1 00:00:01,870 --> 00:00:07,222 Going to have a look at a very simple process. It's called 2 00:00:07,222 --> 00:00:08,560 completing the square. 3 00:00:09,220 --> 00:00:14,414 In order to get to it and to show its potential use, I want 4 00:00:14,414 --> 00:00:18,124 to start with a simple equation. X squared equals 9. 5 00:00:19,900 --> 00:00:24,660 In order to find out what taxes we would take the square root of 6 00:00:24,660 --> 00:00:29,420 both sides, so the square root of X squared is just X and the 7 00:00:29,420 --> 00:00:32,820 square root of 9 is plus three or minus three. 8 00:00:33,350 --> 00:00:38,358 Be'cause minus three all squared is also 9. 9 00:00:38,870 --> 00:00:43,590 So that was relatively straightforward. Both of these 10 00:00:43,590 --> 00:00:46,540 two numbers were square numbers, 11 00:00:46,540 --> 00:00:49,556 complete squares. What if we 12 00:00:49,556 --> 00:00:53,290 got? X squared is equal to 13 00:00:53,290 --> 00:00:59,751 5. Do the same again, so we take the square root of 14 00:00:59,751 --> 00:01:05,283 each side X equals. Now 5 is not a square number, it 15 00:01:05,283 --> 00:01:08,971 does not have an exact square root, but 16 00:01:08,971 --> 00:01:14,042 nevertheless we can write it as root 5 that is exact 17 00:01:14,042 --> 00:01:15,886 or minus Route 5. 18 00:01:17,100 --> 00:01:21,940 So far so good. The same process is working each time. 19 00:01:22,710 --> 00:01:27,539 Let's have a look at something now, like X minus 7. 20 00:01:28,190 --> 00:01:31,090 All squared equals 3. 21 00:01:31,940 --> 00:01:36,857 How can we solve this? Well again, this side of the 22 00:01:36,857 --> 00:01:41,327 equation. We've got something called a complete square, so we 23 00:01:41,327 --> 00:01:47,585 can take the square root of each side. X minus Seven is equal to 24 00:01:47,585 --> 00:01:54,290 the square root of 3 or minus the square root of 3, and so we 25 00:01:54,290 --> 00:02:01,442 can now add 7 to each of these. So X is equal to 7 Plus Route 26 00:02:01,442 --> 00:02:02,783 3 or 7. 27 00:02:02,830 --> 00:02:04,609 Minus Route 3. 28 00:02:06,880 --> 00:02:14,608 I just have a look at another One X plus three, all squared is 29 00:02:14,608 --> 00:02:16,264 equal to 5. 30 00:02:18,670 --> 00:02:24,754 Again, this is a complete square, so again we can take the 31 00:02:24,754 --> 00:02:31,852 square root X +3 is equal to Route 5 or minus Route 5. Now 32 00:02:31,852 --> 00:02:38,443 to get X on its own, we need to take three away from 33 00:02:38,443 --> 00:02:45,541 each side, so we have X equals minus 3 + 5 or minus 3 34 00:02:45,541 --> 00:02:47,062 minus Route 5. 35 00:02:51,010 --> 00:02:57,738 What if we got X squared plus six 36 00:02:57,738 --> 00:03:00,261 X equals 4? 37 00:03:01,590 --> 00:03:07,440 Problem is this X squared plus 6X is not a complete square, so 38 00:03:07,440 --> 00:03:10,140 we can't just take the square 39 00:03:10,140 --> 00:03:15,560 root. So in terms of handling something like this, we've got 40 00:03:15,560 --> 00:03:18,458 to have a way of getting a 41 00:03:18,458 --> 00:03:22,081 complete square. So the process that we're going to 42 00:03:22,081 --> 00:03:24,425 be looking at it's called completing the square. 43 00:03:25,570 --> 00:03:31,330 The sorts of expressions that we have before will like this X 44 00:03:31,330 --> 00:03:33,250 plus a all squared. 45 00:03:34,030 --> 00:03:40,402 Or X minus a all squared, so that's what a complete square 46 00:03:40,402 --> 00:03:43,588 looks like. One of these two. 47 00:03:44,150 --> 00:03:51,038 So let's multiply this out and see what we get. So this is X 48 00:03:51,038 --> 00:03:54,482 plus a times by X plus A. 49 00:03:55,960 --> 00:04:00,160 And we do X times by X. That gives us X squared. 50 00:04:00,930 --> 00:04:05,430 And we do a Times by X, which gives us a X. 51 00:04:06,150 --> 00:04:12,100 And then with X times by a, which again gives us a X and 52 00:04:12,100 --> 00:04:18,475 then at the end a Times by a, which gives us a squared. And so 53 00:04:18,475 --> 00:04:21,450 we've X squared plus 2X plus A 54 00:04:21,450 --> 00:04:28,998 squared. I can do the same with this One X minus a Times by 55 00:04:28,998 --> 00:04:35,662 X minus A and it's going to give very similar results X times by 56 00:04:35,662 --> 00:04:42,802 X will give me X squared X times by minus A minus 8X minus 8 57 00:04:42,802 --> 00:04:49,466 times by X minus 8X minus 8 times by minus A plus a squared. 58 00:04:49,466 --> 00:04:52,322 So tidying up the two middle 59 00:04:52,322 --> 00:04:58,430 bits. Minus X minus X, minus 2X and then plus A squared. 60 00:04:58,430 --> 00:05:03,530 So this is what complete squares look like. They look 61 00:05:03,530 --> 00:05:06,080 like one of these two. 62 00:05:07,210 --> 00:05:13,570 Well. Can I make this look like a complete square in 63 00:05:13,570 --> 00:05:19,394 some way shape or form? If I compare this with this, what is 64 00:05:19,394 --> 00:05:21,186 it that I see? 65 00:05:21,850 --> 00:05:27,238 Well, perhaps one of things I might like to have is this 66 00:05:27,238 --> 00:05:32,626 written as just a function X squared plus 6X minus four? And 67 00:05:32,626 --> 00:05:37,565 let's not worry too much about solving an equation. What we 68 00:05:37,565 --> 00:05:42,055 want to concentrate on is this process of completing the 69 00:05:42,055 --> 00:05:46,545 square, so I'm going to take this quadratic function X 70 00:05:46,545 --> 00:05:52,382 squared plus 6X minus four, and I'm going to compare it with the 71 00:05:52,382 --> 00:05:58,530 complete square. X squared plus 2X plus 72 00:05:58,530 --> 00:06:04,007 A squared. Now The X squared so the same. 73 00:06:05,210 --> 00:06:11,762 6X2A X I've got to have these two terms the same. They've got 74 00:06:11,762 --> 00:06:17,810 to match. They've got to be exactly the same, and that means 75 00:06:17,810 --> 00:06:21,338 that the six has to be equal 76 00:06:21,338 --> 00:06:28,794 to 2A. And of course, that tells us that the A 77 00:06:28,794 --> 00:06:31,378 is equal to 3. 78 00:06:32,750 --> 00:06:39,432 So if I make a equal to three, then I've got plus A 79 00:06:39,432 --> 00:06:42,002 squared on the end +9. 80 00:06:43,300 --> 00:06:49,360 So. I can look at this first bit and I can make it equal to that. 81 00:06:50,070 --> 00:06:56,800 So let's write this down X squared plus 6X minus 82 00:06:56,800 --> 00:07:03,934 4 equals. X plus three all squared. Remember this 83 00:07:03,934 --> 00:07:10,854 is X plus three all squared. Now I'm replacing the 84 00:07:10,854 --> 00:07:12,930 A by three. 85 00:07:13,830 --> 00:07:18,954 Now what more of I got? Well, I've added on a squared, so I've 86 00:07:18,954 --> 00:07:24,444 added on 9. So I've got some how to get rid of that. Well, let's 87 00:07:24,444 --> 00:07:26,640 just take it away, minus 3 88 00:07:26,640 --> 00:07:31,818 squared. And then I can keep this minus four at the end as it 89 00:07:31,818 --> 00:07:39,568 was. So now I've X plus three all squared minus 9 - 90 00:07:39,568 --> 00:07:45,848 4 gives me X plus three all squared minus 30. 91 00:07:47,450 --> 00:07:52,440 So I have completed the square. I made this bit. 92 00:07:53,790 --> 00:07:57,570 Part of a complete square. 93 00:07:58,450 --> 00:08:03,500 And I've done it by comparing the coefficient of X. 94 00:08:04,040 --> 00:08:10,270 With one of the two standard forms and I saw that what I had 95 00:08:10,270 --> 00:08:14,275 to do was take half the coefficient of X. 96 00:08:15,750 --> 00:08:22,900 Let's have a look then at another example, X squared minus 97 00:08:22,900 --> 00:08:30,700 8X plus Seven and I want to write this so it's got 98 00:08:30,700 --> 00:08:33,950 a complete square in it. 99 00:08:34,640 --> 00:08:39,770 Well, one of the standard forms for the complete square 100 00:08:39,770 --> 00:08:45,413 that we had was X squared minus 2X plus A squared. 101 00:08:47,900 --> 00:08:50,994 And I want to make these two 102 00:08:50,994 --> 00:08:52,830 terms. The same. 103 00:08:54,100 --> 00:08:57,948 So again, we can see that the A. 104 00:08:58,800 --> 00:09:05,464 Has got to be 4 because minus 8 is minus 2 times by 4. 105 00:09:08,090 --> 00:09:15,520 So we've got X squared minus 8X plus Seven is 106 00:09:15,520 --> 00:09:21,930 equal to. X minus four all squared. 107 00:09:22,890 --> 00:09:27,954 So I've ensured I've got the X squared. I've ensured that I've 108 00:09:27,954 --> 00:09:33,862 got the minus 8X, but I've also added on a squared, so I've got 109 00:09:33,862 --> 00:09:39,348 a squared too much, so I must take away 4 squared and then 110 00:09:39,348 --> 00:09:45,678 I've got the 7:00 that I need to add on to keep the equal sign. 111 00:09:46,310 --> 00:09:52,538 And so this is now X minus four all 112 00:09:52,538 --> 00:09:58,766 squared minus 16 + 7 X minus four all 113 00:09:58,766 --> 00:10:00,842 squared minus 9. 114 00:10:02,140 --> 00:10:06,090 Let's take one more example. 115 00:10:06,110 --> 00:10:08,350 X 116 00:10:08,350 --> 00:10:14,979 squared Plus 5X plus three and let's see if we 117 00:10:14,979 --> 00:10:19,610 can follow this one through without having to write down the 118 00:10:19,610 --> 00:10:23,399 comparison. In other words, by doing it by inspection. 119 00:10:24,050 --> 00:10:30,342 What do we need? We need a complete square, so we need X 120 00:10:30,342 --> 00:10:37,118 and we look at this number here. The coefficient of the X turn on 121 00:10:37,118 --> 00:10:39,054 this left hand side. 122 00:10:39,770 --> 00:10:45,152 We want half that coefficient, so we want five over 2 and we've 123 00:10:45,152 --> 00:10:50,534 got a plus sign, so it's got to be X +5 over 2. 124 00:10:51,220 --> 00:10:57,382 If we were to multiply out this bracket, we would be adding on 125 00:10:57,382 --> 00:11:03,544 an additional A squared where five over 2 is the a. So we've 126 00:11:03,544 --> 00:11:10,180 got to take that away. Takeaway 5 over 2 squared and then at the 127 00:11:10,180 --> 00:11:12,550 end we've got plus 3. 128 00:11:13,340 --> 00:11:19,812 So this gives us X +5 over 2 129 00:11:19,812 --> 00:11:26,284 or squared minus 25 over 4 + 3. 130 00:11:27,330 --> 00:11:33,523 And of course, we'd like to combine these numbers at this 131 00:11:33,523 --> 00:11:41,405 end X +5 over 2 all squared minus 25 over 4 plus. Now we 132 00:11:41,405 --> 00:11:48,161 need to convert these to quarters as well, so three is 12 133 00:11:48,161 --> 00:11:54,741 quarters. Now we can combine these, since they're both in 134 00:11:54,741 --> 00:12:02,049 terms of quarters X +5 over 2, all squared minus 13 over 135 00:12:02,049 --> 00:12:04,480 4. And we'd leave you like that. 136 00:12:05,510 --> 00:12:09,272 This process now seems to be working quite well. 137 00:12:09,800 --> 00:12:13,140 But of course, we haven't dealt with every kind of 138 00:12:13,140 --> 00:12:16,146 quadratic expression we could have, because so far we've 139 00:12:16,146 --> 00:12:19,820 only had a unit coefficient here in front of the X 140 00:12:19,820 --> 00:12:22,826 squared. We haven't had another number like two or 141 00:12:22,826 --> 00:12:26,834 three or whatever, so let's have a look at what we would 142 00:12:26,834 --> 00:12:28,170 do in that case. 143 00:12:30,900 --> 00:12:35,208 So we have three X squared. 144 00:12:36,250 --> 00:12:43,288 Minus 9X. Plus 50, what do we need to do 145 00:12:43,288 --> 00:12:50,372 to begin with? Well, we know how to do this if we've got a 146 00:12:50,372 --> 00:12:56,444 unit coefficient with the X squared, so let's make it a unit 147 00:12:56,444 --> 00:13:02,010 coefficient by taking out the three as a common factor. So 148 00:13:02,010 --> 00:13:08,082 that's three brackets X squared, minus 3X. Now 50. What are we 149 00:13:08,082 --> 00:13:11,118 going to do with this? Well? 150 00:13:11,140 --> 00:13:15,568 We divide it by three in order that when we do the 151 00:13:15,568 --> 00:13:20,365 multiplication 3 * 50 over three will just give us back the 50. 152 00:13:20,900 --> 00:13:25,671 Now we look at this thing here in the bracket because this is 153 00:13:25,671 --> 00:13:29,341 now exactly the same sort of expression that we've had 154 00:13:29,341 --> 00:13:31,820 before. Equals 3. 155 00:13:32,510 --> 00:13:37,020 Let's have a big Curly bracket. I'm going to make 156 00:13:37,020 --> 00:13:41,981 this going to complete the square around this, so this is 157 00:13:41,981 --> 00:13:47,844 going to be X minus. Look at the coefficient of X and take 158 00:13:47,844 --> 00:13:51,452 half of it 3 over 2 all squared. 159 00:13:52,550 --> 00:13:55,436 By doing that, we've added on. 160 00:13:56,070 --> 00:14:03,714 An additional A squared, so we need to take that off 3 161 00:14:03,714 --> 00:14:11,358 over 2 squared and then finally plus 50 over 3 and close 162 00:14:11,358 --> 00:14:13,269 the big bracket. 163 00:14:14,820 --> 00:14:22,260 3. X minus three over 2 all squared minus 164 00:14:22,260 --> 00:14:29,820 nine over 4 + 50 over 3 and closed the big bracket. 165 00:14:30,460 --> 00:14:34,006 Now all we need to do now is put 166 00:14:34,006 --> 00:14:38,610 these together. And to do that we need a common denominator and 167 00:14:38,610 --> 00:14:41,745 the common denominator. Four and three is going to be 12. 168 00:14:43,040 --> 00:14:50,437 3 the Big Curly Bracket X minus three over 2 all squared minus 169 00:14:50,437 --> 00:14:57,265 over 12. We need to change the nine over 4 into 12. 170 00:14:58,410 --> 00:15:02,810 3/4 gave us 12 so 3 nines give us 27. 171 00:15:03,820 --> 00:15:09,221 Plus we need to change the 50 over 3 into twelfths. 172 00:15:09,820 --> 00:15:15,826 4 * 3 is 12, so 4 * 50 is 200. 173 00:15:16,370 --> 00:15:20,608 And now I've got a little bit of arithmetic to do. Let's just 174 00:15:20,608 --> 00:15:22,238 write back bracket down again. 175 00:15:23,810 --> 00:15:27,722 Equals 3. The Curly 176 00:15:27,722 --> 00:15:34,136 bracket. X minus three over 2 all squared. 177 00:15:35,240 --> 00:15:38,908 Minus 27 over 12. 178 00:15:40,060 --> 00:15:47,740 Plus 200 over 12 and that's the calculation that we need to 179 00:15:47,740 --> 00:15:55,420 do here to simplify 3 Curly bracket X minus three over 2 180 00:15:55,420 --> 00:16:01,860 or squared. Los all over 12 and we're going to take the 27 181 00:16:01,860 --> 00:16:07,460 away from the 200 is going to give us 173 and then we can 182 00:16:07,460 --> 00:16:09,060 close the bracket off. 183 00:16:11,140 --> 00:16:15,020 So despite the fact that the numbers were quite fearsome 184 00:16:15,020 --> 00:16:19,288 there, we've still ended up with a complete square, and we've 185 00:16:19,288 --> 00:16:23,556 automated the process so that what we're doing is looking at 186 00:16:23,556 --> 00:16:28,212 the coefficient of X. First of all, we check that what we've 187 00:16:28,212 --> 00:16:33,644 got the coefficient of X squared is one. If it's not, we take out 188 00:16:33,644 --> 00:16:35,972 the coefficient of X squared as 189 00:16:35,972 --> 00:16:41,298 a factor. Next we check the coefficient of X and we take a 190 00:16:41,298 --> 00:16:45,594 half of it, and that's the number that's going to go here 191 00:16:45,594 --> 00:16:46,668 inside the bracket. 192 00:16:47,590 --> 00:16:52,283 Then we must remember that we've got take off the square of that 193 00:16:52,283 --> 00:16:56,254 number is effectively we've added it back on, and then the 194 00:16:56,254 --> 00:16:57,698 rest is just arithmetic. 195 00:16:59,700 --> 00:17:03,669 So now we've developed this technique of completing the 196 00:17:03,669 --> 00:17:08,520 square. Let's use it to solve our original problem. If you 197 00:17:08,520 --> 00:17:14,694 remember we had X squared plus 6X is equal to four and we chose 198 00:17:14,694 --> 00:17:19,986 to write that as X squared plus 6X minus 4 equals 0. 199 00:17:20,630 --> 00:17:24,350 So first we need to check has it got a unit coefficient. 200 00:17:24,890 --> 00:17:29,960 And it has, so we don't need to take out a common factor. Now we 201 00:17:29,960 --> 00:17:34,354 look at the coefficient of X and it's 6 and it's half that 202 00:17:34,354 --> 00:17:39,086 coefficient that we want. So we need 3. So this is going to be 203 00:17:39,086 --> 00:17:43,328 X. Plus three all squared. 204 00:17:44,340 --> 00:17:51,634 In doing that, we have added on 3 squared, so we need to take 205 00:17:51,634 --> 00:17:53,718 off that 3 squared. 206 00:17:54,260 --> 00:17:57,172 And now we need to include the 207 00:17:57,172 --> 00:18:01,518 minus 4. So that we can maintain the quality of 208 00:18:01,518 --> 00:18:05,278 this xpression with this one and it's equal to 0. 209 00:18:06,290 --> 00:18:10,148 So X plus three all squared. 210 00:18:11,020 --> 00:18:17,840 Minus 3 squared. That's minus nine and minus 4 equals 0, 211 00:18:17,840 --> 00:18:25,280 so we can combine these X plus three all squared minus 13 212 00:18:25,280 --> 00:18:31,518 equals not. At the 13 to each side X plus three, all squared 213 00:18:31,518 --> 00:18:36,692 equals 13, and now we're in a position to take the square root 214 00:18:36,692 --> 00:18:41,866 of both sides. Because here on the left hand side we have a 215 00:18:41,866 --> 00:18:47,683 complete square. And so this is X +3 equals. Now 13 isn't a 216 00:18:47,683 --> 00:18:52,116 complete square. It's not a square number, so we have to 217 00:18:52,116 --> 00:18:57,355 write it as square root of 13. Or remembering when we take a 218 00:18:57,355 --> 00:19:01,788 square root of a number it's plus or minus Route 30. 219 00:19:02,330 --> 00:19:07,730 Now we take the three away from each side and we end up with our 220 00:19:07,730 --> 00:19:12,770 two roots. So we take the three away we have minus 3 + 13. 221 00:19:13,380 --> 00:19:18,330 Or minus 3 minus Route 30 and so that process of 222 00:19:18,330 --> 00:19:23,280 completing the square can be used to help us solve a 223 00:19:23,280 --> 00:19:26,880 quadratic equation. But that's not the real issue 224 00:19:26,880 --> 00:19:30,930 here. You can see another video on solving quadratic 225 00:19:30,930 --> 00:19:34,980 equations. The point is to master this technique of 226 00:19:34,980 --> 00:19:36,330 completing the square.