WEBVTT 00:00:00.550 --> 00:00:03.958 When we have two 00:00:03.958 --> 00:00:11.302 brackets. X +2 times by X +3 and we know how to multiply 00:00:11.302 --> 00:00:13.286 these two brackets out. 00:00:13.970 --> 00:00:21.824 We have X kinds by X that gives us X squared. We have X 00:00:21.824 --> 00:00:25.190 times by three, gives us 3X. 00:00:26.320 --> 00:00:32.937 2 times by X gives us 2X and then two times by three 00:00:32.937 --> 00:00:34.464 gives us 6. 00:00:35.630 --> 00:00:42.923 And we can simplify these two terms. 3X plus 2X gives us 5X. 00:00:44.180 --> 00:00:49.157 This is an example of a quadratic expression or 00:00:49.157 --> 00:00:54.134 quadratic function. It's gotta termine ex squared, which it 00:00:54.134 --> 00:00:57.452 must have to be a quadratic 00:00:57.452 --> 00:01:04.188 expression. It's got a term in X which it might or might not 00:01:04.188 --> 00:01:09.516 have, and it's got a constant term and there are no other 00:01:09.516 --> 00:01:12.624 possibilities, so our most general quadratic expression 00:01:12.624 --> 00:01:15.732 would be AX squared plus BX plus 00:01:15.732 --> 00:01:21.910 C. What we're going to have a look at is how we factorise 00:01:21.910 --> 00:01:27.045 expressions like this in others. How we go back from this kind of 00:01:27.045 --> 00:01:33.210 expression. To this now, why might we want to do that? Well, 00:01:33.210 --> 00:01:39.060 let's just take this X squared plus 5X plus six. And let's say 00:01:39.060 --> 00:01:44.460 it's not just an expression, but it's an equation and it says 00:01:44.460 --> 00:01:47.610 equals 0. What are the values of 00:01:47.610 --> 00:01:54.036 X? That will make it equal to 0 that are answers to that 00:01:54.036 --> 00:01:59.795 equation. One of the things we can do is to rewrite this form. 00:02:00.320 --> 00:02:08.204 By this so we can say X +2 times by X +3 00:02:08.204 --> 00:02:14.009 equals 0. When we have two numbers that multiply 00:02:14.009 --> 00:02:20.967 together to give zero and one of the things that must be true is 00:02:20.967 --> 00:02:27.428 that one of them zero or the other one zero, or they're both 00:02:27.428 --> 00:02:35.380 0. So in this case, X +2 equals 0 or X +3 equals 0, and so 00:02:35.380 --> 00:02:40.847 X would be minus two, or X would be minus three. 00:02:41.460 --> 00:02:47.352 So being able to factorize actually helps us to solve a new 00:02:47.352 --> 00:02:48.825 kind of equation. 00:02:49.360 --> 00:02:56.024 So we're going to be having a look in this video that how you 00:02:56.024 --> 00:03:00.784 factorise this kind of function. This kind of expression a 00:03:00.784 --> 00:03:06.916 quadratic expression. Now I'm going to start by going back to 00:03:06.916 --> 00:03:09.820 this little piece of work again. 00:03:09.830 --> 00:03:13.745 So let's write it down 00:03:13.745 --> 00:03:20.432 X. Plus 3 * 5 X +2 00:03:20.432 --> 00:03:24.586 and again. Will multiply out the 00:03:24.586 --> 00:03:28.378 brackets X times by X is X squared. 00:03:29.550 --> 00:03:36.674 X times by two is 2 X 3 times by X is 3X. 00:03:37.240 --> 00:03:43.884 3 times by two is 6. This simplifies to X squared 00:03:43.884 --> 00:03:46.300 plus 5X or 6. 00:03:47.490 --> 00:03:52.209 So we've gone one way. What happens if we want to go back 00:03:52.209 --> 00:03:53.298 the other way? 00:03:54.070 --> 00:04:00.622 Let's have a look where this six came from. We know it came from 00:04:00.622 --> 00:04:02.494 3 times by two. 00:04:03.730 --> 00:04:11.088 Where did this five come from? Where it came from 2 + 3? 00:04:12.140 --> 00:04:18.536 So if we were to reverse this process, we be looking for two 00:04:18.536 --> 00:04:24.440 numbers that multiply together to give us six an 2 numbers that 00:04:24.440 --> 00:04:27.392 added together to give us 5. 00:04:28.030 --> 00:04:34.660 The obvious ones that go in there are three 2. 00:04:36.550 --> 00:04:38.638 So if we began. 00:04:39.210 --> 00:04:40.380 With this 00:04:42.030 --> 00:04:43.280 We would. 00:04:44.310 --> 00:04:49.133 Be looking to break that 5X down 00:04:49.133 --> 00:04:55.334 as X squared plus 3X plus 2X plus 6. 00:04:56.860 --> 00:05:02.712 Then we could look at these two and see if there was a common 00:05:02.712 --> 00:05:08.564 factor and there is X leaving us with X Plus three. Then we would 00:05:08.564 --> 00:05:13.998 look at these two. Is there a common factor and there is 2. 00:05:14.530 --> 00:05:17.950 Leaving us again with 00:05:17.950 --> 00:05:25.170 X +3. And then we've got this common factor of X plus 00:05:25.170 --> 00:05:31.960 three in each of these two lumps of algebra, so we can take out 00:05:31.960 --> 00:05:38.750 that X +3, and we've got the other factor left X times by X 00:05:38.750 --> 00:05:45.540 plus three and two times by X +3, and so we've arrived at that 00:05:45.540 --> 00:05:49.420 factorization. Those brackets that we started off with. 00:05:50.560 --> 00:05:57.256 Now. That's what we've done and what we need to do is to be able 00:05:57.256 --> 00:06:00.622 to repeat this process of looking for numbers that 00:06:00.622 --> 00:06:04.362 multiply together to give the constant term and numbers that 00:06:04.362 --> 00:06:08.850 will add together to give the exterm. So let's have a little 00:06:08.850 --> 00:06:10.720 bit of practice at that. 00:06:11.820 --> 00:06:18.939 Let's look at X squared minus 7X plus 12. 00:06:19.930 --> 00:06:24.226 So we want two numbers that will multiply together to give us. 00:06:24.820 --> 00:06:32.320 12 Times together to give us 12 and will add together 00:06:32.320 --> 00:06:35.260 to give us minus 7. 00:06:36.350 --> 00:06:42.650 Minus four times minus three is 12 and minus four plus minus 00:06:42.650 --> 00:06:48.950 three gives us minus Seven, so let's just write those in minus 00:06:48.950 --> 00:06:55.250 four times, minus three gives us plus 12 and minus four plus 00:06:55.250 --> 00:07:01.550 minus three gives us minus Seven, so that's given us a way 00:07:01.550 --> 00:07:07.325 of breaking down this minus Seven XX squared minus 4X Minus 00:07:07.325 --> 00:07:12.972 3X. Plus 12 so now we look at these two at the front. 00:07:13.730 --> 00:07:21.052 Take out X as a common factor that gives us X minus four, and 00:07:21.052 --> 00:07:24.190 now we look at these two. 00:07:24.980 --> 00:07:30.160 Well, I want to make sure I get the same factor X minus 4. 00:07:31.470 --> 00:07:36.342 Clearly, in these two terms, that is a factor of three. But 00:07:36.342 --> 00:07:42.026 here I've got minus three, so I think I'm going to have to make 00:07:42.026 --> 00:07:46.492 the factor, not three, but minus three. So that's minus three 00:07:46.492 --> 00:07:50.958 times X, so that insures when I multiply these two together 00:07:50.958 --> 00:07:56.642 minus three times by ex. I do get minus 3X, but now I need 00:07:56.642 --> 00:08:01.514 minus three times by something that's got to give me plus 12. 00:08:01.570 --> 00:08:07.758 So that will have to be minus four and close the bracket. Now 00:08:07.758 --> 00:08:13.946 again I've got two lumps of algebra, and in each one that is 00:08:13.946 --> 00:08:19.658 the same factor. This common factor of X minus 4X minus four, 00:08:19.658 --> 00:08:25.846 so I'll take that as my common factor X minus four. Then I've 00:08:25.846 --> 00:08:32.034 got X minus four times by X&X minus four times by minus three. 00:08:32.370 --> 00:08:35.722 That's my factorization of 00:08:35.722 --> 00:08:39.106 that. Let's take another 00:08:39.106 --> 00:08:41.995 one. X 00:08:41.995 --> 00:08:48.254 squared Minus 5X minus 14. So now 00:08:48.254 --> 00:08:52.448 looking for two numbers to multiply 00:08:52.448 --> 00:08:58.739 together to give minus 14 and add together to 00:08:58.739 --> 00:09:00.836 give minus 5. 00:09:01.860 --> 00:09:08.061 Fairly obvious factors of 14 R. Seven and two. So can we play 00:09:08.061 --> 00:09:14.262 with Seven and two? Well, if we made it minus Seven and plus 00:09:14.262 --> 00:09:20.463 two, then minus Seven times my plus two would give us minus 14 00:09:20.463 --> 00:09:26.187 and minus Seven, plus the two would give us minus five, so 00:09:26.187 --> 00:09:31.911 minus Seven and two look like the two numbers that we need. 00:09:32.530 --> 00:09:39.086 So let's breakdown this minus 5X as minus 7X Plus 2X. 00:09:39.086 --> 00:09:45.642 And let's not forget the minus 14 that we had again. 00:09:45.642 --> 00:09:51.602 Let's look at these two. The front two terms. Common 00:09:51.602 --> 00:09:57.562 factor. Yes, it's X. Take that out X minus 7. 00:09:58.750 --> 00:10:02.698 And here a common factor of +2. 00:10:03.210 --> 00:10:07.056 Let's take that out and X 00:10:07.056 --> 00:10:14.801 minus 7. Two lumps of algebra that one and that one and each 00:10:14.801 --> 00:10:20.752 one's got the same. Factoring this X minus Seven, so we'll 00:10:20.752 --> 00:10:23.998 take that as our common factor. 00:10:24.150 --> 00:10:31.590 So X minus Seven is multiplying X and it's auto 00:10:31.590 --> 00:10:38.152 multiplying +2. So again, there we've arrived at a factorization 00:10:38.152 --> 00:10:45.700 of this X squared minus 5X minus 14 factorizes as X minus 00:10:45.700 --> 00:10:47.587 Seven X +2. 00:10:48.540 --> 00:10:54.676 Type X squared minus 00:10:54.676 --> 00:10:59.278 9X plus 20. 00:11:00.930 --> 00:11:04.472 Now, from what we've got already, it might be that some 00:11:04.472 --> 00:11:08.336 of you watching this might think, well, do I need to do 00:11:08.336 --> 00:11:12.522 that every time? The answer is no. Sometimes you may be able to 00:11:12.522 --> 00:11:15.420 do these by inspection, which means looking at it. 00:11:15.960 --> 00:11:20.424 And doing the working out in your head rather than on the 00:11:20.424 --> 00:11:24.888 paper. So to do it by inspection. What we might do is 00:11:24.888 --> 00:11:28.980 right down the pair of brackets to begin with. Recognize X 00:11:28.980 --> 00:11:33.816 squared means we're going to have to have an X and then X. 00:11:34.690 --> 00:11:41.230 Recognize 20 as being four times by 5 and of course 4 + 5 would 00:11:41.230 --> 00:11:47.334 give me 9, but I want minus nine, so perhaps what I need is 00:11:47.334 --> 00:11:51.694 minus four and minus five, because minus four times by 00:11:51.694 --> 00:11:56.926 minus five is going to give me plus 20 and minus 4X. 00:11:57.510 --> 00:12:04.608 Minus 5X is going to give me minus 9X, so I've done that one 00:12:04.608 --> 00:12:11.097 by inspection. But I could have done it in exactly the same way 00:12:11.097 --> 00:12:16.908 as I did the other two. Let's take X squared minus nine X 00:12:16.908 --> 00:12:23.860 minus 22. And again, let's try this one out by inspection. So 00:12:23.860 --> 00:12:25.477 pair of brackets. 00:12:26.190 --> 00:12:29.940 X&X in front of each bracket. 00:12:31.310 --> 00:12:36.686 Let's have a look at minus 22 two numbers to multiply together 00:12:36.686 --> 00:12:42.062 to give minus 22 will likely candidates are minus 11 and two. 00:12:42.740 --> 00:12:49.668 Or minus 2 and 11, but at the end of the day I need minus 9X 00:12:49.668 --> 00:12:54.864 and that kind of suggests that perhaps we've got to have the 00:12:54.864 --> 00:13:00.493 bigger of 11 and two as being negative and the smaller one as 00:13:00.493 --> 00:13:07.200 being positive. Let's just check minus 11 times +2 gives me minus 00:13:07.200 --> 00:13:14.452 22, and then I have minus 11X and 2X, which gives me minus 9X. 00:13:14.452 --> 00:13:18.078 So again, we've done that one by 00:13:18.078 --> 00:13:23.190 inspection. Again, you don't have to do it by inspection. You 00:13:23.190 --> 00:13:25.100 can use the previous method. 00:13:26.580 --> 00:13:30.120 If we have quadratic expressions which don't have a unit 00:13:30.120 --> 00:13:33.660 coefficient, now this is one that has a unit coefficient, 00:13:33.660 --> 00:13:35.784 'cause this is One X squared. 00:13:37.220 --> 00:13:42.218 It could be 2 X squared. It could be 6 X squared, could be 00:13:42.218 --> 00:13:46.502 11 X squared, could be anything times by X squared. That would 00:13:46.502 --> 00:13:47.930 be harder to do. 00:13:48.450 --> 00:13:54.582 So let's have a look at how we might tackle some of those. So 00:13:54.582 --> 00:13:56.772 we take three X squared. 00:13:57.540 --> 00:14:01.800 Plus 5X minus 2. 00:14:03.340 --> 00:14:08.020 I'm going to use a method that's very similar to the first method 00:14:08.020 --> 00:14:12.340 that we saw. I'm going to look for two numbers that multiply 00:14:12.340 --> 00:14:16.300 together to give, well, let's leave that unsaid for them in 00:14:16.300 --> 00:14:20.980 it, but these two numbers are going to add together to give I. 00:14:20.980 --> 00:14:25.660 They're going to add together to give this +5 the Exterm, so that 00:14:25.660 --> 00:14:29.260 hasn't changed. We're looking for two numbers that will add 00:14:29.260 --> 00:14:32.140 together to give us the coefficient of X. 00:14:33.150 --> 00:14:37.698 What do the two numbers have to multiply together to give us? 00:14:37.698 --> 00:14:42.246 Well, they have to multiply together to give us 3 times by 00:14:42.246 --> 00:14:46.794 minus two, so we don't just take the constant term, we multiply 00:14:46.794 --> 00:14:51.721 it by the coefficient of the X squared and three times by minus 00:14:51.721 --> 00:14:57.027 two is minus 6, and I'll just write that here at the side that 00:14:57.027 --> 00:15:01.575 the minus six came from the three times by the minus two. 00:15:01.575 --> 00:15:05.365 And if you think about it, that's actually consistent with 00:15:05.365 --> 00:15:07.260 what we were doing before. 00:15:07.320 --> 00:15:11.181 Because in the previous examples, this number in front 00:15:11.181 --> 00:15:14.184 of the X squared had been one. 00:15:14.830 --> 00:15:17.231 And so one times by minus two 00:15:17.231 --> 00:15:22.390 would be. The constant term, so we are looking now for two 00:15:22.390 --> 00:15:26.614 numbers that multiply together to give us minus 6 and add 00:15:26.614 --> 00:15:28.534 together to give us 5. 00:15:29.250 --> 00:15:32.560 Well. 3 times by two. 00:15:33.130 --> 00:15:36.370 Well, three times by two would give us plus 6. 00:15:37.450 --> 00:15:41.727 Minus three times by minus two would also give us plus six, so 00:15:41.727 --> 00:15:45.550 that's not good. Six and one. 00:15:46.860 --> 00:15:51.480 Well, if we could have 6 times by minus one, that 00:15:51.480 --> 00:15:56.520 would give us minus six and six AD minus one would give 00:15:56.520 --> 00:16:01.140 us 5. So this looks like the combination that we want. 00:16:02.200 --> 00:16:04.348 So we take three X squared. 00:16:05.170 --> 00:16:09.150 Plus 6X minus X 00:16:09.150 --> 00:16:15.785 minus 2. Let's have a look for a common factor here. 00:16:15.785 --> 00:16:21.700 Well, there's a three X squared and a 6X, so there's obviously a 00:16:21.700 --> 00:16:28.525 tree is a factor, and also an X. So let's take out three X leaves 00:16:28.525 --> 00:16:29.890 me X +2. 00:16:30.820 --> 00:16:36.355 3X times my X gives us the three X squared 3X times by two gives 00:16:36.355 --> 00:16:41.152 us 6X and now want to common factor for these two terms minus 00:16:41.152 --> 00:16:45.211 X minus two. We don't seem to share anything in common. 00:16:45.760 --> 00:16:52.312 I've got a common factor and it's minus 1 - 1 times minus one 00:16:52.312 --> 00:16:58.864 times by something has to give me minus X, so that must be X 00:16:58.864 --> 00:17:04.948 and minus one times by something has to give me minus two. Well, 00:17:04.948 --> 00:17:07.288 that's got to be +2. 00:17:08.300 --> 00:17:14.748 So now I've got these two lumps of algebra again, this one and 00:17:14.748 --> 00:17:21.196 this one, and each lump has the same factor in it. This common 00:17:21.196 --> 00:17:28.636 factor of X +2, so I'll take that one out X +2 and I've got 00:17:28.636 --> 00:17:33.596 X +2. Multiplying 3X and X +2, multiplying minus one. 00:17:33.740 --> 00:17:40.406 And so there's the factorization of the expression that we began 00:17:40.406 --> 00:17:47.565 with. Let's take another one, two X squared. 00:17:48.170 --> 00:17:53.510 Plus 5X minus 7. 00:17:55.220 --> 00:18:00.126 So we're looking for two numbers that will multiply together to 00:18:00.126 --> 00:18:05.478 give us 2 times by minus Seven, so they must multiply together 00:18:05.478 --> 00:18:11.722 to give us minus 14. Just write down again at the side that the 00:18:11.722 --> 00:18:15.736 minus 14 comes from minus Seven times by two. 00:18:16.380 --> 00:18:21.737 And then these two numbers, whatever they are, I've got to 00:18:21.737 --> 00:18:26.607 add together to give us the coefficient of X +5. 00:18:27.810 --> 00:18:32.254 So what are these two numbers? Well, Seven and two seem 00:18:32.254 --> 00:18:36.698 reasonable factors of 14, and they are factors of 14 which 00:18:36.698 --> 00:18:41.950 have a difference. If you like a five, so they seem good options 00:18:41.950 --> 00:18:44.374 7 and 2. Seven and two. 00:18:45.220 --> 00:18:49.588 But we've got to make a balance here. We need +5 and we need 00:18:49.588 --> 00:18:53.956 minus 4T. So one of these is got to be negative, and it looks 00:18:53.956 --> 00:18:58.012 like it's going to have to be negative two in order that 7 00:18:58.012 --> 00:18:59.884 plus negative two should give us 00:18:59.884 --> 00:19:05.219 the five there. So now we can write this down as two 00:19:05.219 --> 00:19:06.061 X squared. 00:19:07.150 --> 00:19:14.470 Breaking up that plus 5X as plus 7X minus 2X and then 00:19:14.470 --> 00:19:17.520 minus Seven at the end. 00:19:18.760 --> 00:19:23.416 What have we got here as a common factor? Well, we've got 00:19:23.416 --> 00:19:29.236 an X in each term, so we can take that out, giving us 2X plus 00:19:29.236 --> 00:19:34.280 Seven. And here again, what have we got for a common factor? Or 00:19:34.280 --> 00:19:39.324 the only thing that's in common is one and there's a minus sign 00:19:39.324 --> 00:19:46.308 with each one, so it's minus 1 * 2 X plus 7 - 1 times by two. X 00:19:46.308 --> 00:19:49.024 gives us minus two X minus one. 00:19:49.090 --> 00:19:54.436 Plus Seven gives us minus Seven close the bracket. 00:19:55.120 --> 00:20:00.856 Two lumps of algebra. Again, this one, and this one. In each 00:20:00.856 --> 00:20:06.592 one. There's this common factor of 2X plus Seven, so we take 00:20:06.592 --> 00:20:11.850 that out 2X plus 7 and that's multiplying X and it's 00:20:11.850 --> 00:20:17.108 multiplying minus one, and so we have got this factorization of 00:20:17.108 --> 00:20:19.498 the expression that we began 00:20:19.498 --> 00:20:22.610 with. Take 00:20:22.610 --> 00:20:29.118 another example. Six X squared. 00:20:29.790 --> 00:20:34.434 Minus 5X minus four. Now what's different here is that this is 00:20:34.434 --> 00:20:40.239 not a prime number. We've had a two and we've had a 3, but this 00:20:40.239 --> 00:20:41.400 is a 6. 00:20:42.600 --> 00:20:47.059 You might have been able to do the other two by inspection, but 00:20:47.059 --> 00:20:51.175 this one is more difficult to do by inspection, and really, we 00:20:51.175 --> 00:20:55.291 perhaps are going to have to depend upon the method we just 00:20:55.291 --> 00:20:58.721 learned, so we're looking again for two numbers that will 00:20:58.721 --> 00:21:02.837 multiply together to give us 6 times by minus four. In other 00:21:02.837 --> 00:21:03.866 words, minus 24. 00:21:04.440 --> 00:21:09.998 OK, I'll just write that down so we can see where it's come from. 00:21:09.998 --> 00:21:15.159 Minus 24 is 6 times by minus four and we want these two 00:21:15.159 --> 00:21:19.129 numbers. Whatever they are. They've also got to add together 00:21:19.129 --> 00:21:24.290 to give us the coefficient of X, so they must add together to 00:21:24.290 --> 00:21:25.878 give us minus 5. 00:21:26.510 --> 00:21:31.892 So 2 numbers that might multiply to give us 24, eight, and three 00:21:31.892 --> 00:21:36.446 good options, and eight and three do have a difference of 00:21:36.446 --> 00:21:41.414 five, so they look options we can use. Now let's juggle the 00:21:41.414 --> 00:21:46.382 signs we need to have minus five, so that would suggest that 00:21:46.382 --> 00:21:52.178 the 8's got to be the negative one. So let's have minus 8 times 00:21:52.178 --> 00:21:56.732 by three. That will give us minus 24 and minus 8. 00:21:56.790 --> 00:22:03.602 Plus three that will give us minus five, so we've got six X 00:22:03.602 --> 00:22:10.932 squared. Minus 8X plus 3X breaking down that 00:22:10.932 --> 00:22:14.620 5X. Minus 4. 00:22:15.160 --> 00:22:17.299 Common factor here. 00:22:18.810 --> 00:22:22.938 Well, the six and the three share a common factor of three. 00:22:22.938 --> 00:22:27.754 And of course we've X squared and X common factor of X, so we 00:22:27.754 --> 00:22:29.474 can take out three X. 00:22:30.050 --> 00:22:34.306 And that will leave us 2X plus one. 00:22:35.710 --> 00:22:40.897 These two terms, what do we got for a common factor? Well, they 00:22:40.897 --> 00:22:46.084 clearly share a common factor of four and also a minus sign. So 00:22:46.084 --> 00:22:48.478 we take minus four times by. 00:22:49.160 --> 00:22:54.110 Now, minus four times by something has to give us minus 00:22:54.110 --> 00:22:59.960 8X, so that's going to be 2X and then minus four times by 00:22:59.960 --> 00:23:05.810 something has to give us minus four, so that's got to be plus 00:23:05.810 --> 00:23:10.760 one again to lump sum algebra sharing. This common factor of 00:23:10.760 --> 00:23:13.910 2X plus one. So we'll take that 00:23:13.910 --> 00:23:20.420 out. And then we have two X plus one multiplying 3X. 00:23:20.670 --> 00:23:23.976 And two X plus one multiplying 00:23:23.976 --> 00:23:31.448 minus 4. Will take 1 final example of 00:23:31.448 --> 00:23:38.410 this kind. So I've got 15 X 00:23:38.410 --> 00:23:42.066 squared. Minus three X 00:23:42.066 --> 00:23:47.602 minus 80. Now in all the others, the thing that we 00:23:47.602 --> 00:23:50.734 haven't checked at the beginning, and perhaps we should 00:23:50.734 --> 00:23:53.866 have done is do the coefficients. The numbers that 00:23:53.866 --> 00:23:55.258 multiply the X squared. 00:23:56.650 --> 00:24:00.290 That multiply the X and the constant term 00:24:00.290 --> 00:24:02.110 share a common factor. 00:24:03.480 --> 00:24:08.088 And in this case they do as a common factor of 3. 00:24:09.420 --> 00:24:13.020 And where there is a common factor, we need to take it 00:24:13.020 --> 00:24:16.020 out to begin with, so will take the three out. 00:24:17.250 --> 00:24:22.094 3 times by something as to give us 15 X squared so three times 00:24:22.094 --> 00:24:24.516 by 5 X squared will do that. 00:24:25.530 --> 00:24:30.738 3 times by something has to give us minus 3X so three times by 00:24:30.738 --> 00:24:32.598 minus X will do that. 00:24:33.590 --> 00:24:39.386 3 times by something has to give us minus 18 and so minus six 00:24:39.386 --> 00:24:40.628 will do that. 00:24:41.130 --> 00:24:45.480 Now we're looking at Factorizing this xpression Here in the 00:24:45.480 --> 00:24:49.830 bracket and we're looking for two numbers that were multiplied 00:24:49.830 --> 00:24:55.485 together to give us five times by minus six, which is minus 30. 00:24:55.485 --> 00:25:00.705 And again, I'll just write down where that came from. Minus 30 00:25:00.705 --> 00:25:06.360 was five times by minus six, and I'm looking for two numbers that 00:25:06.360 --> 00:25:11.145 will add together to give Maine. Now here the number that's 00:25:11.145 --> 00:25:12.885 multiplying the X is. 00:25:12.950 --> 00:25:18.677 Minus one. So I want two numbers to multiply together to 00:25:18.677 --> 00:25:23.393 give me minus 30 and add together to give me minus one, 00:25:23.393 --> 00:25:27.323 well, five and six seem like obvious choices 'cause they've 00:25:27.323 --> 00:25:32.432 got a difference of one and they multiply together to give 30. So 00:25:32.432 --> 00:25:37.148 how can I juggle the signs with the five and the six? 00:25:38.600 --> 00:25:44.600 Well, 5 + 6 has to give me minus one. It looks like the six is 00:25:44.600 --> 00:25:49.850 going to have to carry the minus sign, so 5 plus minus six does 00:25:49.850 --> 00:25:55.100 give me minus one and five times by minus six does give me minus 00:25:55.100 --> 00:25:57.725 30, so I'm going to have three. 00:25:58.530 --> 00:26:05.940 Brackets five X squared plus 5X minus six X, so we broken down 00:26:05.940 --> 00:26:13.920 this minus X into 5X, minus 6X and then the final term on the 00:26:13.920 --> 00:26:17.340 end minus six and close the 00:26:17.340 --> 00:26:23.240 bracket. Keep the three outside. Let's look at the front two 00:26:23.240 --> 00:26:28.135 terms here. There's a common factor of 5X. Let's take that 00:26:28.135 --> 00:26:29.470 out. Five X. 00:26:30.520 --> 00:26:38.020 X plus One 5X times by X gives me the five X squared 5X times 00:26:38.020 --> 00:26:41.020 by one gives me the 5X. 00:26:42.010 --> 00:26:46.630 Common factor here is minus six. Each of these terms shares A6 00:26:46.630 --> 00:26:51.635 and the minus sign, so will take out the factor minus six, and 00:26:51.635 --> 00:26:57.410 then we need minus six times by has to give us minus six X, so 00:26:57.410 --> 00:27:02.030 that's times by X minus six times by something has to give 00:27:02.030 --> 00:27:07.035 us minus six, so that's minus six times by one, and then I 00:27:07.035 --> 00:27:12.040 need to make sure I close the whole bracket with that big one 00:27:12.040 --> 00:27:16.577 there. Two lumps of algebra, each sharing this common factor 00:27:16.577 --> 00:27:19.258 of X plus one. Let's take that 00:27:19.258 --> 00:27:26.300 out three. Bracket X plus one times by the X 00:27:26.300 --> 00:27:29.180 Plus One Times 5X. 00:27:29.190 --> 00:27:36.395 The X plus one also times minus six, and so we've 00:27:36.395 --> 00:27:38.360 completed that factorization. 00:27:39.790 --> 00:27:43.689 OK, we've looked at a series of 00:27:43.689 --> 00:27:46.879 examples. An we've developed. 00:27:47.440 --> 00:27:52.346 A way of handling these that enables us to factorize these 00:27:52.346 --> 00:27:56.916 quadratics. Hasn't really been anything special about them, but 00:27:56.916 --> 00:28:02.402 I want to do now is have a look at three particular special 00:28:02.402 --> 00:28:09.110 cases. Let's begin with the first special case. By having a 00:28:09.110 --> 00:28:12.374 look at X squared minus 9. 00:28:13.520 --> 00:28:16.474 Now it's obviously different about this one. 00:28:17.100 --> 00:28:21.390 From the previous examples is that there's no external, just 00:28:21.390 --> 00:28:23.535 says X squared minus 9. 00:28:24.090 --> 00:28:30.315 So let's do it in the way that we would do we look for two 00:28:30.315 --> 00:28:34.880 numbers that would multiply together to give us minus 9 and 00:28:34.880 --> 00:28:40.275 add together to give us the X coefficient, but there is no X 00:28:40.275 --> 00:28:44.010 coefficient. That means the coefficient has to be 0. 00:28:44.530 --> 00:28:47.930 0 times by XOX is. 00:28:48.930 --> 00:28:53.562 So I've got to find 2 numbers that multiply together to give 00:28:53.562 --> 00:28:56.264 minus 9 and add together to give 00:28:56.264 --> 00:29:03.330 0. Obviously they've got to be the same size but different 00:29:03.330 --> 00:29:10.000 sign, so minus three and three fit the bill perfectly, 00:29:10.000 --> 00:29:16.670 so we have X squared minus 3X plus 3X minus 00:29:16.670 --> 00:29:22.798 9. Look at the front two terms. There's a common factor of X. 00:29:23.320 --> 00:29:30.431 Leaving me with X minus three. The back two terms as a common 00:29:30.431 --> 00:29:36.995 factor of 3 leaving X minus three and now two lumps of 00:29:36.995 --> 00:29:43.559 algebra each sharing this common factor of X minus three X minus 00:29:43.559 --> 00:29:47.935 three, multiplies the X and multiplies the three. 00:29:48.470 --> 00:29:52.254 Well, we compare this 00:29:52.254 --> 00:29:58.420 with this. Not only is this X squared that we get X 00:29:58.420 --> 00:30:00.544 times by X, but this 9. 00:30:01.500 --> 00:30:04.180 Forgetting the minus sign for a moment is 3 squared. 00:30:05.180 --> 00:30:06.628 3 times by three. 00:30:07.770 --> 00:30:15.330 So in fact this expression could be rewritten as X squared minus 00:30:15.330 --> 00:30:20.715 3 squared. In other words, it's the difference of two squares. 00:30:21.420 --> 00:30:23.700 So let's do this again. 00:30:24.590 --> 00:30:29.690 But more generally, In other words, instead of minus nine, 00:30:29.690 --> 00:30:35.810 which is minus 3 squared, let's write minus a squared. So we 00:30:35.810 --> 00:30:38.870 have X squared minus a squared. 00:30:39.530 --> 00:30:46.350 We want to factorize it, so we're looking for two numbers 00:30:46.350 --> 00:30:53.170 that multiply together to give minus a squared and add together 00:30:53.170 --> 00:31:00.610 to give 0. Because there are no access, so minus A and 00:31:00.610 --> 00:31:08.050 a fit the bill. So we're going to have X squared minus 00:31:08.050 --> 00:31:10.530 8X Plus 8X minus. 00:31:10.570 --> 00:31:17.829 A squared. Common factor of X here X minus 00:31:17.829 --> 00:31:25.163 a. Under common factor of a here, X minus a 00:31:25.163 --> 00:31:32.390 again 2 lumps of algebra, each one sharing this common factor 00:31:32.390 --> 00:31:35.018 of X minus a. 00:31:35.030 --> 00:31:42.278 X minus a multiplies X, an multiplies a. 00:31:42.930 --> 00:31:46.770 So we now have. 00:31:47.390 --> 00:31:53.714 What is, in effect a standard result we have what's called the 00:31:53.714 --> 00:31:56.349 difference of two squares and 00:31:56.349 --> 00:32:02.811 its factorization. So let's just write that down again. X squared 00:32:02.811 --> 00:32:09.879 minus a squared is always equal to X minus a X plus 00:32:09.879 --> 00:32:15.225 A. So that if we can identify this number that appears, here 00:32:15.225 --> 00:32:16.645 is a square number. 00:32:17.600 --> 00:32:20.696 We can use this factorization immediately. 00:32:21.920 --> 00:32:28.328 So what if we had something say like X squared minus 25? 00:32:29.100 --> 00:32:33.240 Well, we recognize 25 as being 5 squared, so 00:32:33.240 --> 00:32:38.760 immediately we can write this down as X minus five X +5. 00:32:39.800 --> 00:32:46.224 What if we had something like 2 X squared minus 32? 00:32:47.550 --> 00:32:51.882 Doesn't really look like that, does it? But there is a common 00:32:51.882 --> 00:32:56.214 factor of 2, so as we've said before, take the common factor 00:32:56.214 --> 00:32:57.658 out to begin with. 00:32:58.370 --> 00:33:03.746 Leaving us with X squared minus 16 and 00:33:03.746 --> 00:33:09.794 of course 16 is 4 squared, so this is 00:33:09.794 --> 00:33:13.154 2X minus four X +4. 00:33:16.140 --> 00:33:21.855 Files and we had nine X squared minus 16. 00:33:23.950 --> 00:33:25.078 What about this one? 00:33:25.700 --> 00:33:32.780 Again, look at this term here 9 X squared. It is a 00:33:32.780 --> 00:33:37.500 complete square. It's 3X times by three X. 00:33:38.010 --> 00:33:42.469 So instead of just working with an X, why can't we just work 00:33:42.469 --> 00:33:43.498 with a 3X? 00:33:44.490 --> 00:33:50.524 And of course, that's what we are going to do. This must be 3X 00:33:50.524 --> 00:33:56.989 and 3X and the 16 is 4 squared, so 3X minus four, 3X plus four. 00:33:56.989 --> 00:34:01.730 And again, this is still the difference of two squares, so 00:34:01.730 --> 00:34:06.902 that's one. The first special case, and we really do have to 00:34:06.902 --> 00:34:11.643 learn that one. And remember it 'cause it's a very, very 00:34:11.643 --> 00:34:15.280 important factorization. Let's have a look now. 00:34:15.780 --> 00:34:18.714 Another factorization 00:34:18.714 --> 00:34:25.316 special case. Having just on the difference of two 00:34:25.316 --> 00:34:29.220 squares looking at this quadratic expression, we've got 00:34:29.220 --> 00:34:35.564 a square front X squared and the square at the end 5 squared. 00:34:37.140 --> 00:34:39.198 But we've got 10X in the middle. 00:34:39.890 --> 00:34:44.902 OK, we know how to handle it, so let's not worry too much. We 00:34:44.902 --> 00:34:48.840 want two numbers that multiply together to give 25 and two 00:34:48.840 --> 00:34:51.704 numbers that add together to give us 10. 00:34:52.890 --> 00:34:56.341 The obvious choice for that is 5 00:34:56.341 --> 00:35:02.958 and five. Five times by 5 is 20 five 5 + 5 is 10. 00:35:03.560 --> 00:35:11.108 So we break that middle term down X squared plus 5X Plus 00:35:11.108 --> 00:35:12.995 5X plus 25. 00:35:15.110 --> 00:35:20.921 Look at the front. Two terms are common factor of X, leaving us 00:35:20.921 --> 00:35:22.262 with X +5. 00:35:23.060 --> 00:35:29.729 Look at the back to terms are common factor of +5 leaving us 00:35:29.729 --> 00:35:31.268 with X +5. 00:35:32.210 --> 00:35:37.677 Two lumps of algebra sharing a common factor of X +5. 00:35:38.230 --> 00:35:44.130 X +5 multiplies, X&X, 00:35:44.130 --> 00:35:47.080 +5, multiplies 00:35:47.080 --> 00:35:54.804 +5. These two are the same, so this is 00:35:54.804 --> 00:35:57.436 X +5 all squared. 00:35:58.460 --> 00:36:05.389 In other words, what we've got here X squared plus 10X plus 25 00:36:05.389 --> 00:36:12.318 is a complete square X +5 all squared. We call that a complete 00:36:12.318 --> 00:36:16.180 square. Take another 00:36:16.180 --> 00:36:21.080 example, X squared minus 00:36:21.080 --> 00:36:24.755 8X plus 16. 00:36:25.980 --> 00:36:30.712 We recognize this is a square number X squared and this is a 00:36:30.712 --> 00:36:33.970 square number. 16 is 4 squared. 00:36:35.190 --> 00:36:36.378 And of course. 00:36:37.520 --> 00:36:42.200 Minus 4 plus minus four would give us 8. 00:36:42.830 --> 00:36:48.578 As indeed minus four times, my minus four would give us plus 00:36:48.578 --> 00:36:51.458 16. So we immediately. 00:36:52.210 --> 00:36:58.018 Recognizing this as this complete square X minus 00:36:58.018 --> 00:37:00.196 four all squared. 00:37:02.900 --> 00:37:09.952 One more. 25 X squared minus 00:37:09.952 --> 00:37:13.336 20 X +4. 00:37:16.700 --> 00:37:22.948 25 X squared is a complete square. It's a square number. 00:37:22.948 --> 00:37:27.492 It's the result of multiplying 5X by itself. 00:37:28.150 --> 00:37:34.716 4 is a square number, it's 2 00:37:34.716 --> 00:37:37.530 times by two. 00:37:38.760 --> 00:37:41.812 Minus 20 00:37:41.812 --> 00:37:47.578 X. Well, if I say this is a complete square, 00:37:47.578 --> 00:37:52.042 then I've got to have minus two in their minus two times, 00:37:52.042 --> 00:37:57.622 Y minus two would give me 4 - 2 times by the five. X would 00:37:57.622 --> 00:38:02.086 give me minus 10X and I'm going to have two of them 00:38:02.086 --> 00:38:05.806 minus 20X. So again I recognize this as a complete 00:38:05.806 --> 00:38:06.178 square. 00:38:07.230 --> 00:38:12.078 Might have found that a little bit confusing and a little bit 00:38:12.078 --> 00:38:16.790 quick. That doesn't matter be'cause. If you don't recognize 00:38:16.790 --> 00:38:21.390 it, you can still use the previous method on it. 00:38:22.390 --> 00:38:28.750 Let's just check that 25 X squared minus 20X plus four. So 00:38:28.750 --> 00:38:35.110 if we didn't recognize this as a complete square, we would be 00:38:35.110 --> 00:38:40.410 looking for two numbers that would multiply together to give 00:38:40.410 --> 00:38:47.830 us 100. The 100 is 4 times by 25. Just write this at the 00:38:47.830 --> 00:38:53.130 side four times by 25 and two numbers that would. 00:38:53.150 --> 00:38:58.130 Add together to give us minus 20. The obvious choices are 10 00:38:58.130 --> 00:39:03.525 and 10, well minus 10 and minus 10 because we want minus 10 00:39:03.525 --> 00:39:09.335 times by minus 10 to give plus 100 and minus 10 plus minus 10 00:39:09.335 --> 00:39:11.410 to give us minus 20. 00:39:12.380 --> 00:39:16.916 So we take this expression 25 X squared. 00:39:18.050 --> 00:39:23.810 And we breakdown the minus 20X minus 10X minus 10X. 00:39:24.560 --> 00:39:27.052 And we have the last terms plus 00:39:27.052 --> 00:39:33.100 4. We look at the front two terms for a common factor and 00:39:33.100 --> 00:39:34.740 clearly there is 5X. 00:39:34.750 --> 00:39:40.434 Gives me 5X minus two. Now I look for a common factor in the 00:39:40.434 --> 00:39:46.524 last two terms there is a common factor of 2 here, because 2 * 5 00:39:46.524 --> 00:39:53.020 gives us 10 and 2 * 2 gives us 4. But there is this minus sign, 00:39:53.020 --> 00:39:57.486 so perhaps we better take minus two is our common factor. 00:39:58.000 --> 00:40:02.572 Which will give us minus two times by something has to give 00:40:02.572 --> 00:40:07.525 us minus 10X. That's going to be 5X and minus two times by 00:40:07.525 --> 00:40:12.478 something has to give us plus four which is going to have to 00:40:12.478 --> 00:40:13.621 be minus 2. 00:40:14.190 --> 00:40:19.954 Two lumps of algebra. The common factor is 5X minus 2. 00:40:20.700 --> 00:40:27.870 5X minus two is multiplying 5X. 00:40:27.870 --> 00:40:31.645 And it's also multiplying minus 00:40:31.645 --> 00:40:38.124 2. So we have got this again as a complete square, but not 00:40:38.124 --> 00:40:41.463 by inspection, but using our standard method. 00:40:43.130 --> 00:40:45.836 So that deals with the 2nd 00:40:45.836 --> 00:40:51.660 special case. Let's now have a look at the 3rd and final 00:40:51.660 --> 00:40:56.508 special case, and this is when we don't have a constant term 00:40:56.508 --> 00:41:02.164 and we've got something like 3 X squared minus 8X. What do we do 00:41:02.164 --> 00:41:07.416 with that? Well, clearly there is a common factor of X, so we 00:41:07.416 --> 00:41:12.264 must take that out to begin with. So we take out X. 00:41:14.630 --> 00:41:20.174 We've got three X squared, so X times by three X must give us 00:41:20.174 --> 00:41:21.758 the three X squared. 00:41:22.360 --> 00:41:27.680 And then X times by something to give us the minus 8X. Well it's 00:41:27.680 --> 00:41:33.000 got to be minus 8, and that's all that we need to do. We've 00:41:33.000 --> 00:41:37.940 broken it down into two brackets X times by three X minus 8. 00:41:38.830 --> 00:41:42.580 What if say we had 10 00:41:42.580 --> 00:41:46.110 X squared? Plus 00:41:46.110 --> 00:41:50.319 5X. Again, we look for a common factor. 00:41:51.400 --> 00:41:54.720 Obviously there's an ex as a common factor again, but 00:41:54.720 --> 00:41:57.708 there's more this time because there's ten and five 00:41:57.708 --> 00:42:02.024 which share a common factor of 5, so we need to pull out 00:42:02.024 --> 00:42:04.016 the whole of that as 5X. 00:42:06.050 --> 00:42:11.804 Five times by two will give us the 10, so that's 5X times by 00:42:11.804 --> 00:42:17.147 two. X will give us the 10 X squared plus 5X times by 00:42:17.147 --> 00:42:21.257 something to give us 5X that must just be 1. 00:42:22.290 --> 00:42:27.220 So, so long as we remember to inspect the quadratic 00:42:27.220 --> 00:42:31.164 expression 1st and check for common factors, this 00:42:31.164 --> 00:42:34.615 particular one shouldn't cause us any difficulties.