WEBVTT 00:00:01.870 --> 00:00:07.222 Going to have a look at a very simple process. It's called 00:00:07.222 --> 00:00:08.560 completing the square. 00:00:09.220 --> 00:00:14.414 In order to get to it and to show its potential use, I want 00:00:14.414 --> 00:00:18.124 to start with a simple equation. X squared equals 9. 00:00:19.900 --> 00:00:24.660 In order to find out what taxes we would take the square root of 00:00:24.660 --> 00:00:29.420 both sides, so the square root of X squared is just X and the 00:00:29.420 --> 00:00:32.820 square root of 9 is plus three or minus three. 00:00:33.350 --> 00:00:38.358 Be'cause minus three all squared is also 9. 00:00:38.870 --> 00:00:43.590 So that was relatively straightforward. Both of these 00:00:43.590 --> 00:00:46.540 two numbers were square numbers, 00:00:46.540 --> 00:00:49.556 complete squares. What if we 00:00:49.556 --> 00:00:53.290 got? X squared is equal to 00:00:53.290 --> 00:00:59.751 5. Do the same again, so we take the square root of 00:00:59.751 --> 00:01:05.283 each side X equals. Now 5 is not a square number, it 00:01:05.283 --> 00:01:08.971 does not have an exact square root, but 00:01:08.971 --> 00:01:14.042 nevertheless we can write it as root 5 that is exact 00:01:14.042 --> 00:01:15.886 or minus Route 5. 00:01:17.100 --> 00:01:21.940 So far so good. The same process is working each time. 00:01:22.710 --> 00:01:27.539 Let's have a look at something now, like X minus 7. 00:01:28.190 --> 00:01:31.090 All squared equals 3. 00:01:31.940 --> 00:01:36.857 How can we solve this? Well again, this side of the 00:01:36.857 --> 00:01:41.327 equation. We've got something called a complete square, so we 00:01:41.327 --> 00:01:47.585 can take the square root of each side. X minus Seven is equal to 00:01:47.585 --> 00:01:54.290 the square root of 3 or minus the square root of 3, and so we 00:01:54.290 --> 00:02:01.442 can now add 7 to each of these. So X is equal to 7 Plus Route 00:02:01.442 --> 00:02:02.783 3 or 7. 00:02:02.830 --> 00:02:04.609 Minus Route 3. 00:02:06.880 --> 00:02:14.608 I just have a look at another One X plus three, all squared is 00:02:14.608 --> 00:02:16.264 equal to 5. 00:02:18.670 --> 00:02:24.754 Again, this is a complete square, so again we can take the 00:02:24.754 --> 00:02:31.852 square root X +3 is equal to Route 5 or minus Route 5. Now 00:02:31.852 --> 00:02:38.443 to get X on its own, we need to take three away from 00:02:38.443 --> 00:02:45.541 each side, so we have X equals minus 3 + 5 or minus 3 00:02:45.541 --> 00:02:47.062 minus Route 5. 00:02:51.010 --> 00:02:57.738 What if we got X squared plus six 00:02:57.738 --> 00:03:00.261 X equals 4? 00:03:01.590 --> 00:03:07.440 Problem is this X squared plus 6X is not a complete square, so 00:03:07.440 --> 00:03:10.140 we can't just take the square 00:03:10.140 --> 00:03:15.560 root. So in terms of handling something like this, we've got 00:03:15.560 --> 00:03:18.458 to have a way of getting a 00:03:18.458 --> 00:03:22.081 complete square. So the process that we're going to 00:03:22.081 --> 00:03:24.425 be looking at it's called completing the square. 00:03:25.570 --> 00:03:31.330 The sorts of expressions that we have before will like this X 00:03:31.330 --> 00:03:33.250 plus a all squared. 00:03:34.030 --> 00:03:40.402 Or X minus a all squared, so that's what a complete square 00:03:40.402 --> 00:03:43.588 looks like. One of these two. 00:03:44.150 --> 00:03:51.038 So let's multiply this out and see what we get. So this is X 00:03:51.038 --> 00:03:54.482 plus a times by X plus A. 00:03:55.960 --> 00:04:00.160 And we do X times by X. That gives us X squared. 00:04:00.930 --> 00:04:05.430 And we do a Times by X, which gives us a X. 00:04:06.150 --> 00:04:12.100 And then with X times by a, which again gives us a X and 00:04:12.100 --> 00:04:18.475 then at the end a Times by a, which gives us a squared. And so 00:04:18.475 --> 00:04:21.450 we've X squared plus 2X plus A 00:04:21.450 --> 00:04:28.998 squared. I can do the same with this One X minus a Times by 00:04:28.998 --> 00:04:35.662 X minus A and it's going to give very similar results X times by 00:04:35.662 --> 00:04:42.802 X will give me X squared X times by minus A minus 8X minus 8 00:04:42.802 --> 00:04:49.466 times by X minus 8X minus 8 times by minus A plus a squared. 00:04:49.466 --> 00:04:52.322 So tidying up the two middle 00:04:52.322 --> 00:04:58.430 bits. Minus X minus X, minus 2X and then plus A squared. 00:04:58.430 --> 00:05:03.530 So this is what complete squares look like. They look 00:05:03.530 --> 00:05:06.080 like one of these two. 00:05:07.210 --> 00:05:13.570 Well. Can I make this look like a complete square in 00:05:13.570 --> 00:05:19.394 some way shape or form? If I compare this with this, what is 00:05:19.394 --> 00:05:21.186 it that I see? 00:05:21.850 --> 00:05:27.238 Well, perhaps one of things I might like to have is this 00:05:27.238 --> 00:05:32.626 written as just a function X squared plus 6X minus four? And 00:05:32.626 --> 00:05:37.565 let's not worry too much about solving an equation. What we 00:05:37.565 --> 00:05:42.055 want to concentrate on is this process of completing the 00:05:42.055 --> 00:05:46.545 square, so I'm going to take this quadratic function X 00:05:46.545 --> 00:05:52.382 squared plus 6X minus four, and I'm going to compare it with the 00:05:52.382 --> 00:05:58.530 complete square. X squared plus 2X plus 00:05:58.530 --> 00:06:04.007 A squared. Now The X squared so the same. 00:06:05.210 --> 00:06:11.762 6X2A X I've got to have these two terms the same. They've got 00:06:11.762 --> 00:06:17.810 to match. They've got to be exactly the same, and that means 00:06:17.810 --> 00:06:21.338 that the six has to be equal 00:06:21.338 --> 00:06:28.794 to 2A. And of course, that tells us that the A 00:06:28.794 --> 00:06:31.378 is equal to 3. 00:06:32.750 --> 00:06:39.432 So if I make a equal to three, then I've got plus A 00:06:39.432 --> 00:06:42.002 squared on the end +9. 00:06:43.300 --> 00:06:49.360 So. I can look at this first bit and I can make it equal to that. 00:06:50.070 --> 00:06:56.800 So let's write this down X squared plus 6X minus 00:06:56.800 --> 00:07:03.934 4 equals. X plus three all squared. Remember this 00:07:03.934 --> 00:07:10.854 is X plus three all squared. Now I'm replacing the 00:07:10.854 --> 00:07:12.930 A by three. 00:07:13.830 --> 00:07:18.954 Now what more of I got? Well, I've added on a squared, so I've 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 00:07:24.444 --> 00:07:26.640 just take it away, minus 3 00:07:26.640 --> 00:07:31.818 squared. And then I can keep this minus four at the end as it 00:07:31.818 --> 00:07:39.568 was. So now I've X plus three all squared minus 9 - 00:07:39.568 --> 00:07:45.848 4 gives me X plus three all squared minus 30. 00:07:47.450 --> 00:07:52.440 So I have completed the square. I made this bit. 00:07:53.790 --> 00:07:57.570 Part of a complete square. 00:07:58.450 --> 00:08:03.500 And I've done it by comparing the coefficient of X. 00:08:04.040 --> 00:08:10.270 With one of the two standard forms and I saw that what I had 00:08:10.270 --> 00:08:14.275 to do was take half the coefficient of X. 00:08:15.750 --> 00:08:22.900 Let's have a look then at another example, X squared minus 00:08:22.900 --> 00:08:30.700 8X plus Seven and I want to write this so it's got 00:08:30.700 --> 00:08:33.950 a complete square in it. 00:08:34.640 --> 00:08:39.770 Well, one of the standard forms for the complete square 00:08:39.770 --> 00:08:45.413 that we had was X squared minus 2X plus A squared. 00:08:47.900 --> 00:08:50.994 And I want to make these two 00:08:50.994 --> 00:08:52.830 terms. The same. 00:08:54.100 --> 00:08:57.948 So again, we can see that the A. 00:08:58.800 --> 00:09:05.464 Has got to be 4 because minus 8 is minus 2 times by 4. 00:09:08.090 --> 00:09:15.520 So we've got X squared minus 8X plus Seven is 00:09:15.520 --> 00:09:21.930 equal to. X minus four all squared. 00:09:22.890 --> 00:09:27.954 So I've ensured I've got the X squared. I've ensured that I've 00:09:27.954 --> 00:09:33.862 got the minus 8X, but I've also added on a squared, so I've got 00:09:33.862 --> 00:09:39.348 a squared too much, so I must take away 4 squared and then 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. 00:09:46.310 --> 00:09:52.538 And so this is now X minus four all 00:09:52.538 --> 00:09:58.766 squared minus 16 + 7 X minus four all 00:09:58.766 --> 00:10:00.842 squared minus 9. 00:10:02.140 --> 00:10:06.090 Let's take one more example. 00:10:06.110 --> 00:10:08.350 X 00:10:08.350 --> 00:10:14.979 squared Plus 5X plus three and let's see if we 00:10:14.979 --> 00:10:19.610 can follow this one through without having to write down the 00:10:19.610 --> 00:10:23.399 comparison. In other words, by doing it by inspection. 00:10:24.050 --> 00:10:30.342 What do we need? We need a complete square, so we need X 00:10:30.342 --> 00:10:37.118 and we look at this number here. The coefficient of the X turn on 00:10:37.118 --> 00:10:39.054 this left hand side. 00:10:39.770 --> 00:10:45.152 We want half that coefficient, so we want five over 2 and we've 00:10:45.152 --> 00:10:50.534 got a plus sign, so it's got to be X +5 over 2. 00:10:51.220 --> 00:10:57.382 If we were to multiply out this bracket, we would be adding on 00:10:57.382 --> 00:11:03.544 an additional A squared where five over 2 is the a. So we've 00:11:03.544 --> 00:11:10.180 got to take that away. Takeaway 5 over 2 squared and then at the 00:11:10.180 --> 00:11:12.550 end we've got plus 3. 00:11:13.340 --> 00:11:19.812 So this gives us X +5 over 2 00:11:19.812 --> 00:11:26.284 or squared minus 25 over 4 + 3. 00:11:27.330 --> 00:11:33.523 And of course, we'd like to combine these numbers at this 00:11:33.523 --> 00:11:41.405 end X +5 over 2 all squared minus 25 over 4 plus. Now we 00:11:41.405 --> 00:11:48.161 need to convert these to quarters as well, so three is 12 00:11:48.161 --> 00:11:54.741 quarters. Now we can combine these, since they're both in 00:11:54.741 --> 00:12:02.049 terms of quarters X +5 over 2, all squared minus 13 over 00:12:02.049 --> 00:12:04.480 4. And we'd leave you like that. 00:12:05.510 --> 00:12:09.272 This process now seems to be working quite well. 00:12:09.800 --> 00:12:13.140 But of course, we haven't dealt with every kind of 00:12:13.140 --> 00:12:16.146 quadratic expression we could have, because so far we've 00:12:16.146 --> 00:12:19.820 only had a unit coefficient here in front of the X 00:12:19.820 --> 00:12:22.826 squared. We haven't had another number like two or 00:12:22.826 --> 00:12:26.834 three or whatever, so let's have a look at what we would 00:12:26.834 --> 00:12:28.170 do in that case. 00:12:30.900 --> 00:12:35.208 So we have three X squared. 00:12:36.250 --> 00:12:43.288 Minus 9X. Plus 50, what do we need to do 00:12:43.288 --> 00:12:50.372 to begin with? Well, we know how to do this if we've got a 00:12:50.372 --> 00:12:56.444 unit coefficient with the X squared, so let's make it a unit 00:12:56.444 --> 00:13:02.010 coefficient by taking out the three as a common factor. So 00:13:02.010 --> 00:13:08.082 that's three brackets X squared, minus 3X. Now 50. What are we 00:13:08.082 --> 00:13:11.118 going to do with this? Well? 00:13:11.140 --> 00:13:15.568 We divide it by three in order that when we do the 00:13:15.568 --> 00:13:20.365 multiplication 3 * 50 over three will just give us back the 50. 00:13:20.900 --> 00:13:25.671 Now we look at this thing here in the bracket because this is 00:13:25.671 --> 00:13:29.341 now exactly the same sort of expression that we've had 00:13:29.341 --> 00:13:31.820 before. Equals 3. 00:13:32.510 --> 00:13:37.020 Let's have a big Curly bracket. I'm going to make 00:13:37.020 --> 00:13:41.981 this going to complete the square around this, so this is 00:13:41.981 --> 00:13:47.844 going to be X minus. Look at the coefficient of X and take 00:13:47.844 --> 00:13:51.452 half of it 3 over 2 all squared. 00:13:52.550 --> 00:13:55.436 By doing that, we've added on. 00:13:56.070 --> 00:14:03.714 An additional A squared, so we need to take that off 3 00:14:03.714 --> 00:14:11.358 over 2 squared and then finally plus 50 over 3 and close 00:14:11.358 --> 00:14:13.269 the big bracket. 00:14:14.820 --> 00:14:22.260 3. X minus three over 2 all squared minus 00:14:22.260 --> 00:14:29.820 nine over 4 + 50 over 3 and closed the big bracket. 00:14:30.460 --> 00:14:34.006 Now all we need to do now is put 00:14:34.006 --> 00:14:38.610 these together. And to do that we need a common denominator and 00:14:38.610 --> 00:14:41.745 the common denominator. Four and three is going to be 12. 00:14:43.040 --> 00:14:50.437 3 the Big Curly Bracket X minus three over 2 all squared minus 00:14:50.437 --> 00:14:57.265 over 12. We need to change the nine over 4 into 12. 00:14:58.410 --> 00:15:02.810 3/4 gave us 12 so 3 nines give us 27. 00:15:03.820 --> 00:15:09.221 Plus we need to change the 50 over 3 into twelfths. 00:15:09.820 --> 00:15:15.826 4 * 3 is 12, so 4 * 50 is 200. 00:15:16.370 --> 00:15:20.608 And now I've got a little bit of arithmetic to do. Let's just 00:15:20.608 --> 00:15:22.238 write back bracket down again. 00:15:23.810 --> 00:15:27.722 Equals 3. The Curly 00:15:27.722 --> 00:15:34.136 bracket. X minus three over 2 all squared. 00:15:35.240 --> 00:15:38.908 Minus 27 over 12. 00:15:40.060 --> 00:15:47.740 Plus 200 over 12 and that's the calculation that we need to 00:15:47.740 --> 00:15:55.420 do here to simplify 3 Curly bracket X minus three over 2 00:15:55.420 --> 00:16:01.860 or squared. Los all over 12 and we're going to take the 27 00:16:01.860 --> 00:16:07.460 away from the 200 is going to give us 173 and then we can 00:16:07.460 --> 00:16:09.060 close the bracket off. 00:16:11.140 --> 00:16:15.020 So despite the fact that the numbers were quite fearsome 00:16:15.020 --> 00:16:19.288 there, we've still ended up with a complete square, and we've 00:16:19.288 --> 00:16:23.556 automated the process so that what we're doing is looking at 00:16:23.556 --> 00:16:28.212 the coefficient of X. First of all, we check that what we've 00:16:28.212 --> 00:16:33.644 got the coefficient of X squared is one. If it's not, we take out 00:16:33.644 --> 00:16:35.972 the coefficient of X squared as 00:16:35.972 --> 00:16:41.298 a factor. Next we check the coefficient of X and we take a 00:16:41.298 --> 00:16:45.594 half of it, and that's the number that's going to go here 00:16:45.594 --> 00:16:46.668 inside the bracket. 00:16:47.590 --> 00:16:52.283 Then we must remember that we've got take off the square of that 00:16:52.283 --> 00:16:56.254 number is effectively we've added it back on, and then the 00:16:56.254 --> 00:16:57.698 rest is just arithmetic. 00:16:59.700 --> 00:17:03.669 So now we've developed this technique of completing the 00:17:03.669 --> 00:17:08.520 square. Let's use it to solve our original problem. If you 00:17:08.520 --> 00:17:14.694 remember we had X squared plus 6X is equal to four and we chose 00:17:14.694 --> 00:17:19.986 to write that as X squared plus 6X minus 4 equals 0. 00:17:20.630 --> 00:17:24.350 So first we need to check has it got a unit coefficient. 00:17:24.890 --> 00:17:29.960 And it has, so we don't need to take out a common factor. Now we 00:17:29.960 --> 00:17:34.354 look at the coefficient of X and it's 6 and it's half that 00:17:34.354 --> 00:17:39.086 coefficient that we want. So we need 3. So this is going to be 00:17:39.086 --> 00:17:43.328 X. Plus three all squared. 00:17:44.340 --> 00:17:51.634 In doing that, we have added on 3 squared, so we need to take 00:17:51.634 --> 00:17:53.718 off that 3 squared. 00:17:54.260 --> 00:17:57.172 And now we need to include the 00:17:57.172 --> 00:18:01.518 minus 4. So that we can maintain the quality of 00:18:01.518 --> 00:18:05.278 this xpression with this one and it's equal to 0. 00:18:06.290 --> 00:18:10.148 So X plus three all squared. 00:18:11.020 --> 00:18:17.840 Minus 3 squared. That's minus nine and minus 4 equals 0, 00:18:17.840 --> 00:18:25.280 so we can combine these X plus three all squared minus 13 00:18:25.280 --> 00:18:31.518 equals not. At the 13 to each side X plus three, all squared 00:18:31.518 --> 00:18:36.692 equals 13, and now we're in a position to take the square root 00:18:36.692 --> 00:18:41.866 of both sides. Because here on the left hand side we have a 00:18:41.866 --> 00:18:47.683 complete square. And so this is X +3 equals. Now 13 isn't a 00:18:47.683 --> 00:18:52.116 complete square. It's not a square number, so we have to 00:18:52.116 --> 00:18:57.355 write it as square root of 13. Or remembering when we take a 00:18:57.355 --> 00:19:01.788 square root of a number it's plus or minus Route 30. 00:19:02.330 --> 00:19:07.730 Now we take the three away from each side and we end up with our 00:19:07.730 --> 00:19:12.770 two roots. So we take the three away we have minus 3 + 13. 00:19:13.380 --> 00:19:18.330 Or minus 3 minus Route 30 and so that process of 00:19:18.330 --> 00:19:23.280 completing the square can be used to help us solve a 00:19:23.280 --> 00:19:26.880 quadratic equation. But that's not the real issue 00:19:26.880 --> 00:19:30.930 here. You can see another video on solving quadratic 00:19:30.930 --> 00:19:34.980 equations. The point is to master this technique of 00:19:34.980 --> 00:19:36.330 completing the square.