1
00:00:00,618 --> 00:00:03,818
Use the discriminant to state the number

2
00:00:03,818 --> 00:00:06,481
and type of solutions for the equation

3
00:00:06,527 --> 00:00:07,900
-3<i>x</i>²

4
00:00:07,900 --> 00:00:08,973
-3<i>x</i>² + 5<i>x</i>

5
00:00:08,973 --> 00:00:10,042
-3<i>x</i>² + 5<i>x</i> - 4

6
00:00:10,042 --> 00:00:11,327
-3<i>x</i>² + 5<i>x</i> - 4 = 0.

7
00:00:11,327 --> 00:00:12,581
And so just as a reminder

8
00:00:12,581 --> 00:00:13,250
you're probably wondering

9
00:00:13,250 --> 00:00:14,802
what is the <i>discriminant</i>?

10
00:00:14,802 --> 00:00:16,035
And we can just review it by

11
00:00:16,035 --> 00:00:17,669
looking at the quadratic formula.

12
00:00:17,669 --> 00:00:20,818
So if I have a quadratic equation in standard form

13
00:00:20,818 --> 00:00:22,105
<i>ax</i>²

14
00:00:22,105 --> 00:00:23,555
<i>ax</i>² + <i>bx</i>

15
00:00:23,555 --> 00:00:24,623
<i>ax</i>² + <i>bx</i> + <i>c</i>

16
00:00:24,623 --> 00:00:25,767
<i>ax</i>² + <i>bx</i> + <i>c</i> = 0

17
00:00:25,767 --> 00:00:27,452
We know that the quadratic formula,

18
00:00:27,452 --> 00:00:28,653
which is really just derived from

19
00:00:28,653 --> 00:00:30,477
completing the square right over here,

20
00:00:30,477 --> 00:00:32,296
tells us that the roots of this,

21
00:00:32,296 --> 00:00:34,618
or the solutions of this quadratic equation are

22
00:00:34,618 --> 00:00:35,199
going to be

23
00:00:35,199 --> 00:00:36,366
<i>x</i> =

24
00:00:36,366 --> 00:00:37,249
<i>x</i> = (-<i>b</i>)

25
00:00:37,249 --> 00:00:39,666
<i>x</i> = (-<i>b</i> ± √())

26
00:00:39,666 --> 00:00:40,804
<i>x</i> = (-<i>b</i> ± √(<i>b</i>²))

27
00:00:40,804 --> 00:00:42,216
<i>x</i> = (-<i>b</i> ± √(<i>b</i>² - 4<i>ac</i>))

28
00:00:42,216 --> 00:00:44,959
all of that over (2<i>a</i>).

29
00:00:45,328 --> 00:00:47,635
Now, you might know from experience

30
00:00:47,635 --> 00:00:48,868
applying this a little bit,

31
00:00:48,868 --> 00:00:51,936
we're going to get different types of solutions

32
00:00:51,936 --> 00:00:53,504
depending on what happens

33
00:00:53,504 --> 00:00:56,271
under the radical sign over here.

34
00:00:56,271 --> 00:00:57,868
As you can imagine, if what's

35
00:00:57,868 --> 00:01:00,969
under the radical sign over here is positive,

36
00:01:00,969 --> 00:01:04,118
then we're going to get an actual, real number

37
00:01:04,118 --> 00:01:05,718
as its principal square root.

38
00:01:05,718 --> 00:01:07,152
And when we take the positive and negative

39
00:01:07,152 --> 00:01:10,186
version of it, we're going to get two real solutions.

40
00:01:10,186 --> 00:01:13,635
So if <i>b</i>² - 4<i>ac</i>, and this is

41
00:01:13,635 --> 00:01:15,185
what the discriminant <i>really</i> is,

42
00:01:15,185 --> 00:01:19,402
it's just this expression under the radical sign

43
00:01:19,402 --> 00:01:20,718
of the quadratic formula.

44
00:01:20,718 --> 00:01:22,718
If this is greater than zero,

45
00:01:22,718 --> 00:01:24,688
then we're going to have two real roots,

46
00:01:25,088 --> 00:01:30,487
then we're going to have two real roots,

47
00:01:30,487 --> 00:01:33,701
or two real solutions to this equation right here.

48
00:01:33,701 --> 00:01:39,802
If <i>b</i>² - 4<i>ac</i> = 0,

49
00:01:39,802 --> 00:01:41,016
then this whole thing is

50
00:01:41,016 --> 00:01:42,054
just going to be equal to zero,

51
00:01:42,054 --> 00:01:43,818
so plus or minus the square root of zero,

52
00:01:43,818 --> 00:01:44,944
(which is just zero)

53
00:01:44,944 --> 00:01:46,408
so this is plus or minus zero.

54
00:01:46,408 --> 00:01:48,086
Well, when you add or subtract 0,

55
00:01:48,086 --> 00:01:49,722
that doesn't change the solution,

56
00:01:49,722 --> 00:01:51,318
so the only solution is going to be

57
00:01:51,318 --> 00:01:53,818
-<i>b</i> / 2<i>a</i>

58
00:01:53,818 --> 00:01:57,603
So you're only going to have one real solution.

59
00:01:57,603 --> 00:02:00,200
So this is going to be one--

60
00:02:00,200 --> 00:02:01,703
--I'll just write the number "1"--

61
00:02:01,703 --> 00:02:03,718
--one real solution,

62
00:02:03,718 --> 00:02:04,902
or you could kind of say,

63
00:02:04,902 --> 00:02:06,436
you have a repeated root here.

64
00:02:06,436 --> 00:02:08,735
You're kind of having it twice.

65
00:02:08,735 --> 00:02:13,185
Or you could say one real solution or one real root.

66
00:02:13,354 --> 00:02:17,414
Now if <i>b</i>² - 4<i>ac</i> were negative –

67
00:02:17,414 --> 00:02:20,289
you might already imagine what will happen.

68
00:02:20,289 --> 00:02:22,203
If this expression right over here is negative,

69
00:02:22,203 --> 00:02:24,202
we're taking the square root of a negative number.

70
00:02:24,202 --> 00:02:26,452
So we would then get an imaginary number

71
00:02:26,452 --> 00:02:27,535
right over here.

72
00:02:27,535 --> 00:02:28,820
So we would add or subtract

73
00:02:28,820 --> 00:02:31,280
the same imaginary number.

74
00:02:31,280 --> 00:02:33,185
So we'll have two complex solutions;

75
00:02:33,185 --> 00:02:35,086
not only will we have two complex solutions,

76
00:02:35,086 --> 00:02:37,235
but they will be the <i>conjugates</i> of each other.

77
00:02:37,235 --> 00:02:39,019
So if you have one complex solution

78
00:02:39,019 --> 00:02:41,969
for a quadratic equation, the other solution

79
00:02:41,969 --> 00:02:43,785
will also be a complex solution

80
00:02:43,785 --> 00:02:45,836
and will be its complex conjugate.

81
00:02:45,836 --> 00:02:52,218
So here we would have two complex solutions.

82
00:02:52,218 --> 00:02:53,736
So, numbers that have a real part

83
00:02:53,736 --> 00:02:55,103
and an imaginary part.

84
00:02:55,103 --> 00:02:56,535
And not only are they just complex, but

85
00:02:56,535 --> 00:02:58,118
they are the conjugates of each other.

86
00:02:58,118 --> 00:03:00,785
The imaginary parts have different signs.

87
00:03:00,785 --> 00:03:03,818
So let's look at <i>b</i>² - 4<i>ac</i> over here.

88
00:03:03,818 --> 00:03:05,419
This is our <i>a</i>,

89
00:03:05,419 --> 00:03:06,837
this is our <i>b</i>,

90
00:03:06,837 --> 00:03:07,500
and this is our <i>c</i>.

91
00:03:07,500 --> 00:03:08,368
Let me label them

92
00:03:08,368 --> 00:03:09,606
Let me label them – <i>a</i>

93
00:03:09,606 --> 00:03:10,320
Let me label them – <i>a</i>, <i>b</i>

94
00:03:10,320 --> 00:03:11,157
Let me label them – <i>a</i>, <i>b</i>, <i>c</i>.

95
00:03:11,157 --> 00:03:12,088
I can do that because we've

96
00:03:12,088 --> 00:03:13,319
written it in standard form.

97
00:03:13,319 --> 00:03:15,933
Everything is on one side,

98
00:03:15,933 --> 00:03:17,568
in particular the left-hand side,

99
00:03:17,568 --> 00:03:19,135
we have a zero on the right-hand side,

100
00:03:19,135 --> 00:03:22,535
we've written it in descending power form,

101
00:03:22,535 --> 00:03:24,268
or descending degree,

102
00:03:24,268 --> 00:03:25,736
where we have our 2nd degree term first,

103
00:03:25,736 --> 00:03:26,886
then our 1st degree term,

104
00:03:26,886 --> 00:03:28,686
then our constant term.

105
00:03:28,686 --> 00:03:30,536
And so, we can evaluate the discriminant!

106
00:03:32,367 --> 00:03:33,421
<i>b</i> = 5

107
00:03:33,421 --> 00:03:34,750
<i>b</i> = 5, so <i>b</i>² = 5²

108
00:03:34,750 --> 00:03:36,097
5² - 4

109
00:03:36,097 --> 00:03:37,350
5² - 4 • <i>a</i>

110
00:03:37,350 --> 00:03:38,327
5² - 4 • (-3)

111
00:03:38,327 --> 00:03:39,728
5² - 4 • (-3) • c

112
00:03:39,728 --> 00:03:41,302
5² - 4 • (-3) • (-4)

113
00:03:41,302 --> 00:03:43,506
<i>c</i> is this whole thing, I have to be careful.

114
00:03:43,506 --> 00:03:49,484
<i>c</i> is <i>negative</i> 4, we have to make sure

115
00:03:49,484 --> 00:03:51,917
we take the sign into consideration,

116
00:03:51,917 --> 00:03:55,218
so times <i>c</i>, which is <i>negative</i> 4 over here.

117
00:03:55,218 --> 00:03:55,799
So this is

118
00:03:55,799 --> 00:03:59,827
25 - 4 • (-3) • (-4)

119
00:03:59,827 --> 00:04:02,743
25 - 4 • 12

120
00:04:02,743 --> 00:04:07,705
25 - 48

121
00:04:07,705 --> 00:04:09,904
We don't even have to do the math –

122
00:04:09,904 --> 00:04:12,535
we can just say that this is definitely going to

123
00:04:12,535 --> 00:04:13,678
be less than zero.

124
00:04:13,678 --> 00:04:14,867
You can actually figure it out –

125
00:04:14,867 --> 00:04:22,131
this is equal to negative 23,

126
00:04:22,131 --> 00:04:24,814
negative 23...

127
00:04:24,814 --> 00:04:26,764
which is clearly less than zero.

128
00:04:26,764 --> 00:04:28,495
So our discriminant in this situation is

129
00:04:28,495 --> 00:04:30,996
less than zero, so we are going to have

130
00:04:30,996 --> 00:04:33,245
two complex roots here,

131
00:04:33,245 --> 00:04:35,278
and they are going to be each other's conjugates.