WEBVTT
1
00:00:00.000 --> 00:00:08.000
This is the sixth and the last lecture on factoring, and we're solving equations by factoring.
2
00:00:08.000 --> 00:00:15.000
And this is the actual practical use that factoring really has, is that once we factor something,
3
00:00:15.000 --> 00:00:19.000
then we can use that information to solve for a variable.
4
00:00:19.000 --> 00:00:23.000
So we start off with the zero product property.
5
00:00:23.000 --> 00:00:32.000
All this means is, if a times b equals zero, then what does this mean about a and b?
6
00:00:32.000 --> 00:00:40.000
Well, if a times b equals zero, then either a is zero or b is zero, or both.
7
00:00:40.000 --> 00:00:48.000
There's no way two integers could multiply, or any two numbers, could be multiplied together in equals zero,
8
00:00:48.000 --> 00:00:55.000
unless one of the two was zero. So that's all that property says.
9
00:00:55.000 --> 00:01:00.000
So if we look at this, we're going to apply it to polynomials.
10
00:01:00.000 --> 00:01:09.000
So here's a picture of a diver diving from a cliff along this parabola type trajectory going downward.
11
00:01:09.000 --> 00:01:23.000
And the equation that gives the height of the diver after t seconds is this h equals 87 plus 18 minus 16t squared.
12
00:01:23.000 --> 00:01:29.000
My question is, at what time will the diver be at a height of 87 feet?
13
00:01:29.000 --> 00:01:36.000
So the first thing you might think as well, I mean at time equals zero, if you put zero into these two things,
14
00:01:36.000 --> 00:01:41.000
that would go to zero, that would be zero. At time zero, he starts at 87 feet.
15
00:01:41.000 --> 00:01:49.000
And if you look at the diagram also with this dashed line, yet his starting trajectory, it starts right at 87 feet.
16
00:01:49.000 --> 00:01:56.000
So at time equals zero, that's true. But also at a different time, he also hits 87 feet.
17
00:01:56.000 --> 00:02:01.000
And I also ask, where do you think the maximum height would be?
18
00:02:01.000 --> 00:02:08.000
And it looks like right here, right? And it actually ends up the parabola is very symmetrical.
19
00:02:08.000 --> 00:02:15.000
So yes, halfway between the starting and there are the first time that he hits 87 feet.
20
00:02:15.000 --> 00:02:22.000
And the second time he hits 87 feet, right in between those two, he hits his maximum height.
21
00:02:22.000 --> 00:02:27.000
So if we were to figure out, when does this diver hit 87 feet?
22
00:02:27.000 --> 00:02:34.000
Well, the logical thing to do would be to put 87 in for H and see what T is.
23
00:02:34.000 --> 00:02:43.000
So let's try it. We'll put 87 in for H. So that equals 87 plus 18 minus 16 T squared.
24
00:02:43.000 --> 00:02:53.000
Okay, we'll subtract 87 from both sides. So zero equals 18 minus 16 T squared.
25
00:02:53.000 --> 00:02:58.000
All right, well, can we use our knowledge of factoring to factor that right side?
26
00:02:58.000 --> 00:03:10.000
It looks like we can. We can take an AT out. Well, what's 80 divided by 80? It's 1 minus 2T.
27
00:03:10.000 --> 00:03:22.000
So using our zero product principle now or zero product property, we know that either 80 is 0 or 1 minus 2T is 0.
28
00:03:22.000 --> 00:03:30.000
One or the other must be 0. So if we divide both sides by 8, either T equals 0, which we knew already, right?
29
00:03:30.000 --> 00:03:39.000
At 0 seconds, he started at 87 feet or 1 equals 2T.
30
00:03:39.000 --> 00:03:45.000
At half a second, he's also at 87 feet.
31
00:03:45.000 --> 00:03:54.000
So these are the two times that he's at 87 feet. So that would mean half way in between there, he reaches his maximum height.
32
00:03:54.000 --> 00:04:04.000
So actually at one quarter of a second is where he hits this.
33
00:04:04.000 --> 00:04:14.000
All right, continuing along, we're going to solve each of these equations and then check to make sure our answers are correct.
34
00:04:14.000 --> 00:04:23.000
So here we know either x plus 3 equals 0 or x minus 5 equals 0.
35
00:04:23.000 --> 00:04:34.000
So if we subtract 3, x equals negative 3 or x equals 5.
36
00:04:34.000 --> 00:04:47.000
Now you can write it just like this. You could say there's two answers, x equals negative 3 or x equals 5 or you could write it like this.
37
00:04:47.000 --> 00:05:10.000
So x equals the set negative 3 times 5, where both of those numbers are in the set of possible solutions for this. And then if we check this, well if we put negative 3 in here, whoops, negative 3 plus 3 is 0, 0 times x minus 5, that's going to be 0.
38
00:05:10.000 --> 00:05:25.000
So I put 5 in, I'll get 0. So the only reason I say to check is just to make sure that you did this right because some people hurry along and say if x plus 3 is 0, then x equals 3, where it's actually x equals negative 3.
39
00:05:25.000 --> 00:05:43.000
I just want you to make sure that you're doing it correctly. Okay, this next one, either 2a plus 4 equals 0 or a plus 7 is 0.
40
00:05:43.000 --> 00:06:09.000
If we subtract 4, 2a is negative 4 and a is negative 2 or a is negative 7. And finally, this one's a little tougher, so we'll have to bring the 5x over to this side by subtracting 5x from both sides.
41
00:06:09.000 --> 00:06:20.000
So we got x squared minus 5x minus 36 equals 0. Very, very important thing here is to put equals 0 or to not forget the equals 0.
42
00:06:20.000 --> 00:06:42.000
Alright, so we can factor this using our diamond problem. So negative 36 up here and negative 5 down here. What multiplies to negative 36 and adds to 5 is negative 9 and 4. So x minus 9 times x plus 4 equals 0.
43
00:06:42.000 --> 00:06:58.000
So then either x minus 9 equals 0 or x plus 4 equals 0. So that means x equals 9 or x equals negative 4.
44
00:06:58.000 --> 00:07:11.000
Okay, so you can see how this could be useful to solve for x if you had to factor a trinomial or some sort of polynomial like that. Alright, we just have a few more left.
45
00:07:11.000 --> 00:07:20.000
This is a very similar one to the last one. We'll add 144 to both sides to get everything on the left.
46
00:07:20.000 --> 00:07:44.000
This is a diamond problem where we have to multiply it to 144 and add to negative 24. So let's see if it were 12 and 12 make 144, but those obviously don't work.
47
00:07:44.000 --> 00:08:11.000
If we had 2, it would be 2 and 72. That doesn't work. 4 would be 36. 36 and 4 does not work either. We could use 8, 8 and 18.
48
00:08:11.000 --> 00:08:40.000
That also does not quite work. So how about 6 and 144 divided by 6 is 24. So that also will not work.
49
00:08:40.000 --> 00:08:54.000
So how about 3? Just 3 go into that.
50
00:08:54.000 --> 00:09:18.000
So that's 48 times, but that also doesn't work. So this one's a little tricky. It looks like maybe 7.
51
00:09:18.000 --> 00:09:38.000
Does not work. So I can't find any numbers that multiply to give me 144 and add to give me negative 24.
52
00:09:38.000 --> 00:09:59.000
So it's looking like this one doesn't work at all. So I would say this one's no real solutions because I can't factor this.
53
00:09:59.000 --> 00:10:26.000
Okay. And sorry about that. That was not correct. It actually turns out negative 12 and negative 12, which is perhaps the simplest and most obvious of the two numbers that would work to multiply to give you 144.
54
00:10:26.000 --> 00:10:40.000
If you add negative 12 and negative 12, they do give you negative 24. So I have a squared minus 12, a minus 12, a plus 144. And you can do it by factor and by grouping.
55
00:10:40.000 --> 00:11:01.000
So I know that because this is a 1a squared, it's a minus 12 squared equals zero or a minus 12 times a minus 12. So then I'm going to just have a minus 12 equals zero.
56
00:11:01.000 --> 00:11:15.000
So a equals 12 is my only answer. And that's because this was a perfect square. And so there's only one answer to check. So it's only a equals 12. That's the only answer for that one.
57
00:11:15.000 --> 00:11:39.000
All right. Hopefully this one won't be so tough. So we'll move the 20m to the other to this side. And that equals zero. Okay. We'll do a diamond problem four times 25 is 100. And we want add to negative 20.
58
00:11:39.000 --> 00:11:48.000
So we have to figure out what multiplies to 100 and adds to negative 20.
59
00:11:48.000 --> 00:12:03.000
And let's see 25 and 4 do not quite work. 25 don't work.
60
00:12:03.000 --> 00:12:15.000
So let's see what we could do. It looks like we're going to do negative 10 and negative 10.
61
00:12:15.000 --> 00:12:31.000
So we'll have 4m squared minus 10m minus 10m plus 25 equals zero. And now on this one, because this is not a coefficient of 1, we can't just do m minus 10 times m minus 10.
62
00:12:31.000 --> 00:12:49.000
We'll have to factor by grouping. So with these first two, I can take out 2m and I get 2m minus 5. The next two I'll take out negative 5 and get 2m minus 5 equals zero.
63
00:12:49.000 --> 00:13:06.000
So 2m minus 5 is what's in common. And I'm left with 2m minus 5. So actually this was a special formula that I didn't recognize. This is a perfect square.
64
00:13:06.000 --> 00:13:27.000
And this is 2 times 2m times 5. So it's 2m minus 5 squared. So all we have to check is if 2m minus 5 equals zero or 2m equals 5. So m equals 5 halves.
65
00:13:27.000 --> 00:13:43.000
So 5 over 2 or 2.5 is the only solution. And the last problem, first we'll take out a t. And we get 6t squared plus t minus 5 equals zero.
66
00:13:43.000 --> 00:14:07.000
We have to multiply to negative 30 and add to 1. It's going to be 6 and negative 5. So we'll keep the t and we'll rewrite 6t squared plus 6t minus 5t minus 5 equals zero.
67
00:14:07.000 --> 00:14:20.000
We can take out a 6t out of the first two. And that leaves me with t plus 1. This is a big deal right here. 6t divided by 6t is 1.
68
00:14:20.000 --> 00:14:28.000
You must put something there. You cannot put zero. You cannot leave it blank. 6t divided by 6t is 1.
69
00:14:28.000 --> 00:14:40.000
We'll take out a negative 5 and we're left with t plus 1 also there. So what's in common? We'll continue with the t.
70
00:14:40.000 --> 00:15:02.000
What's in common is t plus 1 and we're left with 6t minus 5 and this all equals zero. So either t equals zero or t plus 1 equals zero or 6t minus 5 equals zero.
71
00:15:02.000 --> 00:15:15.000
So that means t equals zero. That one's done or t equals negative 1. Now that one's done or 6t equals 5 which means t equals 5 over 6.
72
00:15:15.000 --> 00:15:25.000
So there are three possible solutions. t equals zero negative 1 or 5 over 6.
73
00:15:25.000 --> 00:15:37.000
And that's how all of those go. You have to factor. Then you have to set each factor equals zero and see if you can find out what the variable could equal.
74
00:15:37.000 --> 00:16:06.000
Alright that's all for factoring. I hope that cleared a lot of things up. Thanks for watching.