WEBVTT
1
00:00:00.000 --> 00:00:04.000
Hi, this is Julie Harland and I'm your math gal.
2
00:00:04.000 --> 00:00:12.000
Please visit my website at yourmathgal.com where you could search for any of my videos organized by topic.
3
00:00:12.000 --> 00:00:16.000
Let's try this puzzle.
4
00:00:16.000 --> 00:00:24.000
Find two numbers whose difference is 6 and his product is as small as possible.
5
00:00:24.000 --> 00:00:29.000
Hmm, what does that mean? That means when you want two numbers so that when you subtract it you get 6,
6
00:00:29.000 --> 00:00:33.000
which means there's 6 away from each other.
7
00:00:33.000 --> 00:00:39.000
But we also want to make sure when you multiply them together their product is as small as possible.
8
00:00:39.000 --> 00:00:44.000
So let's just look at some possibilities of numbers that subtract out to be 6.
9
00:00:44.000 --> 00:00:52.000
So here's some possibilities.
10
00:00:52.000 --> 00:00:56.000
10 and 4. When you subtract you get 6, right?
11
00:00:56.000 --> 00:01:04.000
And what would their product be?
12
00:01:04.000 --> 00:01:08.000
Well, 10 times 4 is 40.
13
00:01:08.000 --> 00:01:12.000
All right, how about something else? How about 6 and 0?
14
00:01:12.000 --> 00:01:16.000
When you do 6 minus 0 you get 6, so that works and what's 6 times 0?
15
00:01:16.000 --> 00:01:20.000
0. So this product is smaller than that product.
16
00:01:20.000 --> 00:01:23.000
So it's not 10 and 4. That's certainly not the answer.
17
00:01:23.000 --> 00:01:29.000
I wonder if this is the smallest you can get. Well, how about 5 and negative 1?
18
00:01:29.000 --> 00:01:35.000
Now that's tricky, but remember that means the difference means 5 minus negative 1
19
00:01:35.000 --> 00:01:39.000
and you know what that really does equal 6. So that's a possibility.
20
00:01:39.000 --> 00:01:43.000
And what happens when you do 5 times negative 1?
21
00:01:43.000 --> 00:01:45.000
We get negative 5.
22
00:01:45.000 --> 00:01:50.000
Now there's infinitely many possibilities here, like 100 minus 94, et cetera.
23
00:01:50.000 --> 00:01:59.000
You can have some fractions in here, like 6 and a half, and 1 half.
24
00:01:59.000 --> 00:02:03.000
And then you'd have to do this. Well, what's 6 and a half times 1 half?
25
00:02:03.000 --> 00:02:09.000
Well, that would be 13 half times a half. That would be 13 fourths.
26
00:02:09.000 --> 00:02:14.000
Negative 5 is still smaller than 13 fourths, right?
27
00:02:14.000 --> 00:02:22.000
So our goal is to find the combination of two numbers so that when you multiply those two numbers,
28
00:02:22.000 --> 00:02:26.000
you are going to get the smallest number possible.
29
00:02:26.000 --> 00:02:29.000
So now that we have an idea of what we're looking for,
30
00:02:29.000 --> 00:02:40.000
let's see if we do this algebraically. We're trying to find two numbers whose difference is 6.
31
00:02:40.000 --> 00:02:45.000
Alright, so how about we let the first number be x?
32
00:02:45.000 --> 00:02:49.000
Now what would the second number be if their difference is 6?
33
00:02:49.000 --> 00:02:52.000
That means there are 6 units apart from each other.
34
00:02:52.000 --> 00:02:57.000
Well, either a number 6 more than x or 6 less than x will do.
35
00:02:57.000 --> 00:03:02.000
So it's up to you how you want to define it. Either x minus 6 or x plus 6.
36
00:03:02.000 --> 00:03:06.000
Because if it's x plus 6, that's the bigger number, right?
37
00:03:06.000 --> 00:03:10.000
If it's x minus 6, then x would be the bigger number, but really doesn't matter.
38
00:03:10.000 --> 00:03:12.000
We're just trying to find two numbers.
39
00:03:12.000 --> 00:03:14.000
Now what would their product be?
40
00:03:14.000 --> 00:03:17.000
Their product means you multiply the two numbers together.
41
00:03:17.000 --> 00:03:21.000
So x times x plus 6 is the product.
42
00:03:21.000 --> 00:03:26.000
And what we're trying to find is something so the product is as big as possible.
43
00:03:26.000 --> 00:03:32.000
Let's call that product y.
44
00:03:32.000 --> 00:03:34.000
Now what do we have here?
45
00:03:34.000 --> 00:03:39.000
Well, this gives us y equals x squared plus 6x.
46
00:03:39.000 --> 00:03:42.000
Hopefully you recognize that as a quadratic.
47
00:03:42.000 --> 00:03:45.000
And it's a parabola if you graphed it.
48
00:03:45.000 --> 00:03:47.000
And it's a parabola going up.
49
00:03:47.000 --> 00:03:51.000
So we know it's some parabola going up.
50
00:03:51.000 --> 00:03:54.000
And we're trying to find the point.
51
00:03:54.000 --> 00:03:57.000
So the y value is as small as possible.
52
00:03:57.000 --> 00:04:00.000
We're trying to find the product that's as small as possible.
53
00:04:00.000 --> 00:04:02.000
So we want y as small as possible.
54
00:04:02.000 --> 00:04:06.000
So if there's the other numbers on here that have a difference of 6
55
00:04:06.000 --> 00:04:10.000
and their product is going to be bigger, could you see it's up higher.
56
00:04:10.000 --> 00:04:13.000
So we're trying to find that point right here.
57
00:04:13.000 --> 00:04:15.000
The vertex point.
58
00:04:15.000 --> 00:04:19.000
The vertex point will give us usually the maximum or the y value.
59
00:04:19.000 --> 00:04:24.000
Or we'll always look at the maximum or the minimum value for y.
60
00:04:24.000 --> 00:04:29.000
So in this case, it's going to give us a minimum, which is what we want.
61
00:04:29.000 --> 00:04:33.000
And yet the vertex point.
62
00:04:33.000 --> 00:04:38.000
We could do x equals negative b over 2a.
63
00:04:38.000 --> 00:04:43.000
Now on our problem here, b is 6.
64
00:04:43.000 --> 00:04:46.000
So that'll be 6a is 1.
65
00:04:46.000 --> 00:04:50.000
So this becomes negative 3.
66
00:04:50.000 --> 00:04:55.000
So what we're looking for is for the two numbers x and x plus 6.
67
00:04:55.000 --> 00:04:59.000
But for right now, I'm really just trying to find the vertex point here.
68
00:04:59.000 --> 00:05:01.000
I now know what x is.
69
00:05:01.000 --> 00:05:04.000
It's negative 3.
70
00:05:04.000 --> 00:05:07.000
Now how would we figure out what y is?
71
00:05:07.000 --> 00:05:11.000
We would plug negative 3 in to the equation.
72
00:05:11.000 --> 00:05:14.000
Doesn't matter if you do it x times x plus 6.
73
00:05:14.000 --> 00:05:18.000
Or if you do it y equals x squared plus 6x.
74
00:05:18.000 --> 00:05:21.000
How about we do it as x squared plus 6x?
75
00:05:21.000 --> 00:05:27.000
So I'd put a negative 3 for x.
76
00:05:27.000 --> 00:05:32.000
That gives you 9 plus negative 18, which is negative 9.
77
00:05:32.000 --> 00:05:37.000
So the y value of this ordered pairs negative 9.
78
00:05:37.000 --> 00:05:43.000
If you'd put it in the first equation, instead, you would have put a negative 3 for x.
79
00:05:43.000 --> 00:05:46.000
And then wrote negative 3 plus 6.
80
00:05:46.000 --> 00:05:50.000
And that's negative 3 times 3, which also equals negative 9.
81
00:05:50.000 --> 00:05:52.000
It doesn't really matter.
82
00:05:52.000 --> 00:05:59.000
The point is, we now know that the smallest product is going to be negative 9.
83
00:05:59.000 --> 00:06:01.000
But it's not asking for the smallest product.
84
00:06:01.000 --> 00:06:03.000
It says, fine two numbers.
85
00:06:03.000 --> 00:06:06.000
These differences 6.
86
00:06:06.000 --> 00:06:08.000
Antis product is as small as possible.
87
00:06:08.000 --> 00:06:14.000
What we're looking for are these two numbers x and x plus 6.
88
00:06:14.000 --> 00:06:18.000
So we know what x is, right?
89
00:06:18.000 --> 00:06:22.000
x, we've determined, was negative 3.
90
00:06:22.000 --> 00:06:24.000
So what's x plus 6?
91
00:06:24.000 --> 00:06:27.000
6 more than that would be 3.
92
00:06:27.000 --> 00:06:34.000
So that's what we're looking for, the numbers.
93
00:06:34.000 --> 00:06:43.000
R3 and negative 3.
94
00:06:43.000 --> 00:06:46.000
So notice my smallest product was negative 9.
95
00:06:46.000 --> 00:06:51.000
We're looking back up here when we were just trying some possibilities.
96
00:06:51.000 --> 00:06:55.000
We got a product of 40 is 0, a negative 5, a 13 force.
97
00:06:55.000 --> 00:06:58.000
None of these were as small as our answer is.
98
00:06:58.000 --> 00:07:00.000
The answer here was 3 and negative 3.
99
00:07:00.000 --> 00:07:02.000
That's two numbers whose differences 6.
100
00:07:02.000 --> 00:07:04.000
Think about where they are in the number line.
101
00:07:04.000 --> 00:07:06.000
There's six bases apart.
102
00:07:06.000 --> 00:07:10.000
3 minus a negative 3 does equal 6 in other words.
103
00:07:10.000 --> 00:07:12.000
And their product was negative 9.
104
00:07:12.000 --> 00:07:16.000
And that ended up being the very smallest product.
105
00:07:16.000 --> 00:07:21.000
So the answer is 3 and negative 3.
106
00:07:21.000 --> 00:07:27.000
And we were able to solve this puzzle using the idea of the vertex point
107
00:07:27.000 --> 00:07:42.000
because we were able to put this in quadratic form.
108
00:07:42.000 --> 00:07:47.000
Please visit my website at yourmathgal.com where you can view all of my videos
109
00:07:47.000 --> 00:07:50.000
which are organized by topic.
110
00:07:50.000 --> 00:08:05.000
So, please visit my website at www.dhv.com
111
00:08:05.000 --> 00:08:13.000
and visit my website at www.dhv.com