0:00:01.870,0:00:07.222 Going to have a look at a very[br]simple process. It's called 0:00:07.222,0:00:08.560 completing the square. 0:00:09.220,0:00:14.414 In order to get to it and to[br]show its potential use, I want 0:00:14.414,0:00:18.124 to start with a simple equation.[br]X squared equals 9. 0:00:19.900,0:00:24.660 In order to find out what taxes[br]we would take the square root of 0:00:24.660,0:00:29.420 both sides, so the square root[br]of X squared is just X and the 0:00:29.420,0:00:32.820 square root of 9 is plus three[br]or minus three. 0:00:33.350,0:00:38.358 Be'cause minus three all squared[br]is also 9. 0:00:38.870,0:00:43.590 So that was relatively[br]straightforward. Both of these 0:00:43.590,0:00:46.540 two numbers were square numbers, 0:00:46.540,0:00:49.556 complete squares. What if we 0:00:49.556,0:00:53.290 got? X squared is equal to 0:00:53.290,0:00:59.751 5. Do the same again, so[br]we take the square root of 0:00:59.751,0:01:05.283 each side X equals. Now 5[br]is not a square number, it 0:01:05.283,0:01:08.971 does not have an exact[br]square root, but 0:01:08.971,0:01:14.042 nevertheless we can write[br]it as root 5 that is exact 0:01:14.042,0:01:15.886 or minus Route 5. 0:01:17.100,0:01:21.940 So far so good. The same process[br]is working each time. 0:01:22.710,0:01:27.539 Let's have a look at something[br]now, like X minus 7. 0:01:28.190,0:01:31.090 All squared equals 3. 0:01:31.940,0:01:36.857 How can we solve this? Well[br]again, this side of the 0:01:36.857,0:01:41.327 equation. We've got something[br]called a complete square, so we 0:01:41.327,0:01:47.585 can take the square root of each[br]side. X minus Seven is equal to 0:01:47.585,0:01:54.290 the square root of 3 or minus[br]the square root of 3, and so we 0:01:54.290,0:02:01.442 can now add 7 to each of these.[br]So X is equal to 7 Plus Route 0:02:01.442,0:02:02.783 3 or 7. 0:02:02.830,0:02:04.609 Minus Route 3. 0:02:06.880,0:02:14.608 I just have a look at another[br]One X plus three, all squared is 0:02:14.608,0:02:16.264 equal to 5. 0:02:18.670,0:02:24.754 Again, this is a complete[br]square, so again we can take the 0:02:24.754,0:02:31.852 square root X +3 is equal to[br]Route 5 or minus Route 5. Now 0:02:31.852,0:02:38.443 to get X on its own, we[br]need to take three away from 0:02:38.443,0:02:45.541 each side, so we have X equals[br]minus 3 + 5 or minus 3 0:02:45.541,0:02:47.062 minus Route 5. 0:02:51.010,0:02:57.738 What if we got[br]X squared plus six 0:02:57.738,0:03:00.261 X equals 4? 0:03:01.590,0:03:07.440 Problem is this X squared plus[br]6X is not a complete square, so 0:03:07.440,0:03:10.140 we can't just take the square 0:03:10.140,0:03:15.560 root. So in terms of handling[br]something like this, we've got 0:03:15.560,0:03:18.458 to have a way of getting a 0:03:18.458,0:03:22.081 complete square. So the[br]process that we're going to 0:03:22.081,0:03:24.425 be looking at it's called[br]completing the square. 0:03:25.570,0:03:31.330 The sorts of expressions that we[br]have before will like this X 0:03:31.330,0:03:33.250 plus a all squared. 0:03:34.030,0:03:40.402 Or X minus a all squared, so[br]that's what a complete square 0:03:40.402,0:03:43.588 looks like. One of these two. 0:03:44.150,0:03:51.038 So let's multiply this out and[br]see what we get. So this is X 0:03:51.038,0:03:54.482 plus a times by X plus A. 0:03:55.960,0:04:00.160 And we do X times by X. That[br]gives us X squared. 0:04:00.930,0:04:05.430 And we do a Times by X, which[br]gives us a X. 0:04:06.150,0:04:12.100 And then with X times by a,[br]which again gives us a X and 0:04:12.100,0:04:18.475 then at the end a Times by a,[br]which gives us a squared. And so 0:04:18.475,0:04:21.450 we've X squared plus 2X plus A 0:04:21.450,0:04:28.998 squared. I can do the same[br]with this One X minus a Times by 0:04:28.998,0:04:35.662 X minus A and it's going to give[br]very similar results X times by 0:04:35.662,0:04:42.802 X will give me X squared X times[br]by minus A minus 8X minus 8 0:04:42.802,0:04:49.466 times by X minus 8X minus 8[br]times by minus A plus a squared. 0:04:49.466,0:04:52.322 So tidying up the two middle 0:04:52.322,0:04:58.430 bits. Minus X minus X, minus[br]2X and then plus A squared. 0:04:58.430,0:05:03.530 So this is what complete[br]squares look like. They look 0:05:03.530,0:05:06.080 like one of these two. 0:05:07.210,0:05:13.570 Well. Can I make this[br]look like a complete square in 0:05:13.570,0:05:19.394 some way shape or form? If I[br]compare this with this, what is 0:05:19.394,0:05:21.186 it that I see? 0:05:21.850,0:05:27.238 Well, perhaps one of things I[br]might like to have is this 0:05:27.238,0:05:32.626 written as just a function X[br]squared plus 6X minus four? And 0:05:32.626,0:05:37.565 let's not worry too much about[br]solving an equation. What we 0:05:37.565,0:05:42.055 want to concentrate on is this[br]process of completing the 0:05:42.055,0:05:46.545 square, so I'm going to take[br]this quadratic function X 0:05:46.545,0:05:52.382 squared plus 6X minus four, and[br]I'm going to compare it with the 0:05:52.382,0:05:58.530 complete square. X squared[br]plus 2X plus 0:05:58.530,0:06:04.007 A squared. Now The[br]X squared so the same. 0:06:05.210,0:06:11.762 6X2A X I've got to have these[br]two terms the same. They've got 0:06:11.762,0:06:17.810 to match. They've got to be[br]exactly the same, and that means 0:06:17.810,0:06:21.338 that the six has to be equal 0:06:21.338,0:06:28.794 to 2A. And of course,[br]that tells us that the A 0:06:28.794,0:06:31.378 is equal to 3. 0:06:32.750,0:06:39.432 So if I make a equal to[br]three, then I've got plus A 0:06:39.432,0:06:42.002 squared on the end +9. 0:06:43.300,0:06:49.360 So. I can look at this first bit[br]and I can make it equal to that. 0:06:50.070,0:06:56.800 So let's write this down[br]X squared plus 6X minus 0:06:56.800,0:07:03.934 4 equals. X plus[br]three all squared. Remember this 0:07:03.934,0:07:10.854 is X plus three all[br]squared. Now I'm replacing the 0:07:10.854,0:07:12.930 A by three. 0:07:13.830,0:07:18.954 Now what more of I got? Well,[br]I've added on a squared, so I've 0:07:18.954,0:07:24.444 added on 9. So I've got some how[br]to get rid of that. Well, let's 0:07:24.444,0:07:26.640 just take it away, minus 3 0:07:26.640,0:07:31.818 squared. And then I can keep[br]this minus four at the end as it 0:07:31.818,0:07:39.568 was. So now I've X plus[br]three all squared minus 9 - 0:07:39.568,0:07:45.848 4 gives me X plus three[br]all squared minus 30. 0:07:47.450,0:07:52.440 So I have completed the square.[br]I made this bit. 0:07:53.790,0:07:57.570 Part of a complete square. 0:07:58.450,0:08:03.500 And I've done it by comparing[br]the coefficient of X. 0:08:04.040,0:08:10.270 With one of the two standard[br]forms and I saw that what I had 0:08:10.270,0:08:14.275 to do was take half the[br]coefficient of X. 0:08:15.750,0:08:22.900 Let's have a look then at[br]another example, X squared minus 0:08:22.900,0:08:30.700 8X plus Seven and I want[br]to write this so it's got 0:08:30.700,0:08:33.950 a complete square in it. 0:08:34.640,0:08:39.770 Well, one of the standard[br]forms for the complete square 0:08:39.770,0:08:45.413 that we had was X squared[br]minus 2X plus A squared. 0:08:47.900,0:08:50.994 And I want to make these two 0:08:50.994,0:08:52.830 terms. The same. 0:08:54.100,0:08:57.948 So again, we can see that the A. 0:08:58.800,0:09:05.464 Has got to be 4 because minus 8[br]is minus 2 times by 4. 0:09:08.090,0:09:15.520 So we've got X squared[br]minus 8X plus Seven is 0:09:15.520,0:09:21.930 equal to. X minus[br]four all squared. 0:09:22.890,0:09:27.954 So I've ensured I've got the X[br]squared. I've ensured that I've 0:09:27.954,0:09:33.862 got the minus 8X, but I've also[br]added on a squared, so I've got 0:09:33.862,0:09:39.348 a squared too much, so I must[br]take away 4 squared and then 0:09:39.348,0:09:45.678 I've got the 7:00 that I need to[br]add on to keep the equal sign. 0:09:46.310,0:09:52.538 And so this is now[br]X minus four all 0:09:52.538,0:09:58.766 squared minus 16 +[br]7 X minus four all 0:09:58.766,0:10:00.842 squared minus 9. 0:10:02.140,0:10:06.090 Let's take one more example. 0:10:06.110,0:10:08.350 X 0:10:08.350,0:10:14.979 squared Plus 5X[br]plus three and let's see if we 0:10:14.979,0:10:19.610 can follow this one through[br]without having to write down the 0:10:19.610,0:10:23.399 comparison. In other words, by[br]doing it by inspection. 0:10:24.050,0:10:30.342 What do we need? We need a[br]complete square, so we need X 0:10:30.342,0:10:37.118 and we look at this number here.[br]The coefficient of the X turn on 0:10:37.118,0:10:39.054 this left hand side. 0:10:39.770,0:10:45.152 We want half that coefficient,[br]so we want five over 2 and we've 0:10:45.152,0:10:50.534 got a plus sign, so it's got to[br]be X +5 over 2. 0:10:51.220,0:10:57.382 If we were to multiply out this[br]bracket, we would be adding on 0:10:57.382,0:11:03.544 an additional A squared where[br]five over 2 is the a. So we've 0:11:03.544,0:11:10.180 got to take that away. Takeaway[br]5 over 2 squared and then at the 0:11:10.180,0:11:12.550 end we've got plus 3. 0:11:13.340,0:11:19.812 So this gives us[br]X +5 over 2 0:11:19.812,0:11:26.284 or squared minus 25[br]over 4 + 3. 0:11:27.330,0:11:33.523 And of course, we'd like to[br]combine these numbers at this 0:11:33.523,0:11:41.405 end X +5 over 2 all squared[br]minus 25 over 4 plus. Now we 0:11:41.405,0:11:48.161 need to convert these to[br]quarters as well, so three is 12 0:11:48.161,0:11:54.741 quarters. Now we can combine[br]these, since they're both in 0:11:54.741,0:12:02.049 terms of quarters X +5 over[br]2, all squared minus 13 over 0:12:02.049,0:12:04.480 4. And we'd leave you like that. 0:12:05.510,0:12:09.272 This process now seems to be[br]working quite well. 0:12:09.800,0:12:13.140 But of course, we haven't[br]dealt with every kind of 0:12:13.140,0:12:16.146 quadratic expression we could[br]have, because so far we've 0:12:16.146,0:12:19.820 only had a unit coefficient[br]here in front of the X 0:12:19.820,0:12:22.826 squared. We haven't had[br]another number like two or 0:12:22.826,0:12:26.834 three or whatever, so let's[br]have a look at what we would 0:12:26.834,0:12:28.170 do in that case. 0:12:30.900,0:12:35.208 So we have three X[br]squared. 0:12:36.250,0:12:43.288 Minus 9X. Plus[br]50, what do we need to do 0:12:43.288,0:12:50.372 to begin with? Well, we know how[br]to do this if we've got a 0:12:50.372,0:12:56.444 unit coefficient with the X[br]squared, so let's make it a unit 0:12:56.444,0:13:02.010 coefficient by taking out the[br]three as a common factor. So 0:13:02.010,0:13:08.082 that's three brackets X squared,[br]minus 3X. Now 50. What are we 0:13:08.082,0:13:11.118 going to do with this? Well? 0:13:11.140,0:13:15.568 We divide it by three in order[br]that when we do the 0:13:15.568,0:13:20.365 multiplication 3 * 50 over three[br]will just give us back the 50. 0:13:20.900,0:13:25.671 Now we look at this thing here[br]in the bracket because this is 0:13:25.671,0:13:29.341 now exactly the same sort of[br]expression that we've had 0:13:29.341,0:13:31.820 before. Equals 3. 0:13:32.510,0:13:37.020 Let's have a big Curly[br]bracket. I'm going to make 0:13:37.020,0:13:41.981 this going to complete the[br]square around this, so this is 0:13:41.981,0:13:47.844 going to be X minus. Look at[br]the coefficient of X and take 0:13:47.844,0:13:51.452 half of it 3 over 2 all[br]squared. 0:13:52.550,0:13:55.436 By doing that, we've added on. 0:13:56.070,0:14:03.714 An additional A squared, so we[br]need to take that off 3 0:14:03.714,0:14:11.358 over 2 squared and then finally[br]plus 50 over 3 and close 0:14:11.358,0:14:13.269 the big bracket. 0:14:14.820,0:14:22.260 3. X minus[br]three over 2 all squared minus 0:14:22.260,0:14:29.820 nine over 4 + 50 over[br]3 and closed the big bracket. 0:14:30.460,0:14:34.006 Now all we need to do now is put 0:14:34.006,0:14:38.610 these together. And to do that[br]we need a common denominator and 0:14:38.610,0:14:41.745 the common denominator. Four and[br]three is going to be 12. 0:14:43.040,0:14:50.437 3 the Big Curly Bracket X minus[br]three over 2 all squared minus 0:14:50.437,0:14:57.265 over 12. We need to change the[br]nine over 4 into 12. 0:14:58.410,0:15:02.810 3/4 gave us 12 so 3[br]nines give us 27. 0:15:03.820,0:15:09.221 Plus we need to change the 50[br]over 3 into twelfths. 0:15:09.820,0:15:15.826 4 * 3 is 12, so 4[br]* 50 is 200. 0:15:16.370,0:15:20.608 And now I've got a little bit of[br]arithmetic to do. Let's just 0:15:20.608,0:15:22.238 write back bracket down again. 0:15:23.810,0:15:27.722 Equals 3. The Curly 0:15:27.722,0:15:34.136 bracket. X minus three[br]over 2 all squared. 0:15:35.240,0:15:38.908 Minus 27 over 12. 0:15:40.060,0:15:47.740 Plus 200 over 12 and that's[br]the calculation that we need to 0:15:47.740,0:15:55.420 do here to simplify 3 Curly[br]bracket X minus three over 2 0:15:55.420,0:16:01.860 or squared. Los all over 12[br]and we're going to take the 27 0:16:01.860,0:16:07.460 away from the 200 is going to[br]give us 173 and then we can 0:16:07.460,0:16:09.060 close the bracket off. 0:16:11.140,0:16:15.020 So despite the fact that the[br]numbers were quite fearsome 0:16:15.020,0:16:19.288 there, we've still ended up with[br]a complete square, and we've 0:16:19.288,0:16:23.556 automated the process so that[br]what we're doing is looking at 0:16:23.556,0:16:28.212 the coefficient of X. First of[br]all, we check that what we've 0:16:28.212,0:16:33.644 got the coefficient of X squared[br]is one. If it's not, we take out 0:16:33.644,0:16:35.972 the coefficient of X squared as 0:16:35.972,0:16:41.298 a factor. Next we check the[br]coefficient of X and we take a 0:16:41.298,0:16:45.594 half of it, and that's the[br]number that's going to go here 0:16:45.594,0:16:46.668 inside the bracket. 0:16:47.590,0:16:52.283 Then we must remember that we've[br]got take off the square of that 0:16:52.283,0:16:56.254 number is effectively we've[br]added it back on, and then the 0:16:56.254,0:16:57.698 rest is just arithmetic. 0:16:59.700,0:17:03.669 So now we've developed this[br]technique of completing the 0:17:03.669,0:17:08.520 square. Let's use it to solve[br]our original problem. If you 0:17:08.520,0:17:14.694 remember we had X squared plus[br]6X is equal to four and we chose 0:17:14.694,0:17:19.986 to write that as X squared plus[br]6X minus 4 equals 0. 0:17:20.630,0:17:24.350 So first we need to check has it[br]got a unit coefficient. 0:17:24.890,0:17:29.960 And it has, so we don't need to[br]take out a common factor. Now we 0:17:29.960,0:17:34.354 look at the coefficient of X and[br]it's 6 and it's half that 0:17:34.354,0:17:39.086 coefficient that we want. So we[br]need 3. So this is going to be 0:17:39.086,0:17:43.328 X. Plus three all[br]squared. 0:17:44.340,0:17:51.634 In doing that, we have added on[br]3 squared, so we need to take 0:17:51.634,0:17:53.718 off that 3 squared. 0:17:54.260,0:17:57.172 And now we need to include the 0:17:57.172,0:18:01.518 minus 4. So that we can[br]maintain the quality of 0:18:01.518,0:18:05.278 this xpression with this[br]one and it's equal to 0. 0:18:06.290,0:18:10.148 So X plus three all squared. 0:18:11.020,0:18:17.840 Minus 3 squared. That's minus[br]nine and minus 4 equals 0, 0:18:17.840,0:18:25.280 so we can combine these X[br]plus three all squared minus 13 0:18:25.280,0:18:31.518 equals not. At the 13 to each[br]side X plus three, all squared 0:18:31.518,0:18:36.692 equals 13, and now we're in a[br]position to take the square root 0:18:36.692,0:18:41.866 of both sides. Because here on[br]the left hand side we have a 0:18:41.866,0:18:47.683 complete square. And so this is[br]X +3 equals. Now 13 isn't a 0:18:47.683,0:18:52.116 complete square. It's not a[br]square number, so we have to 0:18:52.116,0:18:57.355 write it as square root of 13.[br]Or remembering when we take a 0:18:57.355,0:19:01.788 square root of a number it's[br]plus or minus Route 30. 0:19:02.330,0:19:07.730 Now we take the three away from[br]each side and we end up with our 0:19:07.730,0:19:12.770 two roots. So we take the three[br]away we have minus 3 + 13. 0:19:13.380,0:19:18.330 Or minus 3 minus Route 30[br]and so that process of 0:19:18.330,0:19:23.280 completing the square can be[br]used to help us solve a 0:19:23.280,0:19:26.880 quadratic equation. But[br]that's not the real issue 0:19:26.880,0:19:30.930 here. You can see another[br]video on solving quadratic 0:19:30.930,0:19:34.980 equations. The point is to[br]master this technique of 0:19:34.980,0:19:36.330 completing the square.