0:00:00.290,0:00:04.866
Completing the square is a[br]process that we make use of in a
0:00:04.866,0:00:09.442
number of ways. First, we can[br]make use of it to find maximum
0:00:09.442,0:00:13.314
and minimum values of quadratic[br]functions, second we can make
0:00:13.314,0:00:17.186
use of it to simplify or change[br]algebraic expressions in order
0:00:17.186,0:00:21.410
to be able to calculate the[br]value that they have. Third, we
0:00:21.410,0:00:24.930
can use it for solving quadratic[br]equations. In this particular
0:00:24.930,0:00:29.506
video, we're going to have a[br]look at it for finding max- and
0:00:29.506,0:00:32.210
min-imum values of functions, [br]quadratic functions.
0:00:32.210,0:00:36.138
Let's begin by looking at a very specific[br]example.
0:00:36.138,0:00:39.195
Supposing we've got x squared,
0:00:39.195,0:00:41.950
plus 5x,
0:00:42.460,0:00:45.310
minus 2. Now.
0:00:46.140,0:00:51.090
x squared, it's positive, so one[br]of the things that we do know is
0:00:51.090,0:00:54.170
that if we were to sketch the[br]graph of this function.
0:00:54.980,0:00:57.290
It would look something perhaps
0:00:57.290,0:01:02.729
like that. Question is where's[br]this point down here?
0:01:03.580,0:01:07.771
Where's the minimum value of[br]this function? What value of x
0:01:07.771,0:01:12.724
does it have? Does it actually[br]come below the x-axis as I've
0:01:12.724,0:01:17.677
drawn it, or does it come up[br]here somewhere? At what value of X
0:01:17.677,0:01:22.249
does that minimum value occur?[br]We could use calculus if we knew
0:01:22.249,0:01:26.440
calculus, but sometimes we don't[br]know calculus. We might not have
0:01:26.440,0:01:27.583
reached it yet.
0:01:28.420,0:01:31.907
At other occasions it might be[br]rather like using a sledgehammer
0:01:31.907,0:01:36.662
to crack or not, so let's have a[br]look at how we can deal with
0:01:36.662,0:01:37.930
this kind of function.
0:01:38.720,0:01:42.240
What we're going to do is[br]a process known as
0:01:42.380,0:01:50.110
"completing ... the ... square"
0:01:51.970,0:01:55.267
OK, "completing the square",[br]what does that mean?
0:01:55.447,0:02:00.137
Well, let's have a look at something[br]that is a "complete square".
0:02:00.297,0:02:05.045
That is, an exact square.
0:02:06.110,0:02:12.662
So that's a complete and exact[br]square. If we multiply out the
0:02:12.662,0:02:20.306
brackets, x plus a times by x[br]plus a, what we end up with
0:02:20.306,0:02:21.944
is x squared...
0:02:22.690,0:02:25.070
that's x times by x...
0:02:26.320,0:02:33.488
a times by x, and of course[br]x times by a, so that gives
0:02:33.488,0:02:38.336
us 2ax, and then finally a[br]times by a...
0:02:38.386,0:02:40.656
and that gives us a squared.
0:02:40.656,0:02:46.288
So this expression is[br]a complete square, a complete
0:02:46.288,0:02:51.408
and exact square. Because it's "x[br]plus a" all squared.
0:02:52.410,0:02:57.369
Similarly, we can have "x minus a"[br]all squared.
0:02:58.339,0:03:00.638
And if we[br]multiply out, these brackets
0:03:00.778,0:03:06.798
we will end up with the same[br]result, except, we will have
0:03:06.798,0:03:12.562
minus 2ax plus a squared. And[br]again this is a complete
0:03:12.562,0:03:17.278
square an exact square[br]because it's equal to x minus a...
0:03:17.278,0:03:19.374
all squared.
0:03:20.790,0:03:23.880
So,... we go back to this.
0:03:24.450,0:03:28.861
Expression here x squared, plus[br]5x, minus two and what we're
0:03:28.861,0:03:33.673
going to do is complete the[br]square. In other words we're going to
0:03:33.673,0:03:38.886
try and make it look like this.[br]We're going to try and complete it.
0:03:38.886,0:03:44.099
Make it up so it's a full[br]square. In order to do that,
0:03:44.099,0:03:48.109
what we're going to do is[br]compare that expression directly
0:03:48.109,0:03:49.312
with that one.
0:03:50.080,0:03:55.965
And we've chosen this expression[br]here because that's a plus sign
0:03:55.965,0:04:01.850
plus 5x, and that's a plus sign[br]there plus 2ax.
0:04:01.850,0:04:05.480
So.
0:04:05.500,0:04:11.790
x squared, plus[br]5x, minus 2.
0:04:12.520,0:04:22.142
And we have x squared[br]plus 2ax plus a squared
0:04:22.172,0:04:24.324
These two match up
0:04:24.734,0:04:29.852
Somehow we've got[br]to match these two up.
0:04:29.852,0:04:31.620
Well,... the x's are the same.
0:04:32.260,0:04:38.995
So the 5 and the 2a have got[br]to be the same and that would
0:04:38.995,0:04:44.383
suggest to us that a has got to[br]be 5 / 2.
0:04:44.980,0:04:50.836
So that x squared plus[br]5x minus 2...
0:04:52.510,0:04:59.350
becomes x squared plus 5x...
0:04:59.626,0:05:02.150
now... plus a squared and
0:05:02.690,0:05:06.310
now we decided that 5 was[br]to be equal to 2a
0:05:06.310,0:05:12.520
and so a was equal[br]to 5 over 2.
0:05:13.090,0:05:20.214
So to complete the square, we've[br]got to add on 5 over 2...
0:05:20.214,0:05:21.858
and square it.
0:05:24.400,0:05:26.200
But that isn't equal to that.
0:05:28.360,0:05:32.650
It's equal, this is equal[br]to that, but not to that
0:05:32.650,0:05:36.550
well. Clearly we need to[br]put the minus two on.
0:05:38.600,0:05:42.846
But then it's still not equal,[br]because here we've added on
0:05:42.846,0:05:47.864
something extra 5 over 2 [squared]. So[br]we've got to take off that five
0:05:47.864,0:05:51.724
over 2 all squared. We've got to[br]take that away.
0:05:53.410,0:05:56.278
Now let's look at this bit.
0:05:57.140,0:06:02.207
This is an exact square. It's[br]that expression there.
0:06:03.060,0:06:10.560
No, this began life as x[br]plus a all squared, so this
0:06:10.560,0:06:18.060
bit has got to be the[br]same, x plus (5 over 2) all
0:06:18.060,0:06:25.080
squared. And now we can[br]play with this. We've got minus 2
0:06:25.080,0:06:31.956
minus 25 over 4. We[br]can combine that so we have
0:06:31.956,0:06:35.394
x plus (5 over 2) all squared...
0:06:36.160,0:06:37.390
minus...
0:06:38.600,0:06:43.242
Now we're taking away two, so[br]in terms of quarters, that's
0:06:43.242,0:06:47.462
8 quarters were taking[br]away, and we're taking away 25
0:06:47.462,0:06:50.838
quarters as well, so[br]altogether, that's 33
0:06:50.838,0:06:52.948
quarters that we're taking[br]away.
0:06:54.400,0:06:59.068
Now let's have a look at this[br]expression... x squared plus 5x minus 2.
0:06:59.068,0:07:03.347
Remember what we were[br]asking was "what's its minimum value?"
0:07:03.347,0:07:06.815
Its graph looked like that.[br]We were interested in...
0:07:06.815,0:07:08.004
"where's this point?"
0:07:08.134,0:07:09.803
"where is the lowest point?"
0:07:10.173,0:07:11.734
"what's the x-coordinate?"
0:07:11.905,0:07:13.874
"and what's the y-coordinate?"
0:07:14.529,0:07:16.993
Let's have a look at this[br]expression here.
0:07:17.114,0:07:21.904
This is a square.[br]A square is always
0:07:21.904,0:07:26.720
positive unless it's equal to 0,[br]so its lowest value
0:07:26.930,0:07:30.639
that this expression [can take] is 0.
0:07:30.780,0:07:34.546
So the lowest value of[br]the whole expression...
0:07:34.586,0:07:38.686
is that "minus 33 over 4".
0:07:38.928,0:07:43.070
So therefore we can say [br]that the minimum value...
0:07:48.592,0:07:55.431
of x squared, plus 5x, minus[br]2 equals... minus 33 over 4.
0:07:55.948,0:07:58.556
And we need to be able[br]to say when
0:07:58.876,0:08:00.614
"what's the x-value there?"
0:08:00.864,0:08:04.927
well, it occurs when this bracket[br]is at its lowest value.
0:08:05.194,0:08:11.142
When this bracket is at[br]0. In other words, when x equals...
0:08:11.142,0:08:14.458
minus 5 over 2.
0:08:16.550,0:08:23.380
So we found the minimum[br]value and exactly when it happens.
0:08:25.850,0:08:31.900
Let's take a second example. Our[br]quadratic function this time, f of x,
0:08:31.900,0:08:37.400
is x squared, minus 6[br]x, minus 12.
0:08:38.210,0:08:45.254
We've got a minus sign in here, so[br]let's line this up with the
0:08:45.254,0:08:50.537
complete square: x squared, minus[br]2ax, plus a squared.
0:08:51.210,0:08:56.010
The x squared terms are the same,[br]and we want these two to be the
0:08:56.010,0:09:00.490
same as well. That clearly means[br]that 2a has got to be the same
0:09:00.490,0:09:03.050
as 6, so a has got to be 3.
0:09:03.610,0:09:09.790
So f(x) is equal[br]to x squared, minus 6x,
0:09:09.790,0:09:11.559
plus...
0:09:11.559,0:09:17.338
a squared (which is 3 squared),[br]minus 12, and now we added on
0:09:17.338,0:09:22.462
3 squared. So we've got to take[br]the 3 squared away in order
0:09:22.462,0:09:27.586
to make it equal. To keep the[br]value of the original expression
0:09:27.586,0:09:29.294
that we started with.
0:09:29.930,0:09:37.766
We can now identify this as[br]being (x minus 3) all squared.
0:09:39.060,0:09:41.166
And these numbers at the end...
0:09:41.750,0:09:47.915
minus 12 minus 9, altogether[br]gives us minus 21.
0:09:49.020,0:09:54.610
Again, we can say does it have a[br]maximum value or a minimum value?
0:09:54.610,0:09:57.730
Well what we know that we began[br]with a positive x squared term,
0:09:57.730,0:10:01.820
so the shape of the graph[br]has got to be like that. So we
0:10:01.820,0:10:03.740
know that we're looking for a
0:10:03.740,0:10:08.223
minimum value. We know that that[br]minimum value will occur when
0:10:08.223,0:10:13.607
this bit is 0 because it's a[br]square, it's least value is
0:10:13.607,0:10:21.341
going to be 0, so therefore we[br]can say the minimum value.
0:10:22.090,0:10:25.996
of our quadratic function f of x
0:10:25.996,0:10:31.860
is minus 21, [occurring] when...
0:10:32.870,0:10:38.953
this bit is 0. In other words,[br]when x equals 3.
0:10:40.650,0:10:46.182
The two examples we've taken so far[br]have both had a positive x squared
0:10:46.182,0:10:51.714
and a unit coefficient[br]of x squared, in other words 1 x squared.
0:10:51.714,0:10:57.145
We'll now look at an example [br]where we've got a number here
0:10:57.145,0:10:59.910
in front of the x squared.
0:10:59.920,0:11:04.156
So the example[br]that will take.
0:11:05.980,0:11:11.290
f of x equals 2x squared,
0:11:11.570,0:11:16.550
minus 6x, plus one.
0:11:17.260,0:11:22.852
Our first step is to take out[br]that 2 as a factor.
0:11:23.800,0:11:27.870
2, brackets x squared,
0:11:28.482,0:11:31.020
minus 3x,...
0:11:31.630,0:11:37.368
we've got to take[br]the 2 out of this as well, so
0:11:37.368,0:11:38.700
that's a half.
0:11:39.370,0:11:45.086
And now we do the same as we've[br]done before with this bracket here.
0:11:45.766,0:11:52.654
We line this one up with x squared,[br]minus 2ax, plus a squared.
0:11:53.380,0:11:59.988
When making these two terms the[br]same 3 has to be the same as 2a,
0:11:59.988,0:12:05.652
and so 3 over 2 has to[br]be equal to a.
0:12:06.770,0:12:13.742
So our function f of x is going[br]to be equal to 2 times...
0:12:14.530,0:12:18.438
x squared, minus 3x,...
0:12:19.170,0:12:27.267
now we want plus a squared,[br]so that's plus (3 over 2) all squared
0:12:28.257,0:12:31.743
Plus the half that[br]was there originally and now
0:12:31.743,0:12:35.910
we've added on this, so we've[br]got to take it away,...
0:12:36.144,0:12:42.418
(3 over 2) all squared. And finally we[br]opened a bracket, so we must
0:12:42.418,0:12:44.553
close it at the end.
0:12:45.740,0:12:53.024
Equals... 2, bracket,... now this[br]is going to be our complete square
0:12:53.024,0:12:59.404
(x minus 3 over 2) all squared.
0:12:59.930,0:13:05.371
And then here we've got some[br]calculation to do. We've plus a half,
0:13:05.651,0:13:12.551
take away (3 over 2) squared, [br]so that's plus 1/2
0:13:12.551,0:13:14.955
take away 9 over 4.
0:13:16.620,0:13:21.230
The front bit is going to stay the same
0:13:24.508,0:13:27.230
And now we can juggle with[br]these fractions. At the end,
0:13:27.460,0:13:32.017
we've got plus 1/2 take away 9[br]quarters or 1/2 is 2 quarters,
0:13:32.017,0:13:36.547
so if we're taking[br]away, nine quarters must be
0:13:36.987,0:13:40.095
ultimately taking away 7 quarters.
0:13:40.355,0:13:44.927
So again, what's the[br]minimum value of this function?
0:13:44.927,0:13:49.039
It had a positive 2 in front[br]of the x squared, so again, it
0:13:49.039,0:13:52.396
looks like that. And again,[br]we're asking the question,
0:13:52.396,0:13:56.499
"what's this point down here?"[br]What's the lowest point and that
0:13:56.499,0:13:59.483
lowest point must occur when[br]this is 0.
0:14:00.340,0:14:02.350
So the min ...
0:14:02.890,0:14:10.828
value of f of x must be[br]equal to... now that's going to be 0
0:14:10.828,0:14:18.268
But we're still multiplying[br]by the 2, so it's 2 times
0:14:18.268,0:14:24.724
minus 7 over 4. That's minus[br]14 over 4, which reduces to
0:14:24.724,0:14:27.414
minus 7 over 2. When?
0:14:28.100,0:14:34.756
And that will happen when this[br]is zero. In other words, when x
0:14:34.756,0:14:36.804
equals 3 over 2.
0:14:38.170,0:14:43.955
So a minimum value of minus[br]7 over 2 when x equals 3 over 2.
0:14:43.955,0:14:49.295
Let's take one final[br]example and this time when the
0:14:49.295,0:14:51.520
coefficient of x squared is
0:14:51.520,0:14:53.986
actually negative.
0:14:54.106,0:14:57.991
So for this will take our quadratic function[br]to be f of x
0:14:58.600,0:15:02.084
equals... 3 plus
0:15:02.084,0:15:08.510
8x minus[br]2(x squared).
0:15:09.520,0:15:16.002
We operate in just the same way[br]as we did before. We take out
0:15:16.002,0:15:21.095
the factor that is multiplying[br]the x squared and on this
0:15:21.095,0:15:24.336
occasion it's minus 2. [br]The "- 2" comes out.
0:15:24.336,0:15:31.805
Times by x squared, we[br]take a minus 2 out of the 8x,
0:15:31.805,0:15:39.080
that leaves us minus 4x and the[br]minus 2 out of the 3 is a
0:15:39.080,0:15:42.960
factor which gives us minus[br]three over 2.
0:15:44.760,0:15:50.870
We line this one up with X[br]squared minus 2X plus A squared.
0:15:50.870,0:15:57.920
Those two are the same. We want[br]these two to be the same. 2A is
0:15:57.920,0:16:03.560
equal to four, so a has got to[br]be equal to two.
0:16:04.120,0:16:10.324
So our F of X is going[br]to be minus 2.
0:16:11.280,0:16:17.110
X squared minus 4X plus A[br]squared, so that's +2 squared
0:16:17.110,0:16:23.470
minus the original 3 over 2,[br]but we've added on a 2
0:16:23.470,0:16:30.360
squared, so we need to take it[br]away again to keep the balance
0:16:30.360,0:16:32.480
to keep the equality.
0:16:33.820,0:16:40.398
Minus two, this is now our[br]complete square, so that's X
0:16:40.398,0:16:47.574
minus two all squared. And here[br]we've got minus three over 2
0:16:47.574,0:16:54.444
- 4. Well, let's have it all[br]over too. So minus four is minus
0:16:54.444,0:16:59.141
8 over 2, so altogether we've[br]got minus 11 over 2.
0:17:00.910,0:17:06.188
And we can look at this. We[br]can see that when this is
0:17:06.188,0:17:10.654
zero, we've got. In this case[br]a maximum value, because this
0:17:10.654,0:17:15.526
is a negative X squared term.[br]So we know that we're looking
0:17:15.526,0:17:20.398
for a graph like that. So it's[br]this point that we're looking
0:17:20.398,0:17:24.052
for the maximum point, and so[br]therefore maximum value.
0:17:26.300,0:17:28.308
Solve F of X.
0:17:28.810,0:17:33.828
Will occur when this square term[br]is equal to 0 'cause the square
0:17:33.828,0:17:36.530
term can never be less than 0.
0:17:37.420,0:17:40.577
And so we have minus two times.
0:17:41.080,0:17:46.778
Minus 11. And altogether that[br]gives us plus 11.
0:17:47.300,0:17:51.236
And it will occur when this.
0:17:51.750,0:17:58.933
Is equal to 0. In other[br]words, when X equals 2.