1
00:00:00,440 --> 00:00:02,930
In this video, I'm going to show
you a technique called

2
00:00:02,930 --> 00:00:09,310
completing the square.

3
00:00:09,310 --> 00:00:14,490
And what's neat about this is
that this will work for any

4
00:00:14,490 --> 00:00:18,210
quadratic equation, and it's
actually the basis for the

5
00:00:18,210 --> 00:00:18,750
quadratic formula.

6
00:00:18,750 --> 00:00:21,990
And in the next video or the
video after that I'll prove

7
00:00:21,990 --> 00:00:25,630
the quadratic formula using
completing the square.

8
00:00:25,630 --> 00:00:28,450
But before we do that, we
need to understand even

9
00:00:28,450 --> 00:00:29,470
what it's all about.

10
00:00:29,470 --> 00:00:32,070
And it really just builds off
of what we did in the last

11
00:00:32,070 --> 00:00:33,880
video, where we solved
quadratics

12
00:00:33,880 --> 00:00:36,130
using perfect squares.

13
00:00:36,130 --> 00:00:39,900
So let's say I have the
quadratic equation x squared

14
00:00:39,900 --> 00:00:44,880
minus 4x is equal to 5.

15
00:00:44,880 --> 00:00:47,490
And I put this big space
here for a reason.

16
00:00:47,490 --> 00:00:49,680
In the last video, we saw
that these can be pretty

17
00:00:49,680 --> 00:00:53,200
straightforward to solve if
the left-hand side is a

18
00:00:53,200 --> 00:00:56,500
perfect square.

19
00:00:56,500 --> 00:00:59,050
You see, completing the square
is all about making the

20
00:00:59,050 --> 00:01:01,900
quadratic equation into a
perfect square, engineering

21
00:01:01,900 --> 00:01:05,190
it, adding and subtracting from
both sides so it becomes

22
00:01:05,190 --> 00:01:05,970
a perfect square.

23
00:01:05,970 --> 00:01:07,710
So how can we do that?

24
00:01:07,710 --> 00:01:10,130
Well, in order for this
left-hand side to be a perfect

25
00:01:10,130 --> 00:01:12,990
square, there has to be
some number here.

26
00:01:12,990 --> 00:01:17,510
There has to be some number here
that if I have my number

27
00:01:17,510 --> 00:01:20,910
squared I get that number, and
then if I have two times my

28
00:01:20,910 --> 00:01:22,890
number I get negative 4.

29
00:01:22,890 --> 00:01:24,750
Remember that, and I
think it'll become

30
00:01:24,750 --> 00:01:27,700
clear with a few examples.

31
00:01:27,700 --> 00:01:35,230
I want x squared minus 4x plus
something to be equal to x

32
00:01:35,230 --> 00:01:37,740
minus a squared.

33
00:01:37,740 --> 00:01:41,010
We don't know what a
is just yet, but we

34
00:01:41,010 --> 00:01:42,110
know a couple of things.

35
00:01:42,110 --> 00:01:46,180
When I square things-- so this
is going to be x squared minus

36
00:01:46,180 --> 00:01:49,330
2a plus a squared.

37
00:01:49,330 --> 00:01:53,640
So if you look at this pattern
right here, that has to be--

38
00:01:53,640 --> 00:01:59,880
sorry, x squared minus 2ax--
this right here has to be 2ax.

39
00:01:59,880 --> 00:02:03,530
And this right here would
have to be a squared.

40
00:02:03,530 --> 00:02:07,690
So this number, a is going to
be half of negative 4, a has

41
00:02:07,690 --> 00:02:10,370
to be negative 2, right?

42
00:02:10,370 --> 00:02:13,570
Because 2 times a is going
to be negative 4.

43
00:02:13,570 --> 00:02:18,330
a is negative 2, and if a is
negative 2, what is a squared?

44
00:02:18,330 --> 00:02:21,550
Well, then a squared is going
to be positive 4.

45
00:02:21,550 --> 00:02:24,220
And this might look all
complicated to you right now,

46
00:02:24,220 --> 00:02:25,910
but I'm showing you
the rationale.

47
00:02:25,910 --> 00:02:29,080
You literally just look at this
coefficient right here,

48
00:02:29,080 --> 00:02:32,670
and you say, OK, well what's
half of that coefficient?

49
00:02:32,670 --> 00:02:35,920
Well, half of that coefficient
is negative 2.

50
00:02:35,920 --> 00:02:40,230
So we could say a is equal to
negative 2-- same idea there--

51
00:02:40,230 --> 00:02:41,720
and then you square it.

52
00:02:41,720 --> 00:02:44,100
You square a, you
get positive 4.

53
00:02:44,100 --> 00:02:46,540
So we add positive 4 here.

54
00:02:46,540 --> 00:02:47,630
Add a 4.

55
00:02:47,630 --> 00:02:50,990
Now, from the very first
equation we ever did, you

56
00:02:50,990 --> 00:02:55,240
should know that you can never
do something to just one side

57
00:02:55,240 --> 00:02:55,900
of the equation.

58
00:02:55,900 --> 00:02:58,700
You can't add 4 to just one
side of the equation.

59
00:02:58,700 --> 00:03:02,710
If x squared minus 4x was equal
to 5, then when I add 4

60
00:03:02,710 --> 00:03:04,720
it's not going to be
equal to 5 anymore.

61
00:03:04,720 --> 00:03:07,950
It's going to be equal
to 5 plus 4.

62
00:03:07,950 --> 00:03:11,430
We added 4 on the left-hand side
because we wanted this to

63
00:03:11,430 --> 00:03:12,435
be a perfect square.

64
00:03:12,435 --> 00:03:15,210
But if you add something to the
left-hand side, you've got

65
00:03:15,210 --> 00:03:17,320
to add it to the right-hand
side.

66
00:03:17,320 --> 00:03:20,630
And now, we've gotten ourselves
to a problem that's

67
00:03:20,630 --> 00:03:23,410
just like the problems we
did in the last video.

68
00:03:23,410 --> 00:03:25,960
What is this left-hand side?

69
00:03:25,960 --> 00:03:27,000
Let me rewrite the
whole thing.

70
00:03:27,000 --> 00:03:33,020
We have x squared minus 4x
plus 4 is equal to 9 now.

71
00:03:33,020 --> 00:03:35,380
All we did is add 4 to both
sides of the equation.

72
00:03:35,380 --> 00:03:39,070
But we added 4 on purpose so
that this left-hand side

73
00:03:39,070 --> 00:03:41,080
becomes a perfect square.

74
00:03:41,080 --> 00:03:41,760
Now what is this?

75
00:03:41,760 --> 00:03:45,340
What number when I multiply it
by itself is equal to 4 and

76
00:03:45,340 --> 00:03:47,770
when I add it to itself I'm
equal to negative 2?

77
00:03:47,770 --> 00:03:49,000
Well, we already answered
that question.

78
00:03:49,000 --> 00:03:50,040
It's negative 2.

79
00:03:50,040 --> 00:03:55,310
So we get x minus 2 times
x minus 2 is equal to 9.

80
00:03:55,310 --> 00:03:59,350
Or we could have skipped this
step and written x minus 2

81
00:03:59,350 --> 00:04:02,990
squared is equal to 9.

82
00:04:02,990 --> 00:04:07,280
And then you take the square
root of both sides, you get x

83
00:04:07,280 --> 00:04:10,840
minus 2 is equal to
plus or minus 3.

84
00:04:10,840 --> 00:04:16,870
Add 2 to both sides, you get x
is equal to 2 plus or minus 3.

85
00:04:16,870 --> 00:04:22,440
That tells us that x could be
equal to 2 plus 3, which is 5.

86
00:04:22,440 --> 00:04:28,960
Or x could be equal to 2 minus
3, which is negative 1.

87
00:04:28,960 --> 00:04:30,650
And we are done.

88
00:04:30,650 --> 00:04:31,840
Now I want to be very clear.

89
00:04:31,840 --> 00:04:34,300
You could have done this without
completing the square.

90
00:04:34,300 --> 00:04:37,640
We could've started off
with x squared minus

91
00:04:37,640 --> 00:04:39,850
4x is equal to 5.

92
00:04:39,850 --> 00:04:42,970
We could have subtracted 5 from
both sides and gotten x

93
00:04:42,970 --> 00:04:47,160
squared minus 4x minus
5 is equal to 0.

94
00:04:47,160 --> 00:04:51,940
And you could say, hey, if I
have a negative 5 times a

95
00:04:51,940 --> 00:04:56,190
positive 1, then their product
is negative 5 and their sum is

96
00:04:56,190 --> 00:04:57,000
negative 4.

97
00:04:57,000 --> 00:05:00,800
So I could say this is x
minus 5 times x plus

98
00:05:00,800 --> 00:05:02,480
1 is equal to 0.

99
00:05:02,480 --> 00:05:06,810
And then we would say that x is
equal to 5 or x is equal to

100
00:05:06,810 --> 00:05:07,700
negative 1.

101
00:05:07,700 --> 00:05:10,350
And in this case, this actually
probably would have

102
00:05:10,350 --> 00:05:13,450
been a faster way to
do the problem.

103
00:05:13,450 --> 00:05:16,140
But the neat thing about the
completing the square is it

104
00:05:16,140 --> 00:05:17,770
will always work.

105
00:05:17,770 --> 00:05:21,580
It'll always work no matter what
the coefficients are or

106
00:05:21,580 --> 00:05:23,385
no matter how crazy
the problem is.

107
00:05:23,385 --> 00:05:25,400
And let me prove it to you.

108
00:05:25,400 --> 00:05:28,440
Let's do one that traditionally
would have been

109
00:05:28,440 --> 00:05:31,140
a pretty painful problem if
we just tried to do it by

110
00:05:31,140 --> 00:05:36,200
factoring, especially if we
did it using grouping or

111
00:05:36,200 --> 00:05:37,020
something like that.

112
00:05:37,020 --> 00:05:45,070
Let's say we had 10x squared
minus 30x minus

113
00:05:45,070 --> 00:05:47,530
8 is equal to 0.

114
00:05:47,530 --> 00:05:50,060
Now, right from the get-go, you
could say, hey look, we

115
00:05:50,060 --> 00:05:53,280
could maybe divide
both sides by 2.

116
00:05:53,280 --> 00:05:54,800
That does simplify
a little bit.

117
00:05:54,800 --> 00:05:56,450
Let's divide both sides by 2.

118
00:05:56,450 --> 00:06:02,150
So if you divide everything
by 2, what do you get?

119
00:06:02,150 --> 00:06:11,990
We get 5x squared minus 15x
minus 4 is equal to 0.

120
00:06:11,990 --> 00:06:14,540
But once again, now we have this
crazy 5 in front of this

121
00:06:14,540 --> 00:06:16,810
coefficent and we would have to
solve it by grouping which

122
00:06:16,810 --> 00:06:20,410
is a reasonably painful
process.

123
00:06:20,410 --> 00:06:23,410
But we can now go straight to
completing the square, and to

124
00:06:23,410 --> 00:06:27,500
do that I'm now going to divide
by 5 to get a 1 leading

125
00:06:27,500 --> 00:06:28,870
coefficient here.

126
00:06:28,870 --> 00:06:31,660
And you're going to see why this
is different than what

127
00:06:31,660 --> 00:06:33,010
we've traditionally done.

128
00:06:33,010 --> 00:06:35,730
So if I divide this whole thing
by 5, I could have just

129
00:06:35,730 --> 00:06:38,050
divided by 10 from the get-go
but I wanted to go to this the

130
00:06:38,050 --> 00:06:40,030
step first just to show
you that this really

131
00:06:40,030 --> 00:06:41,800
didn't give us much.

132
00:06:41,800 --> 00:06:43,660
Let's divide everything by 5.

133
00:06:43,660 --> 00:06:52,693
So if you divide everything by
5, you get x squared minus 3x

134
00:06:52,693 --> 00:06:58,720
minus 4/5 is equal to 0.

135
00:06:58,720 --> 00:07:02,020
So, you might say, hey, why did
we ever do that factoring

136
00:07:02,020 --> 00:07:02,630
by grouping?

137
00:07:02,630 --> 00:07:06,140
If we can just always divide by
this leading coefficient,

138
00:07:06,140 --> 00:07:07,220
we can get rid of that.

139
00:07:07,220 --> 00:07:09,840
We can always turn this into a 1
or a negative 1 if we divide

140
00:07:09,840 --> 00:07:10,910
by the right number.

141
00:07:10,910 --> 00:07:14,410
But notice, by doing that we
got this crazy 4/5 here.

142
00:07:14,410 --> 00:07:17,630
So this is super hard to do
just using factoring.

143
00:07:17,630 --> 00:07:19,500
You'd have to say, what two
numbers when I take the

144
00:07:19,500 --> 00:07:22,100
product is equal to
negative 4/5?

145
00:07:22,100 --> 00:07:25,210
It's a fraction and when I take
their sum, is equal to

146
00:07:25,210 --> 00:07:26,140
negative 3?

147
00:07:26,140 --> 00:07:29,310
This is a hard problem
with factoring.

148
00:07:29,310 --> 00:07:36,860
This is hard using factoring.

149
00:07:36,860 --> 00:07:42,080
So, the best thing to do is to
use completing the square.

150
00:07:42,080 --> 00:07:44,720
So let's think a little bit
about how we can turn this

151
00:07:44,720 --> 00:07:45,950
into a perfect square.

152
00:07:45,950 --> 00:07:48,080
What I like to do-- and you'll
see this done some ways and

153
00:07:48,080 --> 00:07:50,040
I'll show you both ways because
you'll see teachers do

154
00:07:50,040 --> 00:07:53,880
it both ways-- I like to get
the 4/5 on the other side.

155
00:07:53,880 --> 00:07:56,900
So let's add 4/5 to both
sides of this equation.

156
00:07:56,900 --> 00:07:59,980
You don't have to do it this
way, but I like to get the 4/5

157
00:07:59,980 --> 00:08:01,160
out of the way.

158
00:08:01,160 --> 00:08:04,010
And then what do we get
if we add 4/5 to both

159
00:08:04,010 --> 00:08:05,250
sides of this equation?

160
00:08:05,250 --> 00:08:08,350
The left-hand hand side of the
equation just becomes x

161
00:08:08,350 --> 00:08:11,800
squared minus 3x,
no 4/5 there.

162
00:08:11,800 --> 00:08:13,660
I'm going to leave a little
bit of space.

163
00:08:13,660 --> 00:08:17,790
And that's going to
be equal to 4/5.

164
00:08:17,790 --> 00:08:19,990
Now, just like the last problem,
we want to turn this

165
00:08:19,990 --> 00:08:23,350
left-hand side into the perfect
square of a binomial.

166
00:08:23,350 --> 00:08:24,740
How do we do that?

167
00:08:24,740 --> 00:08:28,360
Well, we say, well, what number
times 2 is equal to

168
00:08:28,360 --> 00:08:30,110
negative 3?

169
00:08:30,110 --> 00:08:32,309
So some number times
2 is negative 3.

170
00:08:32,309 --> 00:08:35,330
Or we essentially just take
negative 3 and divide it by 2,

171
00:08:35,330 --> 00:08:37,370
which is negative 3/2.

172
00:08:37,370 --> 00:08:39,554
And then we square
negative 3/2.

173
00:08:39,554 --> 00:08:44,840
So in the example, we'll
say a is negative 3/2.

174
00:08:44,840 --> 00:08:48,380
And if we square negative
3/2, what do we get?

175
00:08:48,380 --> 00:08:54,100
We get positive 9/4.

176
00:08:54,100 --> 00:08:56,810
I just took half of this
coefficient, squared it, got

177
00:08:56,810 --> 00:08:58,010
positive 9/4.

178
00:08:58,010 --> 00:09:00,720
The whole purpose of doing that
is to turn this left-hand

179
00:09:00,720 --> 00:09:02,920
side into a perfect square.

180
00:09:02,920 --> 00:09:05,530
Now, anything you do to one side
of the equation, you've

181
00:09:05,530 --> 00:09:06,600
got to do to the other side.

182
00:09:06,600 --> 00:09:11,030
So we added a 9/4 here, let's
add a 9/4 over there.

183
00:09:11,030 --> 00:09:13,850
And what does our
equation become?

184
00:09:13,850 --> 00:09:22,530
We get x squared minus 3x plus
9/4 is equal to-- let's see if

185
00:09:22,530 --> 00:09:24,460
we can get a common
denominator.

186
00:09:24,460 --> 00:09:29,120
So, 4/5 is the same
thing as 16/20.

187
00:09:29,120 --> 00:09:31,880
Just multiply the numerator
and denominator by 4.

188
00:09:31,880 --> 00:09:33,820
Plus over 20.

189
00:09:33,820 --> 00:09:36,960
9/4 is the same thing
if you multiply the

190
00:09:36,960 --> 00:09:42,150
numerator by 5 as 45/20.

191
00:09:42,150 --> 00:09:44,970
And so what is 16 plus 45?

192
00:09:44,970 --> 00:09:47,020
You see, this is kind of getting
kind of hairy, but

193
00:09:47,020 --> 00:09:48,930
that's the fun, I guess, of

194
00:09:48,930 --> 00:09:50,380
completing the square sometimes.

195
00:09:50,380 --> 00:09:53,420
16 plus 45.

196
00:09:53,420 --> 00:09:55,780
See that's 55, 61.

197
00:09:55,780 --> 00:09:59,750
So this is equal to 61/20.

198
00:09:59,750 --> 00:10:02,680
So let me just rewrite it.

199
00:10:02,680 --> 00:10:09,480
x squared minus 3x plus
9/4 is equal to 61/20.

200
00:10:09,480 --> 00:10:11,030
Crazy number.

201
00:10:11,030 --> 00:10:13,630
Now this, at least on
the left hand side,

202
00:10:13,630 --> 00:10:15,970
is a perfect square.

203
00:10:15,970 --> 00:10:21,610
This is the same thing as
x minus 3/2 squared.

204
00:10:21,610 --> 00:10:24,200
And it was by design.

205
00:10:24,200 --> 00:10:27,590
Negative 3/2 times negative
3/2 is positive 9/4.

206
00:10:27,590 --> 00:10:32,790
Negative 3/2 plus negative 3/2
is equal to negative 3.

207
00:10:32,790 --> 00:10:37,960
So this squared is
equal to 61/20.

208
00:10:37,960 --> 00:10:43,090
We can take the square root of
both sides and we get x minus

209
00:10:43,090 --> 00:10:47,820
3/2 is equal to the positive
or the negative

210
00:10:47,820 --> 00:10:53,320
square root of 61/20.

211
00:10:53,320 --> 00:10:57,640
And now, we can add 3/2 to both
sides of this equation

212
00:10:57,640 --> 00:11:03,600
and you get x is equal to
positive 3/2 plus or minus the

213
00:11:03,600 --> 00:11:07,300
square root of 61/20.

214
00:11:07,300 --> 00:11:09,290
And this is a crazy number and
it's hopefully obvious you

215
00:11:09,290 --> 00:11:11,430
would not have been able to-- at
least I would not have been

216
00:11:11,430 --> 00:11:15,250
able to-- get to this number
just by factoring.

217
00:11:15,250 --> 00:11:17,260
And if you want their actual
values, you can get your

218
00:11:17,260 --> 00:11:18,510
calculator out.

219
00:11:20,620 --> 00:11:22,510
And then let me clear
all of this.

220
00:11:25,950 --> 00:11:28,760
And 3/2-- let's do the plus
version first. So we want to

221
00:11:28,760 --> 00:11:33,710
do 3 divided by 2 plus the
second square root.

222
00:11:33,710 --> 00:11:35,050
We want to pick that little
yellow square root.

223
00:11:35,050 --> 00:11:46,480
So the square root of 61 divided
by 20, which is 3.24.

224
00:11:46,480 --> 00:11:52,760
This crazy 3.2464, I'll
just write 3.246.

225
00:11:52,760 --> 00:12:02,230
So this is approximately equal
to 3.246, and that was just

226
00:12:02,230 --> 00:12:03,110
the positive version.

227
00:12:03,110 --> 00:12:06,710
Let's do the subtraction
version.

228
00:12:06,710 --> 00:12:09,180
So we can actually put our
entry-- if you do second and

229
00:12:09,180 --> 00:12:11,535
then entry, that we want that
little yellow entry, that's

230
00:12:11,535 --> 00:12:12,465
why I pressed the
second button.

231
00:12:12,465 --> 00:12:16,130
So I press enter, it puts in
what we just put, we can just

232
00:12:16,130 --> 00:12:23,400
change the positive or the
addition to a subtraction and

233
00:12:23,400 --> 00:12:27,970
you get negative 0.246.

234
00:12:27,970 --> 00:12:33,800
So you get negative 0.246.

235
00:12:33,800 --> 00:12:38,200
And you can actually verify
that these satisfy our

236
00:12:38,200 --> 00:12:39,360
original equation.

237
00:12:39,360 --> 00:12:42,050
Our original equation
was up here.

238
00:12:42,050 --> 00:12:43,840
Let me just verify
for one of them.

239
00:12:47,400 --> 00:12:50,130
So the second answer on your
graphing calculator is the

240
00:12:50,130 --> 00:12:51,760
last answer you use.

241
00:12:51,760 --> 00:12:54,160
So if you use a variable answer,
that's this number

242
00:12:54,160 --> 00:12:55,160
right here.

243
00:12:55,160 --> 00:13:00,090
So if I have my answer squared--
I'm using answer

244
00:13:00,090 --> 00:13:02,380
represents negative 0.24.

245
00:13:02,380 --> 00:13:11,975
Answer squared minus 3 times
answer minus 4/5-- 4 divided

246
00:13:11,975 --> 00:13:16,030
by 5-- it equals--.

247
00:13:16,030 --> 00:13:18,490
And this just a little
bit of explanation.

248
00:13:18,490 --> 00:13:21,860
This doesn't store the entire
number, it goes up to some

249
00:13:21,860 --> 00:13:22,880
level of precision.

250
00:13:22,880 --> 00:13:24,910
It stores some number
of digits.

251
00:13:24,910 --> 00:13:28,930
So when it calculated it using
this stored number right here,

252
00:13:28,930 --> 00:13:32,240
it got 1 times 10 to
the negative 14.

253
00:13:32,240 --> 00:13:34,980
So that is 0.0000.

254
00:13:34,980 --> 00:13:37,100
So that's 13 zeroes
and then a 1.

255
00:13:37,100 --> 00:13:38,870
A decimal, then 13
zeroes and a 1.

256
00:13:38,870 --> 00:13:41,060
So this is pretty much 0.

257
00:13:41,060 --> 00:13:43,550
Or actually, if you got the
exact answer right here, if

258
00:13:43,550 --> 00:13:46,480
you went through an infinite
level of precision here, or

259
00:13:46,480 --> 00:13:49,050
maybe if you kept it in this
radical form, you would get

260
00:13:49,050 --> 00:13:52,390
that it is indeed equal to 0.

261
00:13:52,390 --> 00:13:55,300
So hopefully you found that
helpful, this whole notion of

262
00:13:55,300 --> 00:13:56,160
completing the square.

263
00:13:56,160 --> 00:13:58,670
Now we're going to extend it
to the actual quadratic

264
00:13:58,670 --> 00:14:01,510
formula that we can use, we
can essentially just plug

265
00:14:01,510 --> 00:14:03,610
things into to solve any
quadratic equation.