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

0:00  0:05Completing the square is a
process that we make use of in a 
0:05  0:09number of ways. First, we can
make use of it to find maximum 
0:09  0:13and minimum values of quadratic
functions, second we can make 
0:13  0:17use of it to simplify or change
algebraic expressions in order 
0:17  0:21to be able to calculate the
value that they have. Third, we 
0:21  0:25can use it for solving quadratic
equations. In this particular 
0:25  0:30video, we're going to have a
look at it for finding max and 
0:30  0:32minimum values of functions,
quadratic functions. 
0:32  0:36Let's begin by looking at a very specific
example. 
0:36  0:39Supposing we've got x squared,

0:39  0:42plus 5x,

0:42  0:45minus 2. Now.

0:46  0:51x squared, it's positive, so one
of the things that we do know is 
0:51  0:54that if we were to sketch the
graph of this function. 
0:55  0:57It would look something perhaps

0:57  1:03like that. Question is where's
this point down here? 
1:04  1:08Where's the minimum value of
this function? What value of x 
1:08  1:13does it have? Does it actually
come below the xaxis as I've 
1:13  1:18drawn it, or does it come up
here somewhere? At what value of X 
1:18  1:22does that minimum value occur?
We could use calculus if we knew 
1:22  1:26calculus, but sometimes we don't
know calculus. We might not have 
1:26  1:28reached it yet.

1:28  1:32At other occasions it might be
rather like using a sledgehammer 
1:32  1:37to crack or not, so let's have a
look at how we can deal with 
1:37  1:38this kind of function.

1:39  1:42What we're going to do is
a process known as 
1:42  1:50"completing ... the ... square"

1:52  1:55OK, "completing the square",
what does that mean? 
1:55  2:00Well, let's have a look at something
that is a "complete square". 
2:00  2:05That is, an exact square.

2:06  2:13So that's a complete and exact
square. If we multiply out the 
2:13  2:20brackets, x plus a times by x
plus a, what we end up with 
2:20  2:22is x squared...

2:23  2:25that's x times by x...

2:26  2:33a times by x, and of course
x times by a, so that gives 
2:33  2:38us 2ax, and then finally a
times by a... 
2:38  2:41and that gives us a squared.

2:41  2:46So this expression is
a complete square, a complete 
2:46  2:51and exact square. Because it's "x
plus a" all squared. 
2:52  2:57Similarly, we can have "x minus a"
all squared. 
2:58  3:01And if we
multiply out, these brackets 
3:01  3:07we will end up with the same
result, except, we will have 
3:07  3:13minus 2ax plus a squared. And
again this is a complete 
3:13  3:17square an exact square
because it's equal to x minus a... 
3:17  3:19all squared.

3:21  3:24So,... we go back to this.

3:24  3:29Expression here x squared, plus
5x, minus two and what we're 
3:29  3:34going to do is complete the
square. In other words we're going to 
3:34  3:39try and make it look like this.
We're going to try and complete it. 
3:39  3:44Make it up so it's a full
square. In order to do that, 
3:44  3:48what we're going to do is
compare that expression directly 
3:48  3:49with that one.

3:50  3:56And we've chosen this expression
here because that's a plus sign 
3:56  4:02plus 5x, and that's a plus sign
there plus 2ax. 
4:02  4:05So.

4:06  4:12x squared, plus
5x, minus 2. 
4:13  4:22And we have x squared
plus 2ax plus a squared 
4:22  4:24These two match up

4:25  4:30Somehow we've got
to match these two up. 
4:30  4:32Well,... the x's are the same.

4:32  4:39So the 5 and the 2a have got
to be the same and that would 
4:39  4:44suggest to us that a has got to
be 5 / 2. 
4:45  4:51So that x squared plus
5x minus 2... 
4:53  4:59becomes x squared plus 5x...

5:00  5:02now... plus a squared and

5:03  5:06now we decided that 5 was
to be equal to 2a 
5:06  5:13and so a was equal
to 5 over 2. 
5:13  5:20So to complete the square, we've
got to add on 5 over 2... 
5:20  5:22and square it.

5:24  5:26But that isn't equal to that.

5:28  5:33It's equal, this is equal
to that, but not to that 
5:33  5:37well. Clearly we need to
put the minus two on. 
5:39  5:43But then it's still not equal,
because here we've added on 
5:43  5:48something extra 5 over 2 [squared]. So
we've got to take off that five 
5:48  5:52over 2 all squared. We've got to
take that away. 
5:53  5:56Now let's look at this bit.

5:57  6:02This is an exact square. It's
that expression there. 
6:03  6:11No, this began life as x
plus a all squared, so this 
6:11  6:18bit has got to be the
same, x plus (5 over 2) all 
6:18  6:25squared. And now we can
play with this. We've got minus 2 
6:25  6:32minus 25 over 4. We
can combine that so we have 
6:32  6:35x plus (5 over 2) all squared...

6:36  6:37minus...

6:39  6:43Now we're taking away two, so
in terms of quarters, that's 
6:43  6:478 quarters were taking
away, and we're taking away 25 
6:47  6:51quarters as well, so
altogether, that's 33 
6:51  6:53quarters that we're taking
away. 
6:54  6:59Now let's have a look at this
expression... x squared plus 5x minus 2. 
6:59  7:03Remember what we were
asking was "what's its minimum value?" 
7:03  7:07Its graph looked like that.
We were interested in... 
7:07  7:08"where's this point?"

7:08  7:10"where is the lowest point?"

7:10  7:12"what's the xcoordinate?"

7:12  7:14"and what's the ycoordinate?"

7:15  7:17Let's have a look at this
expression here. 
7:17  7:22This is a square.
A square is always 
7:22  7:27positive unless it's equal to 0,
so its lowest value 
7:27  7:31that this expression [can take] is 0.

7:31  7:35So the lowest value of
the whole expression... 
7:35  7:39is that "minus 33 over 4".

7:39  7:43So therefore we can say
that the minimum value... 
7:49  7:55of x squared, plus 5x, minus
2 equals... minus 33 over 4. 
7:56  7:59And we need to be able
to say when 
7:59  8:01"what's the xvalue there?"

8:01  8:05well, it occurs when this bracket
is at its lowest value. 
8:05  8:11When this bracket is at
0. In other words, when x equals... 
8:11  8:14minus 5 over 2.

8:17  8:23So we found the minimum
value and exactly when it happens. 
8:26  8:32Let's take a second example. Our
quadratic function this time, f of x, 
8:32  8:37is x squared, minus 6
x, minus 12. 
8:38  8:45We've got a minus sign in here, so
let's line this up with the 
8:45  8:51complete square: x squared, minus
2ax, plus a squared. 
8:51  8:56The x squared terms are the same,
and we want these two to be the 
8:56  9:00same as well. That clearly means
that 2a has got to be the same 
9:00  9:03as 6, so a has got to be 3.

9:04  9:10So f(x) is equal
to x squared, minus 6x, 
9:10  9:12plus...

9:12  9:17a squared (which is 3 squared),
minus 12, and now we added on 
9:17  9:223 squared. So we've got to take
the 3 squared away in order 
9:22  9:28to make it equal. To keep the
value of the original expression 
9:28  9:29that we started with.

9:30  9:38We can now identify this as
being (x minus 3) all squared. 
9:39  9:41And these numbers at the end...

9:42  9:48minus 12 minus 9, altogether
gives us minus 21. 
9:49  9:55Again, we can say does it have a
maximum value or a minimum value? 
9:55  9:58Well what we know that we began
with a positive x squared term, 
9:58  10:02so the shape of the graph
has got to be like that. So we 
10:02  10:04know that we're looking for a

10:04  10:08minimum value. We know that that
minimum value will occur when 
10:08  10:14this bit is 0 because it's a
square, it's least value is 
10:14  10:21going to be 0, so therefore we
can say the minimum value. 
10:22  10:26of our quadratic function f of x

10:26  10:32is minus 21, [occurring] when...

10:33  10:39this bit is 0. In other words,
when x equals 3. 
10:41  10:46The two examples we've taken so far
have both had a positive x squared 
10:46  10:52and a unit coefficient
of x squared, in other words 1 x squared. 
10:52  10:57We'll now look at an example
where we've got a number here 
10:57  11:00in front of the x squared.

11:00  11:04So the example
that will take. 
11:06  11:11f of x equals 2x squared,

11:12  11:17minus 6x, plus one.

11:17  11:23Our first step is to take out
that 2 as a factor. 
11:24  11:282, brackets x squared,

11:28  11:31minus 3x,...

11:32  11:37we've got to take
the 2 out of this as well, so 
11:37  11:39that's a half.

11:39  11:45And now we do the same as we've
done before with this bracket here. 
11:46  11:53We line this one up with x squared,
minus 2ax, plus a squared. 
11:53  12:00When making these two terms the
same 3 has to be the same as 2a, 
12:00  12:06and so 3 over 2 has to
be equal to a. 
12:07  12:14So our function f of x is going
to be equal to 2 times... 
12:15  12:18x squared, minus 3x,...

12:19  12:27now we want plus a squared,
so that's plus (3 over 2) all squared 
12:28  12:32Plus the half that
was there originally and now 
12:32  12:36we've added on this, so we've
got to take it away,... 
12:36  12:42(3 over 2) all squared. And finally we
opened a bracket, so we must 
12:42  12:45close it at the end.

12:46  12:53Equals... 2, bracket,... now this
is going to be our complete square 
12:53  12:59(x minus 3 over 2) all squared.

13:00  13:05And then here we've got some
calculation to do. We've plus a half, 
13:06  13:13take away (3 over 2) squared,
so that's plus 1/2 
13:13  13:15take away 9 over 4.

13:17  13:21The front bit is going to stay the same

13:25  13:27And now we can juggle with
these fractions. At the end, 
13:27  13:32we've got plus 1/2 take away 9
quarters or 1/2 is 2 quarters, 
13:32  13:37so if we're taking
away, nine quarters must be 
13:37  13:40ultimately taking away 7 quarters.

13:40  13:45So again, what's the
minimum value of this function? 
13:45  13:49It had a positive 2 in front
of the x squared, so again, it 
13:49  13:52looks like that. And again,
we're asking the question, 
13:52  13:56"what's this point down here?"
What's the lowest point and that 
13:56  13:59lowest point must occur when
this is 0. 
14:00  14:02So the min ...

14:03  14:11value of f of x must be
equal to... now that's going to be 0 
14:11  14:18But we're still multiplying
by the 2, so it's 2 times 
14:18  14:25minus 7 over 4. That's minus
14 over 4, which reduces to 
14:25  14:27minus 7 over 2. When?

14:28  14:35And that will happen when this
is zero. In other words, when x 
14:35  14:37equals 3 over 2.

14:38  14:44So a minimum value of minus
7 over 2 when x equals 3 over 2. 
14:44  14:49Let's take one final
example and this time when the 
14:49  14:52coefficient of x squared is

14:52  14:54actually negative.

14:54  14:58So for this will take our quadratic function
to be f of x 
14:59  15:02equals... 3 plus

15:02  15:098x minus
2(x squared). 
15:10  15:16We operate in just the same way
as we did before. We take out 
15:16  15:21the factor that is multiplying
the x squared and on this 
15:21  15:24occasion it's minus 2.
The " 2" comes out. 
15:24  15:32Times by x squared, we
take a minus 2 out of the 8x, 
15:32  15:39that leaves us minus 4x and the
minus 2 out of the 3 is a 
15:39  15:43factor which gives us minus
three over 2. 
15:45  15:51We line this one up with X
squared minus 2X plus A squared. 
15:51  15:58Those two are the same. We want
these two to be the same. 2A is 
15:58  16:04equal to four, so a has got to
be equal to two. 
16:04  16:10So our F of X is going
to be minus 2. 
16:11  16:17X squared minus 4X plus A
squared, so that's +2 squared 
16:17  16:23minus the original 3 over 2,
but we've added on a 2 
16:23  16:30squared, so we need to take it
away again to keep the balance 
16:30  16:32to keep the equality.

16:34  16:40Minus two, this is now our
complete square, so that's X 
16:40  16:48minus two all squared. And here
we've got minus three over 2 
16:48  16:54 4. Well, let's have it all
over too. So minus four is minus 
16:54  16:598 over 2, so altogether we've
got minus 11 over 2. 
17:01  17:06And we can look at this. We
can see that when this is 
17:06  17:11zero, we've got. In this case
a maximum value, because this 
17:11  17:16is a negative X squared term.
So we know that we're looking 
17:16  17:20for a graph like that. So it's
this point that we're looking 
17:20  17:24for the maximum point, and so
therefore maximum value. 
17:26  17:28Solve F of X.

17:29  17:34Will occur when this square term
is equal to 0 'cause the square 
17:34  17:37term can never be less than 0.

17:37  17:41And so we have minus two times.

17:41  17:47Minus 11. And altogether that
gives us plus 11. 
17:47  17:51And it will occur when this.

17:52  17:59Is equal to 0. In other
words, when X equals 2.
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 www.mathcentre.ac.uk/.../Completing%20the%20Square_maxima_minima.mp4
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