1
00:00:00,000 --> 00:00:00,890


2
00:00:00,890 --> 00:00:03,590
Welcome to part two of the presentation on

3
00:00:03,589 --> 00:00:05,660
quadratic equations.

4
00:00:05,660 --> 00:00:08,470
Well, I think I thoroughly confused you the last time

5
00:00:08,470 --> 00:00:11,170
around, so let me see if I can fix that a bit by doing

6
00:00:11,169 --> 00:00:12,769
several more examples.

7
00:00:12,769 --> 00:00:15,429
So let's just start with a review of what the

8
00:00:15,429 --> 00:00:16,379
quadratic equation is.

9
00:00:16,379 --> 00:00:19,649
The quadratic equation says, if I'm trying to solve the

10
00:00:19,649 --> 00:00:31,589
equation Ax squared plus Bx plus C equals 0, then the

11
00:00:31,589 --> 00:00:35,439
solution or the solutions because there's usually two

12
00:00:35,439 --> 00:00:38,969
times that it intersects the x-axis, or two solutions for

13
00:00:38,969 --> 00:00:47,609
this equation is x equals minus B plus or minus the square root

14
00:00:47,609 --> 00:00:56,390
of B squared minus 4 times A times C.

15
00:00:56,390 --> 00:01:00,270
And all of that over 2A.

16
00:01:00,270 --> 00:01:02,040
So let's do a problem and hopefully this should make

17
00:01:02,039 --> 00:01:02,689
a little more sense.

18
00:01:02,689 --> 00:01:04,620
That's a 2 on the bottom.

19
00:01:04,620 --> 00:01:13,890
So let's say I had the equation minus 9x squared minus

20
00:01:13,890 --> 00:01:19,950
9x plus 6 equals 0.

21
00:01:19,950 --> 00:01:22,230
So in this example what's A?

22
00:01:22,230 --> 00:01:25,410
Well, A is the coefficient on the x squared term.

23
00:01:25,409 --> 00:01:29,819
The x squared term is here, the coefficient is minus 9.

24
00:01:29,819 --> 00:01:30,619
So let's write that.

25
00:01:30,620 --> 00:01:34,120
A equals minus 9.

26
00:01:34,120 --> 00:01:35,400
What does B equal?

27
00:01:35,400 --> 00:01:39,180
B is the coefficient on the x term, so that's this term here.

28
00:01:39,180 --> 00:01:43,220
So B is also equal to minus 9.

29
00:01:43,219 --> 00:01:47,140
And C is the constant term, which in this example is 6.

30
00:01:47,140 --> 00:01:49,549
So C is equal to 6.

31
00:01:49,549 --> 00:01:52,069
Now we just substitute these values into the actual

32
00:01:52,069 --> 00:01:53,259
quadratic equation.

33
00:01:53,260 --> 00:01:59,600
So negative B, so it's negative times negative 9.

34
00:01:59,599 --> 00:02:00,780
That's B.

35
00:02:00,780 --> 00:02:08,110
Plus or minus the square root of B squared, so that's 81.

36
00:02:08,110 --> 00:02:08,390
Right?

37
00:02:08,389 --> 00:02:10,029
Negative 9 squared.

38
00:02:10,030 --> 00:02:14,719
Minus 4 times negative 9.

39
00:02:14,719 --> 00:02:16,139
That's A.

40
00:02:16,139 --> 00:02:19,479
Times C, which is 6.

41
00:02:19,479 --> 00:02:23,949
And all of that over 2 times negative 9, which

42
00:02:23,949 --> 00:02:25,629
is minus 18, right?

43
00:02:25,629 --> 00:02:26,719
2 times negative 9-- 2A.

44
00:02:26,719 --> 00:02:29,229


45
00:02:29,229 --> 00:02:33,759
Let's try to simplify this up here.

46
00:02:33,759 --> 00:02:37,929
Well, negative negative 9, that's positive 9.

47
00:02:37,930 --> 00:02:46,480
Plus or minus the square root of 81.

48
00:02:46,479 --> 00:02:47,899
Let's see.

49
00:02:47,900 --> 00:02:50,270
This is negative 4 times negative 9.

50
00:02:50,270 --> 00:02:53,469
Negative 4 times negative 9 is positive 36.

51
00:02:53,469 --> 00:02:58,310
And then positive 36 times 6 is-- let's see.

52
00:02:58,310 --> 00:03:01,330
30 times 6 is 180.

53
00:03:01,330 --> 00:03:07,890
And then 180 plus another 36 is 216.

54
00:03:07,889 --> 00:03:10,979
Plus 216, is that right?

55
00:03:10,979 --> 00:03:14,489
180 plus 36 is 216.

56
00:03:14,490 --> 00:03:16,840
All of that over 2A.

57
00:03:16,840 --> 00:03:19,569
2A we already said is minus 19.

58
00:03:19,569 --> 00:03:20,739
So we simplify that more.

59
00:03:20,740 --> 00:03:28,090
That's 9 plus or minus the square root 81 plus 216.

60
00:03:28,090 --> 00:03:30,400
That's 80 plus 217.

61
00:03:30,400 --> 00:03:38,039
That's 297.

62
00:03:38,039 --> 00:03:41,900
And all of that over minus 18.

63
00:03:41,900 --> 00:03:45,020
Now, this is actually-- the hardest part with the quadratic

64
00:03:45,020 --> 00:03:47,719
equation is oftentimes just simplifying this expression.

65
00:03:47,719 --> 00:03:50,859
We have to figure out if we can simplify this radical.

66
00:03:50,860 --> 00:03:53,090
Well, let's see.

67
00:03:53,090 --> 00:03:56,490
One way to figure out if a number is divisible by 9 is to

68
00:03:56,490 --> 00:03:58,320
actually add up the digits and see if the digits

69
00:03:58,319 --> 00:03:59,259
are divisible by 9.

70
00:03:59,259 --> 00:03:59,949
In this case, it is.

71
00:03:59,949 --> 00:04:02,509
2 plus 9 plus 7 is equal to 18.

72
00:04:02,509 --> 00:04:04,599
So let's see how many times 9 goes into that.

73
00:04:04,599 --> 00:04:07,150
I'll do it on the side here; I don't want to be too messy.

74
00:04:07,150 --> 00:04:09,450
9 goes into 2 97.

75
00:04:09,449 --> 00:04:13,629


76
00:04:13,629 --> 00:04:16,189
3 times 27.

77
00:04:16,189 --> 00:04:19,040
27-- it goes 33 times, right?

78
00:04:19,040 --> 00:04:24,290
So this is the same thing as 9 plus or minus the square root

79
00:04:24,290 --> 00:04:31,110
of 9 times 33 over minus 18.

80
00:04:31,110 --> 00:04:32,470
And 9 is a perfect square.

81
00:04:32,470 --> 00:04:34,650
That's why I actually wanted to see if 9 would work because

82
00:04:34,649 --> 00:04:36,389
that's the only way I could get it out of the radical, if

83
00:04:36,389 --> 00:04:37,389
it's a perfect square.

84
00:04:37,389 --> 00:04:40,409
As you learned in that exponent rules number one module.

85
00:04:40,410 --> 00:04:46,140
So this is equal to 9 plus or minus 3 times the square

86
00:04:46,139 --> 00:04:53,229
root of 33, and all of that over minus 18.

87
00:04:53,230 --> 00:04:54,569
We're almost done.

88
00:04:54,569 --> 00:04:57,839
We can actually simplify it because 9, 3, and minus 18

89
00:04:57,839 --> 00:05:00,649
are all divisible by 3.

90
00:05:00,649 --> 00:05:02,269
Let's divide everything by 3.

91
00:05:02,269 --> 00:05:14,370
3 plus or minus the square root of 33 over minus 6.

92
00:05:14,370 --> 00:05:15,610
And we are done.

93
00:05:15,610 --> 00:05:17,009
So as you see, the hardest thing with the quadratic

94
00:05:17,009 --> 00:05:20,110
equation is often just simplifying the expression.

95
00:05:20,110 --> 00:05:22,750
But what we've said, I know you might have lost track-- we did

96
00:05:22,750 --> 00:05:27,120
all this math-- is we said, this equation: minus 9x

97
00:05:27,120 --> 00:05:30,550
squared minus 9x plus 6.

98
00:05:30,550 --> 00:05:34,199
Now we found two x values that would satisfy this equation

99
00:05:34,199 --> 00:05:35,969
and make it equal to 0.

100
00:05:35,970 --> 00:05:39,830
One x value is x equals 3 plus the square root

101
00:05:39,829 --> 00:05:42,099
of 33 over minus 6.

102
00:05:42,100 --> 00:05:45,860
And the second value is 3 minus the square root

103
00:05:45,860 --> 00:05:50,160
of 33 over minus 6.

104
00:05:50,160 --> 00:05:52,250
And you might want to think about why we have

105
00:05:52,250 --> 00:05:53,370
that plus or minus.

106
00:05:53,370 --> 00:05:55,490
We have that plus or minus because a square root could

107
00:05:55,490 --> 00:05:59,550
actually be a positive or a negative number.

108
00:05:59,550 --> 00:06:02,180
Let's do another problem.

109
00:06:02,180 --> 00:06:05,889
Hopefully this one will be a little bit simpler.

110
00:06:05,889 --> 00:06:09,209


111
00:06:09,209 --> 00:06:16,779
Let's say I wanted to solve minus 8x squared

112
00:06:16,779 --> 00:06:21,000
plus 5x plus 9.

113
00:06:21,000 --> 00:06:23,149
Now I'm going to assume that you've memorized the quadratic

114
00:06:23,149 --> 00:06:25,310
equation because that's something you should do.

115
00:06:25,310 --> 00:06:26,629
Or you should write it down on a piece of paper.

116
00:06:26,629 --> 00:06:31,629
But the quadratic equation is negative B-- So b is 5, right?

117
00:06:31,629 --> 00:06:34,159
We're trying to solve that equal to 0, so negative B.

118
00:06:34,160 --> 00:06:39,790
So negative 5, plus or minus the square root of B squared-

119
00:06:39,790 --> 00:06:44,030
that's 5 squared, 25.

120
00:06:44,029 --> 00:06:50,469
Minus 4 times A, which is minus 8.

121
00:06:50,470 --> 00:06:53,820
Times C, which is 9.

122
00:06:53,819 --> 00:06:56,399
And all of that over 2 times A.

123
00:06:56,399 --> 00:07:00,319
Well, A is minus 8, so all of that is over minus 16.

124
00:07:00,319 --> 00:07:04,089
So let's simplify this expression up here.

125
00:07:04,089 --> 00:07:09,439
Well, that's equal to minus 5 plus or minus

126
00:07:09,439 --> 00:07:13,629
the square root of 25.

127
00:07:13,629 --> 00:07:14,620
Let's see.

128
00:07:14,620 --> 00:07:18,220
4 times 8 is 32 and the negatives cancel out, so

129
00:07:18,220 --> 00:07:21,520
that's positive 32 times 9.

130
00:07:21,519 --> 00:07:24,479
Positive 32 times 9, let's see.

131
00:07:24,480 --> 00:07:26,720
30 times 9 is 270.

132
00:07:26,720 --> 00:07:31,110
It's 288.

133
00:07:31,110 --> 00:07:31,569
I think.

134
00:07:31,569 --> 00:07:31,800
Right?

135
00:07:31,800 --> 00:07:36,129


136
00:07:36,129 --> 00:07:37,490
288.

137
00:07:37,490 --> 00:07:40,590
We have all of that over minus 16.

138
00:07:40,589 --> 00:07:42,560
Now simplify it more.

139
00:07:42,560 --> 00:07:47,759
Minus 5 plus or minus the square root-- 25 plus

140
00:07:47,759 --> 00:07:51,339
288 is 313 I believe.

141
00:07:51,339 --> 00:07:56,949


142
00:07:56,949 --> 00:08:00,229
And all of that over minus 16.

143
00:08:00,230 --> 00:08:03,430
And I think, I'm not 100% sure, although I'm pretty sure.

144
00:08:03,430 --> 00:08:04,569
I haven't checked it.

145
00:08:04,569 --> 00:08:10,370
That 313 can't be factored into a product of a perfect

146
00:08:10,370 --> 00:08:11,689
square and another number.

147
00:08:11,689 --> 00:08:13,670
In fact, it actually might be a prime number.

148
00:08:13,670 --> 00:08:15,600
That's something that you might want to check out.

149
00:08:15,600 --> 00:08:18,200
So if that is the case and we've got it in completely

150
00:08:18,199 --> 00:08:21,839
simplified form, and we say there are two solutions, two

151
00:08:21,839 --> 00:08:24,939
x values that will make this equation true.

152
00:08:24,939 --> 00:08:30,750
One of them is x is equal to minus 5 plus the square

153
00:08:30,750 --> 00:08:35,830
root of 313 over minus 16.

154
00:08:35,830 --> 00:08:44,110
And the other one is x is equal to minus 5 minus the square

155
00:08:44,110 --> 00:08:49,659
root of 313 over minus 16.

156
00:08:49,659 --> 00:08:51,759
Hopefully those two examples will give you a good

157
00:08:51,759 --> 00:08:53,939
sense of how to use the quadratic equation.

158
00:08:53,940 --> 00:08:55,860
I might add some more modules.

159
00:08:55,860 --> 00:08:58,230
And then, once you master this, I'll actually teach you how to

160
00:08:58,230 --> 00:09:00,370
solve quadratic equations when you actually get a negative

161
00:09:00,370 --> 00:09:01,909
number under the radical.

162
00:09:01,909 --> 00:09:03,139
Very interesting.

163
00:09:03,139 --> 00:09:06,759
Anyway, I hope you can do the module now and maybe I'll add a

164
00:09:06,759 --> 00:09:10,370
few more presentations because this isn't the easiest module.

165
00:09:10,370 --> 00:09:11,840
But I hope you have fun.

166
00:09:11,840 --> 00:09:13,139
Bye.

167
00:09:13,139 --> 00:09:13,481