1 00:00:00,550 --> 00:00:03,958 When we have two 2 00:00:03,958 --> 00:00:11,302 brackets. X +2 times by X +3 and we know how to multiply 3 00:00:11,302 --> 00:00:13,286 these two brackets out. 4 00:00:13,970 --> 00:00:21,824 We have X kinds by X that gives us X squared. We have X 5 00:00:21,824 --> 00:00:25,190 times by three, gives us 3X. 6 00:00:26,320 --> 00:00:32,937 2 times by X gives us 2X and then two times by three 7 00:00:32,937 --> 00:00:34,464 gives us 6. 8 00:00:35,630 --> 00:00:42,923 And we can simplify these two terms. 3X plus 2X gives us 5X. 9 00:00:44,180 --> 00:00:49,157 This is an example of a quadratic expression or 10 00:00:49,157 --> 00:00:54,134 quadratic function. It's gotta termine ex squared, which it 11 00:00:54,134 --> 00:00:57,452 must have to be a quadratic 12 00:00:57,452 --> 00:01:04,188 expression. It's got a term in X which it might or might not 13 00:01:04,188 --> 00:01:09,516 have, and it's got a constant term and there are no other 14 00:01:09,516 --> 00:01:12,624 possibilities, so our most general quadratic expression 15 00:01:12,624 --> 00:01:15,732 would be AX squared plus BX plus 16 00:01:15,732 --> 00:01:21,910 C. What we're going to have a look at is how we factorise 17 00:01:21,910 --> 00:01:27,045 expressions like this in others. How we go back from this kind of 18 00:01:27,045 --> 00:01:33,210 expression. To this now, why might we want to do that? Well, 19 00:01:33,210 --> 00:01:39,060 let's just take this X squared plus 5X plus six. And let's say 20 00:01:39,060 --> 00:01:44,460 it's not just an expression, but it's an equation and it says 21 00:01:44,460 --> 00:01:47,610 equals 0. What are the values of 22 00:01:47,610 --> 00:01:54,036 X? That will make it equal to 0 that are answers to that 23 00:01:54,036 --> 00:01:59,795 equation. One of the things we can do is to rewrite this form. 24 00:02:00,320 --> 00:02:08,204 By this so we can say X +2 times by X +3 25 00:02:08,204 --> 00:02:14,009 equals 0. When we have two numbers that multiply 26 00:02:14,009 --> 00:02:20,967 together to give zero and one of the things that must be true is 27 00:02:20,967 --> 00:02:27,428 that one of them zero or the other one zero, or they're both 28 00:02:27,428 --> 00:02:35,380 0. So in this case, X +2 equals 0 or X +3 equals 0, and so 29 00:02:35,380 --> 00:02:40,847 X would be minus two, or X would be minus three. 30 00:02:41,460 --> 00:02:47,352 So being able to factorize actually helps us to solve a new 31 00:02:47,352 --> 00:02:48,825 kind of equation. 32 00:02:49,360 --> 00:02:56,024 So we're going to be having a look in this video that how you 33 00:02:56,024 --> 00:03:00,784 factorise this kind of function. This kind of expression a 34 00:03:00,784 --> 00:03:06,916 quadratic expression. Now I'm going to start by going back to 35 00:03:06,916 --> 00:03:09,820 this little piece of work again. 36 00:03:09,830 --> 00:03:13,745 So let's write it down 37 00:03:13,745 --> 00:03:20,432 X. Plus 3 * 5 X +2 38 00:03:20,432 --> 00:03:24,586 and again. Will multiply out the 39 00:03:24,586 --> 00:03:28,378 brackets X times by X is X squared. 40 00:03:29,550 --> 00:03:36,674 X times by two is 2 X 3 times by X is 3X. 41 00:03:37,240 --> 00:03:43,884 3 times by two is 6. This simplifies to X squared 42 00:03:43,884 --> 00:03:46,300 plus 5X or 6. 43 00:03:47,490 --> 00:03:52,209 So we've gone one way. What happens if we want to go back 44 00:03:52,209 --> 00:03:53,298 the other way? 45 00:03:54,070 --> 00:04:00,622 Let's have a look where this six came from. We know it came from 46 00:04:00,622 --> 00:04:02,494 3 times by two. 47 00:04:03,730 --> 00:04:11,088 Where did this five come from? Where it came from 2 + 3? 48 00:04:12,140 --> 00:04:18,536 So if we were to reverse this process, we be looking for two 49 00:04:18,536 --> 00:04:24,440 numbers that multiply together to give us six an 2 numbers that 50 00:04:24,440 --> 00:04:27,392 added together to give us 5. 51 00:04:28,030 --> 00:04:34,660 The obvious ones that go in there are three 2. 52 00:04:36,550 --> 00:04:38,638 So if we began. 53 00:04:39,210 --> 00:04:40,380 With this 54 00:04:42,030 --> 00:04:43,280 We would. 55 00:04:44,310 --> 00:04:49,133 Be looking to break that 5X down 56 00:04:49,133 --> 00:04:55,334 as X squared plus 3X plus 2X plus 6. 57 00:04:56,860 --> 00:05:02,712 Then we could look at these two and see if there was a common 58 00:05:02,712 --> 00:05:08,564 factor and there is X leaving us with X Plus three. Then we would 59 00:05:08,564 --> 00:05:13,998 look at these two. Is there a common factor and there is 2. 60 00:05:14,530 --> 00:05:17,950 Leaving us again with 61 00:05:17,950 --> 00:05:25,170 X +3. And then we've got this common factor of X plus 62 00:05:25,170 --> 00:05:31,960 three in each of these two lumps of algebra, so we can take out 63 00:05:31,960 --> 00:05:38,750 that X +3, and we've got the other factor left X times by X 64 00:05:38,750 --> 00:05:45,540 plus three and two times by X +3, and so we've arrived at that 65 00:05:45,540 --> 00:05:49,420 factorization. Those brackets that we started off with. 66 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 67 00:05:57,256 --> 00:06:00,622 to repeat this process of looking for numbers that 68 00:06:00,622 --> 00:06:04,362 multiply together to give the constant term and numbers that 69 00:06:04,362 --> 00:06:08,850 will add together to give the exterm. So let's have a little 70 00:06:08,850 --> 00:06:10,720 bit of practice at that. 71 00:06:11,820 --> 00:06:18,939 Let's look at X squared minus 7X plus 12. 72 00:06:19,930 --> 00:06:24,226 So we want two numbers that will multiply together to give us. 73 00:06:24,820 --> 00:06:32,320 12 Times together to give us 12 and will add together 74 00:06:32,320 --> 00:06:35,260 to give us minus 7. 75 00:06:36,350 --> 00:06:42,650 Minus four times minus three is 12 and minus four plus minus 76 00:06:42,650 --> 00:06:48,950 three gives us minus Seven, so let's just write those in minus 77 00:06:48,950 --> 00:06:55,250 four times, minus three gives us plus 12 and minus four plus 78 00:06:55,250 --> 00:07:01,550 minus three gives us minus Seven, so that's given us a way 79 00:07:01,550 --> 00:07:07,325 of breaking down this minus Seven XX squared minus 4X Minus 80 00:07:07,325 --> 00:07:12,972 3X. Plus 12 so now we look at these two at the front. 81 00:07:13,730 --> 00:07:21,052 Take out X as a common factor that gives us X minus four, and 82 00:07:21,052 --> 00:07:24,190 now we look at these two. 83 00:07:24,980 --> 00:07:30,160 Well, I want to make sure I get the same factor X minus 4. 84 00:07:31,470 --> 00:07:36,342 Clearly, in these two terms, that is a factor of three. But 85 00:07:36,342 --> 00:07:42,026 here I've got minus three, so I think I'm going to have to make 86 00:07:42,026 --> 00:07:46,492 the factor, not three, but minus three. So that's minus three 87 00:07:46,492 --> 00:07:50,958 times X, so that insures when I multiply these two together 88 00:07:50,958 --> 00:07:56,642 minus three times by ex. I do get minus 3X, but now I need 89 00:07:56,642 --> 00:08:01,514 minus three times by something that's got to give me plus 12. 90 00:08:01,570 --> 00:08:07,758 So that will have to be minus four and close the bracket. Now 91 00:08:07,758 --> 00:08:13,946 again I've got two lumps of algebra, and in each one that is 92 00:08:13,946 --> 00:08:19,658 the same factor. This common factor of X minus 4X minus four, 93 00:08:19,658 --> 00:08:25,846 so I'll take that as my common factor X minus four. Then I've 94 00:08:25,846 --> 00:08:32,034 got X minus four times by X&X minus four times by minus three. 95 00:08:32,370 --> 00:08:35,722 That's my factorization of 96 00:08:35,722 --> 00:08:39,106 that. Let's take another 97 00:08:39,106 --> 00:08:41,995 one. X 98 00:08:41,995 --> 00:08:48,254 squared Minus 5X minus 14. So now 99 00:08:48,254 --> 00:08:52,448 looking for two numbers to multiply 100 00:08:52,448 --> 00:08:58,739 together to give minus 14 and add together to 101 00:08:58,739 --> 00:09:00,836 give minus 5. 102 00:09:01,860 --> 00:09:08,061 Fairly obvious factors of 14 R. Seven and two. So can we play 103 00:09:08,061 --> 00:09:14,262 with Seven and two? Well, if we made it minus Seven and plus 104 00:09:14,262 --> 00:09:20,463 two, then minus Seven times my plus two would give us minus 14 105 00:09:20,463 --> 00:09:26,187 and minus Seven, plus the two would give us minus five, so 106 00:09:26,187 --> 00:09:31,911 minus Seven and two look like the two numbers that we need. 107 00:09:32,530 --> 00:09:39,086 So let's breakdown this minus 5X as minus 7X Plus 2X. 108 00:09:39,086 --> 00:09:45,642 And let's not forget the minus 14 that we had again. 109 00:09:45,642 --> 00:09:51,602 Let's look at these two. The front two terms. Common 110 00:09:51,602 --> 00:09:57,562 factor. Yes, it's X. Take that out X minus 7. 111 00:09:58,750 --> 00:10:02,698 And here a common factor of +2. 112 00:10:03,210 --> 00:10:07,056 Let's take that out and X 113 00:10:07,056 --> 00:10:14,801 minus 7. Two lumps of algebra that one and that one and each 114 00:10:14,801 --> 00:10:20,752 one's got the same. Factoring this X minus Seven, so we'll 115 00:10:20,752 --> 00:10:23,998 take that as our common factor. 116 00:10:24,150 --> 00:10:31,590 So X minus Seven is multiplying X and it's auto 117 00:10:31,590 --> 00:10:38,152 multiplying +2. So again, there we've arrived at a factorization 118 00:10:38,152 --> 00:10:45,700 of this X squared minus 5X minus 14 factorizes as X minus 119 00:10:45,700 --> 00:10:47,587 Seven X +2. 120 00:10:48,540 --> 00:10:54,676 Type X squared minus 121 00:10:54,676 --> 00:10:59,278 9X plus 20. 122 00:11:00,930 --> 00:11:04,472 Now, from what we've got already, it might be that some 123 00:11:04,472 --> 00:11:08,336 of you watching this might think, well, do I need to do 124 00:11:08,336 --> 00:11:12,522 that every time? The answer is no. Sometimes you may be able to 125 00:11:12,522 --> 00:11:15,420 do these by inspection, which means looking at it. 126 00:11:15,960 --> 00:11:20,424 And doing the working out in your head rather than on the 127 00:11:20,424 --> 00:11:24,888 paper. So to do it by inspection. What we might do is 128 00:11:24,888 --> 00:11:28,980 right down the pair of brackets to begin with. Recognize X 129 00:11:28,980 --> 00:11:33,816 squared means we're going to have to have an X and then X. 130 00:11:34,690 --> 00:11:41,230 Recognize 20 as being four times by 5 and of course 4 + 5 would 131 00:11:41,230 --> 00:11:47,334 give me 9, but I want minus nine, so perhaps what I need is 132 00:11:47,334 --> 00:11:51,694 minus four and minus five, because minus four times by 133 00:11:51,694 --> 00:11:56,926 minus five is going to give me plus 20 and minus 4X. 134 00:11:57,510 --> 00:12:04,608 Minus 5X is going to give me minus 9X, so I've done that one 135 00:12:04,608 --> 00:12:11,097 by inspection. But I could have done it in exactly the same way 136 00:12:11,097 --> 00:12:16,908 as I did the other two. Let's take X squared minus nine X 137 00:12:16,908 --> 00:12:23,860 minus 22. And again, let's try this one out by inspection. So 138 00:12:23,860 --> 00:12:25,477 pair of brackets. 139 00:12:26,190 --> 00:12:29,940 X&X in front of each bracket. 140 00:12:31,310 --> 00:12:36,686 Let's have a look at minus 22 two numbers to multiply together 141 00:12:36,686 --> 00:12:42,062 to give minus 22 will likely candidates are minus 11 and two. 142 00:12:42,740 --> 00:12:49,668 Or minus 2 and 11, but at the end of the day I need minus 9X 143 00:12:49,668 --> 00:12:54,864 and that kind of suggests that perhaps we've got to have the 144 00:12:54,864 --> 00:13:00,493 bigger of 11 and two as being negative and the smaller one as 145 00:13:00,493 --> 00:13:07,200 being positive. Let's just check minus 11 times +2 gives me minus 146 00:13:07,200 --> 00:13:14,452 22, and then I have minus 11X and 2X, which gives me minus 9X. 147 00:13:14,452 --> 00:13:18,078 So again, we've done that one by 148 00:13:18,078 --> 00:13:23,190 inspection. Again, you don't have to do it by inspection. You 149 00:13:23,190 --> 00:13:25,100 can use the previous method. 150 00:13:26,580 --> 00:13:30,120 If we have quadratic expressions which don't have a unit 151 00:13:30,120 --> 00:13:33,660 coefficient, now this is one that has a unit coefficient, 152 00:13:33,660 --> 00:13:35,784 'cause this is One X squared. 153 00:13:37,220 --> 00:13:42,218 It could be 2 X squared. It could be 6 X squared, could be 154 00:13:42,218 --> 00:13:46,502 11 X squared, could be anything times by X squared. That would 155 00:13:46,502 --> 00:13:47,930 be harder to do. 156 00:13:48,450 --> 00:13:54,582 So let's have a look at how we might tackle some of those. So 157 00:13:54,582 --> 00:13:56,772 we take three X squared. 158 00:13:57,540 --> 00:14:01,800 Plus 5X minus 2. 159 00:14:03,340 --> 00:14:08,020 I'm going to use a method that's very similar to the first method 160 00:14:08,020 --> 00:14:12,340 that we saw. I'm going to look for two numbers that multiply 161 00:14:12,340 --> 00:14:16,300 together to give, well, let's leave that unsaid for them in 162 00:14:16,300 --> 00:14:20,980 it, but these two numbers are going to add together to give I. 163 00:14:20,980 --> 00:14:25,660 They're going to add together to give this +5 the Exterm, so that 164 00:14:25,660 --> 00:14:29,260 hasn't changed. We're looking for two numbers that will add 165 00:14:29,260 --> 00:14:32,140 together to give us the coefficient of X. 166 00:14:33,150 --> 00:14:37,698 What do the two numbers have to multiply together to give us? 167 00:14:37,698 --> 00:14:42,246 Well, they have to multiply together to give us 3 times by 168 00:14:42,246 --> 00:14:46,794 minus two, so we don't just take the constant term, we multiply 169 00:14:46,794 --> 00:14:51,721 it by the coefficient of the X squared and three times by minus 170 00:14:51,721 --> 00:14:57,027 two is minus 6, and I'll just write that here at the side that 171 00:14:57,027 --> 00:15:01,575 the minus six came from the three times by the minus two. 172 00:15:01,575 --> 00:15:05,365 And if you think about it, that's actually consistent with 173 00:15:05,365 --> 00:15:07,260 what we were doing before. 174 00:15:07,320 --> 00:15:11,181 Because in the previous examples, this number in front 175 00:15:11,181 --> 00:15:14,184 of the X squared had been one. 176 00:15:14,830 --> 00:15:17,231 And so one times by minus two 177 00:15:17,231 --> 00:15:22,390 would be. The constant term, so we are looking now for two 178 00:15:22,390 --> 00:15:26,614 numbers that multiply together to give us minus 6 and add 179 00:15:26,614 --> 00:15:28,534 together to give us 5. 180 00:15:29,250 --> 00:15:32,560 Well. 3 times by two. 181 00:15:33,130 --> 00:15:36,370 Well, three times by two would give us plus 6. 182 00:15:37,450 --> 00:15:41,727 Minus three times by minus two would also give us plus six, so 183 00:15:41,727 --> 00:15:45,550 that's not good. Six and one. 184 00:15:46,860 --> 00:15:51,480 Well, if we could have 6 times by minus one, that 185 00:15:51,480 --> 00:15:56,520 would give us minus six and six AD minus one would give 186 00:15:56,520 --> 00:16:01,140 us 5. So this looks like the combination that we want. 187 00:16:02,200 --> 00:16:04,348 So we take three X squared. 188 00:16:05,170 --> 00:16:09,150 Plus 6X minus X 189 00:16:09,150 --> 00:16:15,785 minus 2. Let's have a look for a common factor here. 190 00:16:15,785 --> 00:16:21,700 Well, there's a three X squared and a 6X, so there's obviously a 191 00:16:21,700 --> 00:16:28,525 tree is a factor, and also an X. So let's take out three X leaves 192 00:16:28,525 --> 00:16:29,890 me X +2. 193 00:16:30,820 --> 00:16:36,355 3X times my X gives us the three X squared 3X times by two gives 194 00:16:36,355 --> 00:16:41,152 us 6X and now want to common factor for these two terms minus 195 00:16:41,152 --> 00:16:45,211 X minus two. We don't seem to share anything in common. 196 00:16:45,760 --> 00:16:52,312 I've got a common factor and it's minus 1 - 1 times minus one 197 00:16:52,312 --> 00:16:58,864 times by something has to give me minus X, so that must be X 198 00:16:58,864 --> 00:17:04,948 and minus one times by something has to give me minus two. Well, 199 00:17:04,948 --> 00:17:07,288 that's got to be +2. 200 00:17:08,300 --> 00:17:14,748 So now I've got these two lumps of algebra again, this one and 201 00:17:14,748 --> 00:17:21,196 this one, and each lump has the same factor in it. This common 202 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 203 00:17:28,636 --> 00:17:33,596 X +2. Multiplying 3X and X +2, multiplying minus one. 204 00:17:33,740 --> 00:17:40,406 And so there's the factorization of the expression that we began 205 00:17:40,406 --> 00:17:47,565 with. Let's take another one, two X squared. 206 00:17:48,170 --> 00:17:53,510 Plus 5X minus 7. 207 00:17:55,220 --> 00:18:00,126 So we're looking for two numbers that will multiply together to 208 00:18:00,126 --> 00:18:05,478 give us 2 times by minus Seven, so they must multiply together 209 00:18:05,478 --> 00:18:11,722 to give us minus 14. Just write down again at the side that the 210 00:18:11,722 --> 00:18:15,736 minus 14 comes from minus Seven times by two. 211 00:18:16,380 --> 00:18:21,737 And then these two numbers, whatever they are, I've got to 212 00:18:21,737 --> 00:18:26,607 add together to give us the coefficient of X +5. 213 00:18:27,810 --> 00:18:32,254 So what are these two numbers? Well, Seven and two seem 214 00:18:32,254 --> 00:18:36,698 reasonable factors of 14, and they are factors of 14 which 215 00:18:36,698 --> 00:18:41,950 have a difference. If you like a five, so they seem good options 216 00:18:41,950 --> 00:18:44,374 7 and 2. Seven and two. 217 00:18:45,220 --> 00:18:49,588 But we've got to make a balance here. We need +5 and we need 218 00:18:49,588 --> 00:18:53,956 minus 4T. So one of these is got to be negative, and it looks 219 00:18:53,956 --> 00:18:58,012 like it's going to have to be negative two in order that 7 220 00:18:58,012 --> 00:18:59,884 plus negative two should give us 221 00:18:59,884 --> 00:19:05,219 the five there. So now we can write this down as two 222 00:19:05,219 --> 00:19:06,061 X squared. 223 00:19:07,150 --> 00:19:14,470 Breaking up that plus 5X as plus 7X minus 2X and then 224 00:19:14,470 --> 00:19:17,520 minus Seven at the end. 225 00:19:18,760 --> 00:19:23,416 What have we got here as a common factor? Well, we've got 226 00:19:23,416 --> 00:19:29,236 an X in each term, so we can take that out, giving us 2X plus 227 00:19:29,236 --> 00:19:34,280 Seven. And here again, what have we got for a common factor? Or 228 00:19:34,280 --> 00:19:39,324 the only thing that's in common is one and there's a minus sign 229 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 230 00:19:46,308 --> 00:19:49,024 gives us minus two X minus one. 231 00:19:49,090 --> 00:19:54,436 Plus Seven gives us minus Seven close the bracket. 232 00:19:55,120 --> 00:20:00,856 Two lumps of algebra. Again, this one, and this one. In each 233 00:20:00,856 --> 00:20:06,592 one. There's this common factor of 2X plus Seven, so we take 234 00:20:06,592 --> 00:20:11,850 that out 2X plus 7 and that's multiplying X and it's 235 00:20:11,850 --> 00:20:17,108 multiplying minus one, and so we have got this factorization of 236 00:20:17,108 --> 00:20:19,498 the expression that we began 237 00:20:19,498 --> 00:20:22,610 with. Take 238 00:20:22,610 --> 00:20:29,118 another example. Six X squared. 239 00:20:29,790 --> 00:20:34,434 Minus 5X minus four. Now what's different here is that this is 240 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 241 00:20:40,239 --> 00:20:41,400 is a 6. 242 00:20:42,600 --> 00:20:47,059 You might have been able to do the other two by inspection, but 243 00:20:47,059 --> 00:20:51,175 this one is more difficult to do by inspection, and really, we 244 00:20:51,175 --> 00:20:55,291 perhaps are going to have to depend upon the method we just 245 00:20:55,291 --> 00:20:58,721 learned, so we're looking again for two numbers that will 246 00:20:58,721 --> 00:21:02,837 multiply together to give us 6 times by minus four. In other 247 00:21:02,837 --> 00:21:03,866 words, minus 24. 248 00:21:04,440 --> 00:21:09,998 OK, I'll just write that down so we can see where it's come from. 249 00:21:09,998 --> 00:21:15,159 Minus 24 is 6 times by minus four and we want these two 250 00:21:15,159 --> 00:21:19,129 numbers. Whatever they are. They've also got to add together 251 00:21:19,129 --> 00:21:24,290 to give us the coefficient of X, so they must add together to 252 00:21:24,290 --> 00:21:25,878 give us minus 5. 253 00:21:26,510 --> 00:21:31,892 So 2 numbers that might multiply to give us 24, eight, and three 254 00:21:31,892 --> 00:21:36,446 good options, and eight and three do have a difference of 255 00:21:36,446 --> 00:21:41,414 five, so they look options we can use. Now let's juggle the 256 00:21:41,414 --> 00:21:46,382 signs we need to have minus five, so that would suggest that 257 00:21:46,382 --> 00:21:52,178 the 8's got to be the negative one. So let's have minus 8 times 258 00:21:52,178 --> 00:21:56,732 by three. That will give us minus 24 and minus 8. 259 00:21:56,790 --> 00:22:03,602 Plus three that will give us minus five, so we've got six X 260 00:22:03,602 --> 00:22:10,932 squared. Minus 8X plus 3X breaking down that 261 00:22:10,932 --> 00:22:14,620 5X. Minus 4. 262 00:22:15,160 --> 00:22:17,299 Common factor here. 263 00:22:18,810 --> 00:22:22,938 Well, the six and the three share a common factor of three. 264 00:22:22,938 --> 00:22:27,754 And of course we've X squared and X common factor of X, so we 265 00:22:27,754 --> 00:22:29,474 can take out three X. 266 00:22:30,050 --> 00:22:34,306 And that will leave us 2X plus one. 267 00:22:35,710 --> 00:22:40,897 These two terms, what do we got for a common factor? Well, they 268 00:22:40,897 --> 00:22:46,084 clearly share a common factor of four and also a minus sign. So 269 00:22:46,084 --> 00:22:48,478 we take minus four times by. 270 00:22:49,160 --> 00:22:54,110 Now, minus four times by something has to give us minus 271 00:22:54,110 --> 00:22:59,960 8X, so that's going to be 2X and then minus four times by 272 00:22:59,960 --> 00:23:05,810 something has to give us minus four, so that's got to be plus 273 00:23:05,810 --> 00:23:10,760 one again to lump sum algebra sharing. This common factor of 274 00:23:10,760 --> 00:23:13,910 2X plus one. So we'll take that 275 00:23:13,910 --> 00:23:20,420 out. And then we have two X plus one multiplying 3X. 276 00:23:20,670 --> 00:23:23,976 And two X plus one multiplying 277 00:23:23,976 --> 00:23:31,448 minus 4. Will take 1 final example of 278 00:23:31,448 --> 00:23:38,410 this kind. So I've got 15 X 279 00:23:38,410 --> 00:23:42,066 squared. Minus three X 280 00:23:42,066 --> 00:23:47,602 minus 80. Now in all the others, the thing that we 281 00:23:47,602 --> 00:23:50,734 haven't checked at the beginning, and perhaps we should 282 00:23:50,734 --> 00:23:53,866 have done is do the coefficients. The numbers that 283 00:23:53,866 --> 00:23:55,258 multiply the X squared. 284 00:23:56,650 --> 00:24:00,290 That multiply the X and the constant term 285 00:24:00,290 --> 00:24:02,110 share a common factor. 286 00:24:03,480 --> 00:24:08,088 And in this case they do as a common factor of 3. 287 00:24:09,420 --> 00:24:13,020 And where there is a common factor, we need to take it 288 00:24:13,020 --> 00:24:16,020 out to begin with, so will take the three out. 289 00:24:17,250 --> 00:24:22,094 3 times by something as to give us 15 X squared so three times 290 00:24:22,094 --> 00:24:24,516 by 5 X squared will do that. 291 00:24:25,530 --> 00:24:30,738 3 times by something has to give us minus 3X so three times by 292 00:24:30,738 --> 00:24:32,598 minus X will do that. 293 00:24:33,590 --> 00:24:39,386 3 times by something has to give us minus 18 and so minus six 294 00:24:39,386 --> 00:24:40,628 will do that. 295 00:24:41,130 --> 00:24:45,480 Now we're looking at Factorizing this xpression Here in the 296 00:24:45,480 --> 00:24:49,830 bracket and we're looking for two numbers that were multiplied 297 00:24:49,830 --> 00:24:55,485 together to give us five times by minus six, which is minus 30. 298 00:24:55,485 --> 00:25:00,705 And again, I'll just write down where that came from. Minus 30 299 00:25:00,705 --> 00:25:06,360 was five times by minus six, and I'm looking for two numbers that 300 00:25:06,360 --> 00:25:11,145 will add together to give Maine. Now here the number that's 301 00:25:11,145 --> 00:25:12,885 multiplying the X is. 302 00:25:12,950 --> 00:25:18,677 Minus one. So I want two numbers to multiply together to 303 00:25:18,677 --> 00:25:23,393 give me minus 30 and add together to give me minus one, 304 00:25:23,393 --> 00:25:27,323 well, five and six seem like obvious choices 'cause they've 305 00:25:27,323 --> 00:25:32,432 got a difference of one and they multiply together to give 30. So 306 00:25:32,432 --> 00:25:37,148 how can I juggle the signs with the five and the six? 307 00:25:38,600 --> 00:25:44,600 Well, 5 + 6 has to give me minus one. It looks like the six is 308 00:25:44,600 --> 00:25:49,850 going to have to carry the minus sign, so 5 plus minus six does 309 00:25:49,850 --> 00:25:55,100 give me minus one and five times by minus six does give me minus 310 00:25:55,100 --> 00:25:57,725 30, so I'm going to have three. 311 00:25:58,530 --> 00:26:05,940 Brackets five X squared plus 5X minus six X, so we broken down 312 00:26:05,940 --> 00:26:13,920 this minus X into 5X, minus 6X and then the final term on the 313 00:26:13,920 --> 00:26:17,340 end minus six and close the 314 00:26:17,340 --> 00:26:23,240 bracket. Keep the three outside. Let's look at the front two 315 00:26:23,240 --> 00:26:28,135 terms here. There's a common factor of 5X. Let's take that 316 00:26:28,135 --> 00:26:29,470 out. Five X. 317 00:26:30,520 --> 00:26:38,020 X plus One 5X times by X gives me the five X squared 5X times 318 00:26:38,020 --> 00:26:41,020 by one gives me the 5X. 319 00:26:42,010 --> 00:26:46,630 Common factor here is minus six. Each of these terms shares A6 320 00:26:46,630 --> 00:26:51,635 and the minus sign, so will take out the factor minus six, and 321 00:26:51,635 --> 00:26:57,410 then we need minus six times by has to give us minus six X, so 322 00:26:57,410 --> 00:27:02,030 that's times by X minus six times by something has to give 323 00:27:02,030 --> 00:27:07,035 us minus six, so that's minus six times by one, and then I 324 00:27:07,035 --> 00:27:12,040 need to make sure I close the whole bracket with that big one 325 00:27:12,040 --> 00:27:16,577 there. Two lumps of algebra, each sharing this common factor 326 00:27:16,577 --> 00:27:19,258 of X plus one. Let's take that 327 00:27:19,258 --> 00:27:26,300 out three. Bracket X plus one times by the X 328 00:27:26,300 --> 00:27:29,180 Plus One Times 5X. 329 00:27:29,190 --> 00:27:36,395 The X plus one also times minus six, and so we've 330 00:27:36,395 --> 00:27:38,360 completed that factorization. 331 00:27:39,790 --> 00:27:43,689 OK, we've looked at a series of 332 00:27:43,689 --> 00:27:46,879 examples. An we've developed. 333 00:27:47,440 --> 00:27:52,346 A way of handling these that enables us to factorize these 334 00:27:52,346 --> 00:27:56,916 quadratics. Hasn't really been anything special about them, but 335 00:27:56,916 --> 00:28:02,402 I want to do now is have a look at three particular special 336 00:28:02,402 --> 00:28:09,110 cases. Let's begin with the first special case. By having a 337 00:28:09,110 --> 00:28:12,374 look at X squared minus 9. 338 00:28:13,520 --> 00:28:16,474 Now it's obviously different about this one. 339 00:28:17,100 --> 00:28:21,390 From the previous examples is that there's no external, just 340 00:28:21,390 --> 00:28:23,535 says X squared minus 9. 341 00:28:24,090 --> 00:28:30,315 So let's do it in the way that we would do we look for two 342 00:28:30,315 --> 00:28:34,880 numbers that would multiply together to give us minus 9 and 343 00:28:34,880 --> 00:28:40,275 add together to give us the X coefficient, but there is no X 344 00:28:40,275 --> 00:28:44,010 coefficient. That means the coefficient has to be 0. 345 00:28:44,530 --> 00:28:47,930 0 times by XOX is. 346 00:28:48,930 --> 00:28:53,562 So I've got to find 2 numbers that multiply together to give 347 00:28:53,562 --> 00:28:56,264 minus 9 and add together to give 348 00:28:56,264 --> 00:29:03,330 0. Obviously they've got to be the same size but different 349 00:29:03,330 --> 00:29:10,000 sign, so minus three and three fit the bill perfectly, 350 00:29:10,000 --> 00:29:16,670 so we have X squared minus 3X plus 3X minus 351 00:29:16,670 --> 00:29:22,798 9. Look at the front two terms. There's a common factor of X. 352 00:29:23,320 --> 00:29:30,431 Leaving me with X minus three. The back two terms as a common 353 00:29:30,431 --> 00:29:36,995 factor of 3 leaving X minus three and now two lumps of 354 00:29:36,995 --> 00:29:43,559 algebra each sharing this common factor of X minus three X minus 355 00:29:43,559 --> 00:29:47,935 three, multiplies the X and multiplies the three. 356 00:29:48,470 --> 00:29:52,254 Well, we compare this 357 00:29:52,254 --> 00:29:58,420 with this. Not only is this X squared that we get X 358 00:29:58,420 --> 00:30:00,544 times by X, but this 9. 359 00:30:01,500 --> 00:30:04,180 Forgetting the minus sign for a moment is 3 squared. 360 00:30:05,180 --> 00:30:06,628 3 times by three. 361 00:30:07,770 --> 00:30:15,330 So in fact this expression could be rewritten as X squared minus 362 00:30:15,330 --> 00:30:20,715 3 squared. In other words, it's the difference of two squares. 363 00:30:21,420 --> 00:30:23,700 So let's do this again. 364 00:30:24,590 --> 00:30:29,690 But more generally, In other words, instead of minus nine, 365 00:30:29,690 --> 00:30:35,810 which is minus 3 squared, let's write minus a squared. So we 366 00:30:35,810 --> 00:30:38,870 have X squared minus a squared. 367 00:30:39,530 --> 00:30:46,350 We want to factorize it, so we're looking for two numbers 368 00:30:46,350 --> 00:30:53,170 that multiply together to give minus a squared and add together 369 00:30:53,170 --> 00:31:00,610 to give 0. Because there are no access, so minus A and 370 00:31:00,610 --> 00:31:08,050 a fit the bill. So we're going to have X squared minus 371 00:31:08,050 --> 00:31:10,530 8X Plus 8X minus. 372 00:31:10,570 --> 00:31:17,829 A squared. Common factor of X here X minus 373 00:31:17,829 --> 00:31:25,163 a. Under common factor of a here, X minus a 374 00:31:25,163 --> 00:31:32,390 again 2 lumps of algebra, each one sharing this common factor 375 00:31:32,390 --> 00:31:35,018 of X minus a. 376 00:31:35,030 --> 00:31:42,278 X minus a multiplies X, an multiplies a. 377 00:31:42,930 --> 00:31:46,770 So we now have. 378 00:31:47,390 --> 00:31:53,714 What is, in effect a standard result we have what's called the 379 00:31:53,714 --> 00:31:56,349 difference of two squares and 380 00:31:56,349 --> 00:32:02,811 its factorization. So let's just write that down again. X squared 381 00:32:02,811 --> 00:32:09,879 minus a squared is always equal to X minus a X plus 382 00:32:09,879 --> 00:32:15,225 A. So that if we can identify this number that appears, here 383 00:32:15,225 --> 00:32:16,645 is a square number. 384 00:32:17,600 --> 00:32:20,696 We can use this factorization immediately. 385 00:32:21,920 --> 00:32:28,328 So what if we had something say like X squared minus 25? 386 00:32:29,100 --> 00:32:33,240 Well, we recognize 25 as being 5 squared, so 387 00:32:33,240 --> 00:32:38,760 immediately we can write this down as X minus five X +5. 388 00:32:39,800 --> 00:32:46,224 What if we had something like 2 X squared minus 32? 389 00:32:47,550 --> 00:32:51,882 Doesn't really look like that, does it? But there is a common 390 00:32:51,882 --> 00:32:56,214 factor of 2, so as we've said before, take the common factor 391 00:32:56,214 --> 00:32:57,658 out to begin with. 392 00:32:58,370 --> 00:33:03,746 Leaving us with X squared minus 16 and 393 00:33:03,746 --> 00:33:09,794 of course 16 is 4 squared, so this is 394 00:33:09,794 --> 00:33:13,154 2X minus four X +4. 395 00:33:16,140 --> 00:33:21,855 Files and we had nine X squared minus 16. 396 00:33:23,950 --> 00:33:25,078 What about this one? 397 00:33:25,700 --> 00:33:32,780 Again, look at this term here 9 X squared. It is a 398 00:33:32,780 --> 00:33:37,500 complete square. It's 3X times by three X. 399 00:33:38,010 --> 00:33:42,469 So instead of just working with an X, why can't we just work 400 00:33:42,469 --> 00:33:43,498 with a 3X? 401 00:33:44,490 --> 00:33:50,524 And of course, that's what we are going to do. This must be 3X 402 00:33:50,524 --> 00:33:56,989 and 3X and the 16 is 4 squared, so 3X minus four, 3X plus four. 403 00:33:56,989 --> 00:34:01,730 And again, this is still the difference of two squares, so 404 00:34:01,730 --> 00:34:06,902 that's one. The first special case, and we really do have to 405 00:34:06,902 --> 00:34:11,643 learn that one. And remember it 'cause it's a very, very 406 00:34:11,643 --> 00:34:15,280 important factorization. Let's have a look now. 407 00:34:15,780 --> 00:34:18,714 Another factorization 408 00:34:18,714 --> 00:34:25,316 special case. Having just on the difference of two 409 00:34:25,316 --> 00:34:29,220 squares looking at this quadratic expression, we've got 410 00:34:29,220 --> 00:34:35,564 a square front X squared and the square at the end 5 squared. 411 00:34:37,140 --> 00:34:39,198 But we've got 10X in the middle. 412 00:34:39,890 --> 00:34:44,902 OK, we know how to handle it, so let's not worry too much. We 413 00:34:44,902 --> 00:34:48,840 want two numbers that multiply together to give 25 and two 414 00:34:48,840 --> 00:34:51,704 numbers that add together to give us 10. 415 00:34:52,890 --> 00:34:56,341 The obvious choice for that is 5 416 00:34:56,341 --> 00:35:02,958 and five. Five times by 5 is 20 five 5 + 5 is 10. 417 00:35:03,560 --> 00:35:11,108 So we break that middle term down X squared plus 5X Plus 418 00:35:11,108 --> 00:35:12,995 5X plus 25. 419 00:35:15,110 --> 00:35:20,921 Look at the front. Two terms are common factor of X, leaving us 420 00:35:20,921 --> 00:35:22,262 with X +5. 421 00:35:23,060 --> 00:35:29,729 Look at the back to terms are common factor of +5 leaving us 422 00:35:29,729 --> 00:35:31,268 with X +5. 423 00:35:32,210 --> 00:35:37,677 Two lumps of algebra sharing a common factor of X +5. 424 00:35:38,230 --> 00:35:44,130 X +5 multiplies, X&X, 425 00:35:44,130 --> 00:35:47,080 +5, multiplies 426 00:35:47,080 --> 00:35:54,804 +5. These two are the same, so this is 427 00:35:54,804 --> 00:35:57,436 X +5 all squared. 428 00:35:58,460 --> 00:36:05,389 In other words, what we've got here X squared plus 10X plus 25 429 00:36:05,389 --> 00:36:12,318 is a complete square X +5 all squared. We call that a complete 430 00:36:12,318 --> 00:36:16,180 square. Take another 431 00:36:16,180 --> 00:36:21,080 example, X squared minus 432 00:36:21,080 --> 00:36:24,755 8X plus 16. 433 00:36:25,980 --> 00:36:30,712 We recognize this is a square number X squared and this is a 434 00:36:30,712 --> 00:36:33,970 square number. 16 is 4 squared. 435 00:36:35,190 --> 00:36:36,378 And of course. 436 00:36:37,520 --> 00:36:42,200 Minus 4 plus minus four would give us 8. 437 00:36:42,830 --> 00:36:48,578 As indeed minus four times, my minus four would give us plus 438 00:36:48,578 --> 00:36:51,458 16. So we immediately. 439 00:36:52,210 --> 00:36:58,018 Recognizing this as this complete square X minus 440 00:36:58,018 --> 00:37:00,196 four all squared. 441 00:37:02,900 --> 00:37:09,952 One more. 25 X squared minus 442 00:37:09,952 --> 00:37:13,336 20 X +4. 443 00:37:16,700 --> 00:37:22,948 25 X squared is a complete square. It's a square number. 444 00:37:22,948 --> 00:37:27,492 It's the result of multiplying 5X by itself. 445 00:37:28,150 --> 00:37:34,716 4 is a square number, it's 2 446 00:37:34,716 --> 00:37:37,530 times by two. 447 00:37:38,760 --> 00:37:41,812 Minus 20 448 00:37:41,812 --> 00:37:47,578 X. Well, if I say this is a complete square, 449 00:37:47,578 --> 00:37:52,042 then I've got to have minus two in their minus two times, 450 00:37:52,042 --> 00:37:57,622 Y minus two would give me 4 - 2 times by the five. X would 451 00:37:57,622 --> 00:38:02,086 give me minus 10X and I'm going to have two of them 452 00:38:02,086 --> 00:38:05,806 minus 20X. So again I recognize this as a complete 453 00:38:05,806 --> 00:38:06,178 square. 454 00:38:07,230 --> 00:38:12,078 Might have found that a little bit confusing and a little bit 455 00:38:12,078 --> 00:38:16,790 quick. That doesn't matter be'cause. If you don't recognize 456 00:38:16,790 --> 00:38:21,390 it, you can still use the previous method on it. 457 00:38:22,390 --> 00:38:28,750 Let's just check that 25 X squared minus 20X plus four. So 458 00:38:28,750 --> 00:38:35,110 if we didn't recognize this as a complete square, we would be 459 00:38:35,110 --> 00:38:40,410 looking for two numbers that would multiply together to give 460 00:38:40,410 --> 00:38:47,830 us 100. The 100 is 4 times by 25. Just write this at the 461 00:38:47,830 --> 00:38:53,130 side four times by 25 and two numbers that would. 462 00:38:53,150 --> 00:38:58,130 Add together to give us minus 20. The obvious choices are 10 463 00:38:58,130 --> 00:39:03,525 and 10, well minus 10 and minus 10 because we want minus 10 464 00:39:03,525 --> 00:39:09,335 times by minus 10 to give plus 100 and minus 10 plus minus 10 465 00:39:09,335 --> 00:39:11,410 to give us minus 20. 466 00:39:12,380 --> 00:39:16,916 So we take this expression 25 X squared. 467 00:39:18,050 --> 00:39:23,810 And we breakdown the minus 20X minus 10X minus 10X. 468 00:39:24,560 --> 00:39:27,052 And we have the last terms plus 469 00:39:27,052 --> 00:39:33,100 4. We look at the front two terms for a common factor and 470 00:39:33,100 --> 00:39:34,740 clearly there is 5X. 471 00:39:34,750 --> 00:39:40,434 Gives me 5X minus two. Now I look for a common factor in the 472 00:39:40,434 --> 00:39:46,524 last two terms there is a common factor of 2 here, because 2 * 5 473 00:39:46,524 --> 00:39:53,020 gives us 10 and 2 * 2 gives us 4. But there is this minus sign, 474 00:39:53,020 --> 00:39:57,486 so perhaps we better take minus two is our common factor. 475 00:39:58,000 --> 00:40:02,572 Which will give us minus two times by something has to give 476 00:40:02,572 --> 00:40:07,525 us minus 10X. That's going to be 5X and minus two times by 477 00:40:07,525 --> 00:40:12,478 something has to give us plus four which is going to have to 478 00:40:12,478 --> 00:40:13,621 be minus 2. 479 00:40:14,190 --> 00:40:19,954 Two lumps of algebra. The common factor is 5X minus 2. 480 00:40:20,700 --> 00:40:27,870 5X minus two is multiplying 5X. 481 00:40:27,870 --> 00:40:31,645 And it's also multiplying minus 482 00:40:31,645 --> 00:40:38,124 2. So we have got this again as a complete square, but not 483 00:40:38,124 --> 00:40:41,463 by inspection, but using our standard method. 484 00:40:43,130 --> 00:40:45,836 So that deals with the 2nd 485 00:40:45,836 --> 00:40:51,660 special case. Let's now have a look at the 3rd and final 486 00:40:51,660 --> 00:40:56,508 special case, and this is when we don't have a constant term 487 00:40:56,508 --> 00:41:02,164 and we've got something like 3 X squared minus 8X. What do we do 488 00:41:02,164 --> 00:41:07,416 with that? Well, clearly there is a common factor of X, so we 489 00:41:07,416 --> 00:41:12,264 must take that out to begin with. So we take out X. 490 00:41:14,630 --> 00:41:20,174 We've got three X squared, so X times by three X must give us 491 00:41:20,174 --> 00:41:21,758 the three X squared. 492 00:41:22,360 --> 00:41:27,680 And then X times by something to give us the minus 8X. Well it's 493 00:41:27,680 --> 00:41:33,000 got to be minus 8, and that's all that we need to do. We've 494 00:41:33,000 --> 00:41:37,940 broken it down into two brackets X times by three X minus 8. 495 00:41:38,830 --> 00:41:42,580 What if say we had 10 496 00:41:42,580 --> 00:41:46,110 X squared? Plus 497 00:41:46,110 --> 00:41:50,319 5X. Again, we look for a common factor. 498 00:41:51,400 --> 00:41:54,720 Obviously there's an ex as a common factor again, but 499 00:41:54,720 --> 00:41:57,708 there's more this time because there's ten and five 500 00:41:57,708 --> 00:42:02,024 which share a common factor of 5, so we need to pull out 501 00:42:02,024 --> 00:42:04,016 the whole of that as 5X. 502 00:42:06,050 --> 00:42:11,804 Five times by two will give us the 10, so that's 5X times by 503 00:42:11,804 --> 00:42:17,147 two. X will give us the 10 X squared plus 5X times by 504 00:42:17,147 --> 00:42:21,257 something to give us 5X that must just be 1. 505 00:42:22,290 --> 00:42:27,220 So, so long as we remember to inspect the quadratic 506 00:42:27,220 --> 00:42:31,164 expression 1st and check for common factors, this 507 00:42:31,164 --> 00:42:34,615 particular one shouldn't cause us any difficulties.