by x, and of course
    x times by a, so that gives
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    us 2ax, and then finally a
    times by a...
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    and that gives us a squared.
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    So this expression is
    a complete square, a complete
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    and exact square. Because it's "x
    plus a" all squared.
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    Similarly, we can have "x minus a"
    all squared.
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    And if we
    multiply out, these brackets
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    we will end up with the same
    result, except, we will have
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    minus 2ax plus a squared. And
    again this is a complete
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    square an exact square
    because it's equal to x minus a...
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    all squared.
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    So,... we go back to this.
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    Expression here x squared, plus
    5x, minus two and what we're
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    going to do is complete the
    square. In other words we're going to
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    try and make it look like this.
    We're going to try and complete it.
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    Make it up so it's a full
    square. In order to do that,
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    what we're going to do is
    compare that expression directly
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    with that one.
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    And we've chosen this expression
    here because that's a plus sign
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    plus 5x, and that's a plus sign
    there plus 2ax.
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    So.
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    x squared, plus
    5x, minus 2.
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    And we have x squared
    plus 2ax plus a squared
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    These two match up
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    Somehow we've got
    to match these two up.
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    Well,... the x's are the same.
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    So the 5 and the 2a have got
    to be the same and that would
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    suggest to us that a has got to
    be 5 / 2.
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    So that x squared plus
    5x minus 2...
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    becomes x squared plus 5x...
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    now... plus a squared and
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    now we decided that 5 was
    to be equal to 2a
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    and so a was equal
    to 5 over 2.
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    So to complete the square, we've
    got to add on 5 over 2...
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    and square it.
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    But that isn't equal to that.
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    It's equal, this is equal
    to that, but not to that
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    well. Clearly we need to
    put the minus two on.
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    But then it's still not equal,
    because here we've added on
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    something extra 5 over 2 [squared]. So
    we've got to take off that five
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    over 2 all squared. We've got to
    take that away.
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    Now let's look at this bit.
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    This is an exact square. It's
    that expression there.
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    No, this began life as x
    plus a all squared, so this
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    bit has got to be the
    same, x plus (5 over 2) all
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    squared. And now we can
    play with this. We've got minus 2
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    minus 25 over 4. We
    can combine that so we have
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    x plus (5 over 2) all squared...
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    minus...
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    Now we're taking away two, so
    in terms of quarters, that's
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    8 quarters were taking
    away, and we're taking away 25
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    quarters as well, so
    altogether, that's 33
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    quarters that we're taking
    away.
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    Now let's have a look at this
    expression... x squared plus 5x minus 2.
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    Remember what we were
    asking was "what's its minimum value?"
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    Its graph looked like that.
    We were interested in...
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    "where's this point?"
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    "where is the lowest point?"
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    "what's the x-coordinate?"
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    "and what's the y-coordinate?"
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    Let's have a look at this
    expression here.
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    This is a square.
    A square is always
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    positive unless it's equal to 0,
    so its lowest value
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    that this expression [can take] is 0.
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    So the lowest value of
    the whole expression...
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    is that "minus 33 over 4".
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    So therefore we can say
    that the minimum value...
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    of x squared, plus 5x, minus
    2 equals... minus 33 over 4.
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    And we need to be able
    to say when
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    "what's the x-value there?"
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    well, it occurs when this bracket
    is at its lowest value.
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    When this bracket is at
    0. In other words, when x equals...
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    minus 5 over 2.
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    So we found the minimum
    value and exactly when it happens.
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    Let's take a second example. Our
    quadratic function this time, f of x,
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    is x squared, minus 6
    x, minus 12.
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    We've got a minus sign in here, so
    let's line this up with the
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    complete square: x squared, minus
    2ax, plus a squared.
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    The x squared terms are the same,
    and we want these two to be the
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    same as well. That clearly means
    that 2a has got to be the same
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    as 6, so a has got to be 3.
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    So f(x) is equal
    to x squared, minus 6x,
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    plus...
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    a squared (which is 3 squared),
    minus 12, and now we added on
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    3 squared. So we've got to take
    the 3 squared away in order
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    to make it equal. To keep the
    value of the original expression
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    that we started with.
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    We can now identify this as
    being (x minus 3) all squared.
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    And these numbers at the end...
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    minus 12 minus 9, altogether
    gives us minus 21.
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    Again, we can say does it have a
    maximum value or a minimum value?
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    Well what we know that we began
    with a positive x squared term,
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    so the shape of the graph
    has got to be like that. So we
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    know that we're looking for a
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    minimum value. We know that that
    minimum value will occur when
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    this bit is 0 because it's a
    square, it's least value is
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    going to be 0, so therefore we
    can say the minimum value.
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    of our quadratic function f of x
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    is minus 21, [occurring] when...
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    this bit is 0. In other words,
    when x equals 3.
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    The two examples we've taken so far
    have both had a positive x squared
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    and a unit coefficient
    of x squared, in other words 1 x squared.
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    We'll now look at an example
    where we've got a number here
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    in front of the x squared.
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    So the example
    that will take.
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    f of x equals 2x squared,
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    minus 6x, plus one.
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    Our first step is to take out
    that 2 as a factor.
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    2, brackets x squared,
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    minus 3x,...
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    we've got to take
    the 2 out of this as well, so
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    that's a half.
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    And now we do the same as we've
    done before with this bracket here.
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    We line this one up with x squared,
    minus 2ax, plus a squared.
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    When making these two terms the
    same 3 has to be the same as 2a,
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    and so 3 over 2 has to
    be equal to a.
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    So our function f of x is going
    to be equal to 2 times...
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    x squared, minus 3x,...
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    now we want plus a squared,
    so that's plus (3 over 2) all squared
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    Plus the half that
    was there originally and now
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    we've added on this, so we've
    got to take it away,...
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    (3 over 2) all squared. And finally we
    opened a bracket, so we must
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    close it at the end.
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    Equals... 2, bracket,... now this
    is going to be our complete square
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    (x minus 3 over 2) all squared.
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    And then here we've got some
    calculation to do. We've plus a half,
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    take away (3 over 2) squared,
    so that's plus 1/2
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    take away 9 over 4.
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    The front bit is going to stay the same
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    And now we can juggle with
    these fractions. At the end,
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    we've got plus 1/2 take away 9
    quarters or 1/2 is 2 quarters,
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    so if we're taking
    away, nine quarters must be
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    ultimately taking away 7 quarters.
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    So again, what's the
    minimum value of this function?
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    It had a positive 2 in front
    of the x squared, so again, it
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    looks like that. And again,
    we're asking the question,
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    "what's this point down here?"
    What's the lowest point and that
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    lowest point must occur when
    this is 0.
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    So the min ...
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    value of f of x must be
    equal to... now that's going to be 0
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    But we're still multiplying
    by the 2, so it's 2 times
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    minus 7 over 4. That's minus
    14 over 4, which reduces to
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    minus 7 over 2. When?
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    And that will happen when this
    is zero. In other words, when x
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    equals 3 over 2.
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    So a minimum value of minus
    7 over 2 when x equals 3 over 2.
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    Let's take one final
    example and this time when the
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    coefficient of x squared is
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    actually negative.
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    So for this will take our quadratic function
    to be f of x
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    equals... 3 plus
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    8x minus
    2(x squared).
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    We operate in just the same way
    as we did before. We take out
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    the factor that is multiplying
    the x squared and on this
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    occasion it's minus 2.
    The "- 2" comes out.
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    Times by x squared, we
    take a minus 2 out of the 8x,
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    that leaves us minus 4x and the
    minus 2 out of the 3 is a
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    factor which gives us minus
    three over 2.
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    We line this one up with X
    squared minus 2X plus A squared.
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    Those two are the same. We want
    these two to be the same. 2A is
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    equal to four, so a has got to
    be equal to two.
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    So our F of X is going
    to be minus 2.
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    X squared minus 4X plus A
    squared, so that's +2 squared
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    minus the original 3 over 2,
    but we've added on a 2
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    squared, so we need to take it
    away again to keep the balance
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    to keep the equality.
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    Minus two, this is now our
    complete square, so that's X
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    minus two all squared. And here
    we've got minus three over 2
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    - 4. Well, let's have it all
    over too. So minus four is minus
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    8 over 2, so altogether we've
    got minus 11 over 2.
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    And we can look at this. We
    can see that when this is
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    zero, we've got. In this case
    a maximum value, because this
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    is a negative X squared term.
    So we know that we're looking
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    for a graph like that. So it's
    this point that we're looking
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    for the maximum point, and so
    therefore maximum value.
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    Solve F of X.
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    Will occur when this square term
    is equal to 0 'cause the square
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    term can never be less than 0.
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    And so we have minus two times.
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    Minus 11. And altogether that
    gives us plus 11.
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    And it will occur when this.
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    Is equal to 0. In other
    words, when X equals 2.
Title:
www.mathcentre.ac.uk/.../Completing%20the%20Square_maxima_minima.mp4
Video Language:
English
Duration:
18:02

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