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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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00:07:38,928 --> 00:07:43,070
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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00:14:43,955 --> 00:14:49,295
Let's take one final
example and this time when the
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coefficient of x squared is
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00:14:51,520 --> 00:14:53,986
actually negative.
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00:14:54,106 --> 00:14:57,991
So for this will take our quadratic function
to be f of x
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equals... 3 plus
186
00:15:02,084 --> 00:15:08,510
8x minus
2(x squared).
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00:15:09,520 --> 00:15:16,002
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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00:15:21,095 --> 00:15:24,336
occasion it's minus 2.
The "- 2" comes out.
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00:15:24,336 --> 00:15:31,805
Times by x squared, we
take a minus 2 out of the 8x,
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00:15:31,805 --> 00:15:39,080
that leaves us minus 4x and the
minus 2 out of the 3 is a
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00:15:39,080 --> 00:15:42,960
factor which gives us minus
three over 2.
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00:15:44,760 --> 00:15:50,870
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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00:15:57,920 --> 00:16:03,560
equal to four, so a has got to
be equal to two.
196
00:16:04,120 --> 00:16:10,324
So our F of X is going
to be minus 2.
197
00:16:11,280 --> 00:16:17,110
X squared minus 4X plus A
squared, so that's +2 squared
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00:16:17,110 --> 00:16:23,470
minus the original 3 over 2,
but we've added on a 2
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00:16:23,470 --> 00:16:30,360
squared, so we need to take it
away again to keep the balance
200
00:16:30,360 --> 00:16:32,480
to keep the equality.
201
00:16:33,820 --> 00:16:40,398
Minus two, this is now our
complete square, so that's X
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00:16:40,398 --> 00:16:47,574
minus two all squared. And here
we've got minus three over 2
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00:16:47,574 --> 00:16:54,444
- 4. Well, let's have it all
over too. So minus four is minus
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00:16:54,444 --> 00:16:59,141
8 over 2, so altogether we've
got minus 11 over 2.
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00:17:00,910 --> 00:17:06,188
And we can look at this. We
can see that when this is
206
00:17:06,188 --> 00:17:10,654
zero, we've got. In this case
a maximum value, because this
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00:17:10,654 --> 00:17:15,526
is a negative X squared term.
So we know that we're looking
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00:17:15,526 --> 00:17:20,398
for a graph like that. So it's
this point that we're looking
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00:17:20,398 --> 00:17:24,052
for the maximum point, and so
therefore maximum value.
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00:17:26,300 --> 00:17:28,308
Solve F of X.
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00:17:28,810 --> 00:17:33,828
Will occur when this square term
is equal to 0 'cause the square
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00:17:33,828 --> 00:17:36,530
term can never be less than 0.
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00:17:37,420 --> 00:17:40,577
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.