1 00:00:01,290 --> 00:00:05,658 The general strategy for solving a cubic equation is to reduce it 2 00:00:05,658 --> 00:00:10,026 to a quadratic and then solve the quadratic by the usual means 3 00:00:10,026 --> 00:00:11,846 either by Factorizing or using 4 00:00:11,846 --> 00:00:18,995 the formula. A cubic equation has the form a X cubed 5 00:00:18,995 --> 00:00:24,710 plus BX squared plus CX plus D equals not. 6 00:00:25,280 --> 00:00:28,673 It must have a Turman X cubed, or it wouldn't be a cubic. 7 00:00:29,350 --> 00:00:33,292 But any or all of BC&D can be 0. 8 00:00:34,890 --> 00:00:37,630 So for instance X cubed. 9 00:00:39,010 --> 00:00:45,360 Minus six X squared plus 11X minus six equals note 10 00:00:45,360 --> 00:00:47,265 is a cubic. 11 00:00:48,530 --> 00:00:53,578 So is 4X cubed plus 57 equals not. 12 00:00:55,620 --> 00:00:58,970 So is. X cubed 13 00:00:59,560 --> 00:01:03,040 +9 X equals not. 14 00:01:04,950 --> 00:01:08,770 Just as a quadratic equation may have two real roots. 15 00:01:09,580 --> 00:01:12,275 So a cubic equation possibly has three. 16 00:01:13,290 --> 00:01:17,510 But unlike a quadratic equation which main have no real 17 00:01:17,510 --> 00:01:21,730 solution, a cubic equation always has at least one real 18 00:01:21,730 --> 00:01:23,840 root. I'll explain why later. 19 00:01:24,740 --> 00:01:28,592 If a cubic does have 3 Routes, 2 or even all three 20 00:01:28,592 --> 00:01:30,197 of them may be repeated. 21 00:01:31,430 --> 00:01:37,790 This gives us four 22 00:01:37,790 --> 00:01:39,380 possibilities. 23 00:01:41,410 --> 00:01:48,960 The equation X cubed minus six X squared plus 11X. 24 00:01:49,850 --> 00:01:56,700 Minus 6 equals 0 Factorizes 2X minus one times X 25 00:01:56,700 --> 00:02:02,180 minus two times X minus three equals 0. 26 00:02:02,990 --> 00:02:06,110 This equation has three real roots, all different. 27 00:02:06,840 --> 00:02:11,350 Solutions X equals 1 X equals 2 or X equals 3. 28 00:02:11,890 --> 00:02:16,832 I'd like to show you the graph of the curve Y equals X cubed 29 00:02:16,832 --> 00:02:21,421 minus six X squared plus 11X minus six. I'm not very good at 30 00:02:21,421 --> 00:02:22,480 drawing freehand graphs. 31 00:02:23,010 --> 00:02:25,140 So here's one I prepared earlier. 32 00:02:27,980 --> 00:02:33,146 Notice that it starts slow down to the left. 33 00:02:33,860 --> 00:02:37,976 Because as X gets larger, negative, so does X cubed and it 34 00:02:37,976 --> 00:02:41,749 finishes high to the right because there's X gets large and 35 00:02:41,749 --> 00:02:43,464 positive. So does X cubed. 36 00:02:44,420 --> 00:02:48,944 And the curve crosses the X axis three times, once where X equals 37 00:02:48,944 --> 00:02:53,468 1 once, where X equals 2 and once where X equals 3. This 38 00:02:53,468 --> 00:02:55,208 gives us our three separate 39 00:02:55,208 --> 00:03:01,856 solutions. The equation X 40 00:03:01,856 --> 00:03:05,790 cubed. Minus five 41 00:03:05,790 --> 00:03:11,396 X squared. Plus 8X minus 4 equals 0 42 00:03:11,396 --> 00:03:16,805 Factorizes 2X minus one times X minus two all 43 00:03:16,805 --> 00:03:21,012 squared, and that is equal to 0. 44 00:03:22,630 --> 00:03:26,770 In this case we have. We do have 3 routes, but two of them are 45 00:03:26,770 --> 00:03:30,082 the same. We have X minus 2 squared, so we only actually 46 00:03:30,082 --> 00:03:35,550 have two solutions. Again, I'll show you the graph of Y equals X 47 00:03:35,550 --> 00:03:39,818 cubed minus five X squared plus 8X minus 40 equals 0. 48 00:03:45,470 --> 00:03:49,502 Again, the curve starts load to the left and goes high to 49 00:03:49,502 --> 00:03:53,198 the right. It crosses the X axis once and then just 50 00:03:53,198 --> 00:03:57,902 touches it. So we have our two roots, X equals 1 and X equals 51 00:03:57,902 --> 00:04:01,598 2 and it touches at the repeated root X equals 2. 52 00:04:03,220 --> 00:04:10,692 The equation X cubed minus three X squared 53 00:04:10,692 --> 00:04:14,428 plus 3X minus one 54 00:04:14,428 --> 00:04:20,900 equals note. The left hand side factorizes to X minus 55 00:04:20,900 --> 00:04:23,300 one or cubed equals not. 56 00:04:23,830 --> 00:04:27,220 So there there are three factors. They're all the same 57 00:04:27,220 --> 00:04:30,610 and we only have a single solution. X equals 1. 58 00:04:31,840 --> 00:04:36,064 The corresponding curve is Y equals X cubed, minus three X 59 00:04:36,064 --> 00:04:37,984 squared plus 3X minus one. 60 00:04:38,700 --> 00:04:41,350 And it looks like this. 61 00:04:44,870 --> 00:04:47,738 As with all the Cubix I've shown you so far, it starts 62 00:04:47,738 --> 00:04:50,606 slow down on the left and goes high up to the right. 63 00:04:51,960 --> 00:04:55,964 Notice that the curve does cross the X axis at the point X 64 00:04:55,964 --> 00:05:00,276 equals 1, but it is also a tangent. X axis is a tangent to 65 00:05:00,276 --> 00:05:03,048 the curve at this point, indicating the three repeated 66 00:05:03,048 --> 00:05:03,356 roots. 67 00:05:06,800 --> 00:05:14,780 Now look at the equation X cubed plus X squared 68 00:05:14,780 --> 00:05:19,950 plus X. Minus 3 equals 0. 69 00:05:21,000 --> 00:05:23,792 This expression Factorizes 2X 70 00:05:23,792 --> 00:05:31,418 minus one. X squared plus 2X plus three, so we can put this 71 00:05:31,418 --> 00:05:32,972 equal to 0. 72 00:05:35,080 --> 00:05:40,792 The quadratic X squared +2 X +3 equals not has no real 73 00:05:40,792 --> 00:05:44,426 solutions. So the only solution to the cubic 74 00:05:44,426 --> 00:05:48,962 equation is to put X minus one equal to 0, giving this 75 00:05:48,962 --> 00:05:51,230 single real solution X equals 1. 76 00:05:53,400 --> 00:05:59,715 The graph Y equals X cubed plus X squared plus 6 - 3 looks like 77 00:05:59,715 --> 00:06:07,292 this. And you can see that it only 78 00:06:07,292 --> 00:06:13,774 crosses the X axis in one place. 79 00:06:14,380 --> 00:06:16,006 From the graphs that I've shown 80 00:06:16,006 --> 00:06:19,942 you. You can see why a cubic equation always has at least 81 00:06:19,942 --> 00:06:20,788 one real root. 82 00:06:22,110 --> 00:06:25,656 The graph either starts large, negative, and finishes large 83 00:06:25,656 --> 00:06:30,384 positive. If the coefficient of X cubed is positive or it will 84 00:06:30,384 --> 00:06:33,536 start large positive and finished down here. Large 85 00:06:33,536 --> 00:06:38,264 negative if the coefficient of X cubed is negative, the graph of 86 00:06:38,264 --> 00:06:43,780 a cubic must cross the X axis, giving you one real root. So any 87 00:06:43,780 --> 00:06:47,720 problem you get that involves solving a cubic equation will 88 00:06:47,720 --> 00:06:49,296 have a real solution. 89 00:06:52,720 --> 00:06:59,056 Now let's move on to how we solve cubics. 90 00:06:59,090 --> 00:07:05,310 Like a quadratic, cubic should always be rearranged into the 91 00:07:05,310 --> 00:07:12,774 form X cubed plus BX squared plus CX plus D equals 0. 92 00:07:13,750 --> 00:07:20,442 The equation X squared plus 4X minus one equals 6 over X is a 93 00:07:20,442 --> 00:07:26,178 cubic, but I wouldn't like to try and solve it in this 94 00:07:26,178 --> 00:07:31,436 particular form. We need to multiply through by X, giving us 95 00:07:31,436 --> 00:07:36,216 X cubed plus four X squared minus X equals 6. 96 00:07:36,740 --> 00:07:40,940 And then we subtract 6 from both sides, giving us X cubed. 97 00:07:41,540 --> 00:07:48,250 Plus four X squared minus X minus 6 equals 0. 98 00:07:49,880 --> 00:07:54,740 When solving cubics, it helps if you know or 99 00:07:54,740 --> 00:07:59,600 think you know one route to start with. For 100 00:07:59,600 --> 00:08:02,840 instance, take the equation X cubed. 101 00:08:04,760 --> 00:08:11,304 Minus five X squared minus 2X plus 24 102 00:08:11,304 --> 00:08:18,188 equals not. Given that X equals minus two is a 103 00:08:18,188 --> 00:08:22,376 solution. There is a theorem called the factor theorem which 104 00:08:22,376 --> 00:08:26,328 I'm not going to attempt to prove here that says that if X 105 00:08:26,328 --> 00:08:28,152 equals minus two is a solution 106 00:08:28,152 --> 00:08:35,410 of this equation. Then X +2 is a 107 00:08:35,410 --> 00:08:39,350 factor. Of this whole expression. 108 00:08:40,590 --> 00:08:47,077 This means that X cubed minus five X squared minus 2X plus 24 109 00:08:47,077 --> 00:08:49,572 is equal to X +2. 110 00:08:50,240 --> 00:08:56,135 Times some quadratic which will call X squared plus 111 00:08:56,135 --> 00:08:58,100 8X Plus B. 112 00:08:59,220 --> 00:09:01,196 And then all that is equal to, not. 113 00:09:03,060 --> 00:09:08,505 So our task now is to find A&B and we do this by a process 114 00:09:08,505 --> 00:09:09,594 called synthetic division. 115 00:09:12,440 --> 00:09:15,646 This involves looking at the coefficients of. 116 00:09:16,680 --> 00:09:18,168 The original expression. 117 00:09:19,550 --> 00:09:22,610 So for instance, the coefficient of X cubed is one. 118 00:09:23,190 --> 00:09:28,570 The coefficient of X squared is minus 5. The coefficient 119 00:09:28,570 --> 00:09:31,260 of X is minus 2. 120 00:09:32,370 --> 00:09:34,870 And the constant is 24. 121 00:09:35,650 --> 00:09:38,128 And we just right in that we're 122 00:09:38,128 --> 00:09:39,898 synthetically dividing by minus 2. 123 00:09:41,620 --> 00:09:43,060 I leave a line. 124 00:09:44,740 --> 00:09:51,019 And then bring the one down one times minus two is minus 2. 125 00:09:52,210 --> 00:09:55,170 Minus 5 plus minus 2. 126 00:09:55,830 --> 00:09:57,039 Is minus 7. 127 00:09:58,190 --> 00:10:01,178 Minus Seven times minus two is 128 00:10:01,178 --> 00:10:04,834 14. 14 plus minus two 129 00:10:04,834 --> 00:10:08,435 is 12. 12 times minus 130 00:10:08,435 --> 00:10:11,709 2. Is minus 24? 131 00:10:12,220 --> 00:10:15,916 And 24 plus minus 24 is 0. 132 00:10:16,710 --> 00:10:22,422 The zero tells us that X +2 is indeed a factor, and the numbers 133 00:10:22,422 --> 00:10:26,910 we have here give us the coefficients of the quadratic. A 134 00:10:26,910 --> 00:10:32,622 is equal to minus Seven and B is equal to 12. So the quadratic 135 00:10:32,622 --> 00:10:36,294 that we're looking for is X squared minus 7X. 136 00:10:37,080 --> 00:10:42,300 Plus 12. And synthetic division is explained fully in the 137 00:10:42,300 --> 00:10:48,413 accompanying notes. So we've reduced our cubic 2X plus 138 00:10:48,413 --> 00:10:55,103 two times X squared minus 7X plus 12 equals zero 139 00:10:55,103 --> 00:11:01,793 X squared minus 7X plus 12 can be factorized into 140 00:11:01,793 --> 00:11:08,483 X minus three times X minus four. So we have 141 00:11:08,483 --> 00:11:15,173 X +2 times X minus three times X minus 4. 142 00:11:15,810 --> 00:11:18,390 Equals 0. Giving us. 143 00:11:19,300 --> 00:11:21,540 X equals minus 2. 144 00:11:22,690 --> 00:11:25,900 3 or 4. 145 00:11:28,330 --> 00:11:35,000 If you don't know a route, it's always worth trying 146 00:11:35,000 --> 00:11:40,336 a few simple values. Let's solve X cubed. 147 00:11:40,930 --> 00:11:41,710 Minus. 148 00:11:43,010 --> 00:11:44,220 7X. 149 00:11:45,620 --> 00:11:48,460 Minus 6. Equals 0. 150 00:11:49,040 --> 00:11:51,128 The simplest value should try is 151 00:11:51,128 --> 00:11:53,919 one. 1 - 7. 152 00:11:54,820 --> 00:12:00,462 Minus 6 - 12 so that doesn't work. Let's try minus 1 - 1 153 00:12:00,462 --> 00:12:06,910 cubed is minus 1 + 7 - 6 is 0, so minus one is a route. 154 00:12:07,870 --> 00:12:10,444 Which means that X plus one is a factor. 155 00:12:12,000 --> 00:12:18,105 If minus one is a route, we can synthetically divide through 156 00:12:18,105 --> 00:12:20,880 this expression by minus one. 157 00:12:21,680 --> 00:12:26,289 Coefficient of X squared, sorry coefficient of X cubed is one. 158 00:12:27,240 --> 00:12:31,644 The coefficient of X squared is 0, there's no Turman X squared. 159 00:12:32,600 --> 00:12:38,480 The coefficient of X is minus Seven and the constant is minus 160 00:12:38,480 --> 00:12:42,400 6 and were synthetically dividing by minus one. 161 00:12:43,260 --> 00:12:48,132 I bring the one down one times minus one is minus one. 162 00:12:48,650 --> 00:12:50,946 Minus one and zero is minus one. 163 00:12:51,650 --> 00:12:54,032 Minus one times minus one is 164 00:12:54,032 --> 00:12:58,459 one. Minus 7 Plus One is minus 6. 165 00:12:59,510 --> 00:13:06,478 Minus six times minus one is 6 - 6 at 6 is 0. 166 00:13:10,340 --> 00:13:13,994 These numbers give us the coefficients of the quadratic 167 00:13:13,994 --> 00:13:16,430 equation, and we have X squared. 168 00:13:17,420 --> 00:13:23,129 Minus X minus 6 equals 0 and we now need to 169 00:13:23,129 --> 00:13:24,686 solve this equation. 170 00:13:26,400 --> 00:13:29,732 So we have X 171 00:13:29,732 --> 00:13:33,929 plus one. Times X squared. 172 00:13:35,990 --> 00:13:43,036 Minus X minus 6 equal to zero X squared minus X minus 6, 173 00:13:43,036 --> 00:13:50,082 factorizes to X minus three X +2, so we have X Plus one 174 00:13:50,082 --> 00:13:56,586 times X, minus three times X +2 equals 0, and the three 175 00:13:56,586 --> 00:14:02,548 solutions to the cubic equation RX equals minus 2 - 1. 176 00:14:03,090 --> 00:14:04,680 Or three. 177 00:14:06,930 --> 00:14:12,186 Sometimes you may be able to spot a factor. 178 00:14:12,750 --> 00:14:15,480 In the equation X cubed. 179 00:14:16,440 --> 00:14:23,096 Minus four X squared minus 9X plus 36 180 00:14:23,096 --> 00:14:24,760 equals 0. 181 00:14:25,840 --> 00:14:30,262 The coefficient of X squared is minus four times the coefficient 182 00:14:30,262 --> 00:14:31,468 of X cubed. 183 00:14:32,200 --> 00:14:34,792 And the constant is minus four 184 00:14:34,792 --> 00:14:37,478 times. The coefficient of X. 185 00:14:38,850 --> 00:14:43,575 This means that we're going to be able to take out X minus 4 as 186 00:14:43,575 --> 00:14:47,040 a factor from that those two terms and those two terms. 187 00:14:47,830 --> 00:14:52,858 Those two terms are X squared times X minus four X squared 188 00:14:52,858 --> 00:14:57,886 times, X is X cubed X squared times minus four is minus 189 00:14:57,886 --> 00:14:59,143 four X squared. 190 00:15:00,360 --> 00:15:03,090 Those two terms are minus. 191 00:15:04,510 --> 00:15:08,400 9 times X minus 4. 192 00:15:10,400 --> 00:15:14,564 Minus nine times X is minus 9X and minus nine times minus 193 00:15:14,564 --> 00:15:18,381 four is plus 36 and all that is equal to 0. 194 00:15:20,130 --> 00:15:25,374 And we can now factorize again because we have a common factor 195 00:15:25,374 --> 00:15:31,055 X minus four, so we have X squared minus nine times X minus 196 00:15:31,055 --> 00:15:32,366 4 equals 0. 197 00:15:34,140 --> 00:15:39,756 X squared minus nine is the difference of two squares, so we 198 00:15:39,756 --> 00:15:45,840 can write that in as X plus three times X minus three times 199 00:15:45,840 --> 00:15:48,180 X minus 4 equals 0. 200 00:15:48,820 --> 00:15:56,090 Giving a solutions X equals minus 3, three or four. 201 00:15:58,420 --> 00:16:01,060 You may have noticed that in each example that we've done. 202 00:16:02,050 --> 00:16:07,780 Every root. There's a factor of the constant term in the 203 00:16:07,780 --> 00:16:11,740 original equation. For instance, three and four, both divided 204 00:16:11,740 --> 00:16:16,440 into 36. As long as the coefficient of X cubed is one, 205 00:16:16,440 --> 00:16:18,030 this must be the case. 206 00:16:18,780 --> 00:16:25,286 Because. The constant is simply the product of the roots 3 * 3 * 207 00:16:25,286 --> 00:16:27,196 4 is equal to 36. 208 00:16:27,710 --> 00:16:30,310 This gives us another possible 209 00:16:30,310 --> 00:16:36,985 approach. Consider the equation X cubed 210 00:16:36,985 --> 00:16:40,198 minus six X 211 00:16:40,198 --> 00:16:46,888 squared. Minus six X minus 7 equals 0. 212 00:16:47,840 --> 00:16:50,980 Now it's possible for every solution to be irrational, but 213 00:16:50,980 --> 00:16:54,434 if there is a rational solution, then because the coefficient of 214 00:16:54,434 --> 00:16:59,144 X cubed is one, it's going to be an integer, and it's going to be 215 00:16:59,144 --> 00:17:00,400 a factor of 7. 216 00:17:01,010 --> 00:17:05,462 This only leaves us with four possibilities, 1 - 1 Seven and 217 00:17:05,462 --> 00:17:09,543 minus Seven, so we can try each of them in turn. 218 00:17:10,250 --> 00:17:14,839 You can see fairly quickly that one and minus one don't work, so 219 00:17:14,839 --> 00:17:15,898 let's try 7. 220 00:17:17,000 --> 00:17:20,130 Rather than substituting 7 into this expression and having to 221 00:17:20,130 --> 00:17:24,199 workout 7 cubed and so on, what I'm going to do is synthetically 222 00:17:24,199 --> 00:17:29,207 divide by 7, because if Seven is a route I'll end up with a 0 and 223 00:17:29,207 --> 00:17:32,650 What's more, the division will give me the quadratic that I'm 224 00:17:32,650 --> 00:17:35,900 looking for. So let's synthetically divide this 225 00:17:35,900 --> 00:17:37,136 expression by 7. 226 00:17:38,790 --> 00:17:40,410 The coefficient of X cubed is 227 00:17:40,410 --> 00:17:43,335 one. Coefficient of X squared is 228 00:17:43,335 --> 00:17:48,614 minus 6. The coefficient of X is minus 6 and the constant is 229 00:17:48,614 --> 00:17:51,526 minus Seven and were synthetically dividing by 7. 230 00:17:54,600 --> 00:18:00,102 Bring the one down 1 * 7 is Seven 7 - 6 is one. 231 00:18:00,950 --> 00:18:04,424 1 * 7 is 7 Seven and minus six 232 00:18:04,424 --> 00:18:07,330 is one. 1 * 7 is 7. 233 00:18:08,130 --> 00:18:10,470 7 and minus 70. 234 00:18:10,970 --> 00:18:16,767 So 7 is indeed a route, and the resulting quadratic is 235 00:18:16,767 --> 00:18:20,983 X squared plus X plus one equals not. 236 00:18:22,600 --> 00:18:25,912 Now you'll find if you try to solve it that the quadratic 237 00:18:25,912 --> 00:18:27,568 equation X squared plus X plus 238 00:18:27,568 --> 00:18:31,950 one equals not. Has no real solutions, so the 239 00:18:31,950 --> 00:18:36,800 only possible solution to this cubic is X equals 7. 240 00:18:39,520 --> 00:18:44,510 Certain basic identity's which you may wish to learn can 241 00:18:44,510 --> 00:18:49,001 help. In fact, rising both cubics and quadratics. I'll 242 00:18:49,001 --> 00:18:51,496 just give you 1 example. 243 00:18:52,730 --> 00:18:56,700 We have the equation X cubed plus three X squared. 244 00:18:57,530 --> 00:19:02,270 Plus 3X plus one equals 0. 245 00:19:04,330 --> 00:19:11,370 1331 is the standard expansion of X plus one cubed. 246 00:19:12,900 --> 00:19:18,269 So the solution to this equation is X plus one cubed equals 0. 247 00:19:18,800 --> 00:19:23,656 We have The root minus One X equals minus one 248 00:19:23,656 --> 00:19:25,072 repeated three times. 249 00:19:27,690 --> 00:19:34,962 If you can't find a factor by these methods, then draw an 250 00:19:34,962 --> 00:19:37,992 accurate graph of the cubic 251 00:19:37,992 --> 00:19:41,960 expression. The points where it crosses the X axis. 252 00:19:42,670 --> 00:19:44,266 Will give you the solutions to 253 00:19:44,266 --> 00:19:48,504 the equation. But their accuracy will be limited to the accuracy 254 00:19:48,504 --> 00:19:49,572 of your graph. 255 00:19:50,600 --> 00:19:54,656 You might indeed find the graph crosses the X axis at a point 256 00:19:54,656 --> 00:19:58,400 that would suggest a factor. For instance, if you draw craft a 257 00:19:58,400 --> 00:20:01,520 graph that appears to cross the X axis would say. 258 00:20:02,080 --> 00:20:06,461 X equals 1/2. Then it's worth trying to find out if X minus 259 00:20:06,461 --> 00:20:08,146 1/2 is indeed a factor. 260 00:20:10,130 --> 00:20:12,080 Let's look at the equation. 261 00:20:12,590 --> 00:20:14,290 X cubed 262 00:20:15,580 --> 00:20:18,188 Plus four X squared. 263 00:20:19,030 --> 00:20:23,638 Plus X minus 5 equals 0. 264 00:20:26,900 --> 00:20:30,090 Now this equation won't yield affected by any of the methods 265 00:20:30,090 --> 00:20:32,990 that I've discussed, so it's time to draw a graph. 266 00:20:36,270 --> 00:20:40,586 And this is the graph of Y equals X cubed plus four X 267 00:20:40,586 --> 00:20:42,246 squared plus 6 - 5. 268 00:20:44,450 --> 00:20:49,476 It crosses the X axis at three places, 1 near X equals minus 3, 269 00:20:49,476 --> 00:20:51,630 one near X equals minus 2. 270 00:20:52,160 --> 00:20:54,320 And one near X equals 1. 271 00:20:56,010 --> 00:20:59,718 Pending on how accurate your graph is, you may be able to 272 00:20:59,718 --> 00:21:03,117 pinpoint it a bit closer than this, and we get solutions. 273 00:21:03,750 --> 00:21:07,090 X is approximately equal to 274 00:21:07,090 --> 00:21:11,206 minus 3.2. Minus 1.7 275 00:21:11,206 --> 00:21:14,410 and not .9. 276 00:21:16,230 --> 00:21:18,950 These solutions may be accurate enough for your 277 00:21:18,950 --> 00:21:22,350 needs, but if you require more accurate answers then you 278 00:21:22,350 --> 00:21:24,730 should use a numerical algorithm using the 279 00:21:24,730 --> 00:21:28,130 approximate answers. If you obtain from the graph as a 280 00:21:28,130 --> 00:21:28,810 starting point.