WEBVTT 00:00:00.290 --> 00:00:04.866 Completing the square is a process that we make use of in a 00:00:04.866 --> 00:00:09.442 number of ways. First, we can make use of it to find maximum 00:00:09.442 --> 00:00:13.314 and minimum values of quadratic functions, second we can make 00:00:13.314 --> 00:00:17.186 use of it to simplify or change algebraic expressions in order 00:00:17.186 --> 00:00:21.410 to be able to calculate the value that they have. Third, we 00:00:21.410 --> 00:00:24.930 can use it for solving quadratic equations. In this particular 00:00:24.930 --> 00:00:29.506 video, we're going to have a look at it for finding max- and 00:00:29.506 --> 00:00:32.210 min-imum values of functions, quadratic functions. 00:00:32.210 --> 00:00:36.138 Let's begin by looking at a very specific example. 00:00:36.138 --> 00:00:39.195 Supposing we've got x squared, 00:00:39.195 --> 00:00:41.950 plus 5x, 00:00:42.460 --> 00:00:45.310 minus 2. Now. 00:00:46.140 --> 00:00:51.090 x squared, it's positive, so one of the things that we do know is 00:00:51.090 --> 00:00:54.170 that if we were to sketch the graph of this function. 00:00:54.980 --> 00:00:57.290 It would look something perhaps 00:00:57.290 --> 00:01:02.729 like that. Question is where's this point down here? 00:01:03.580 --> 00:01:07.771 Where's the minimum value of this function? What value of x 00:01:07.771 --> 00:01:12.724 does it have? Does it actually come below the x-axis as I've 00:01:12.724 --> 00:01:17.677 drawn it, or does it come up here somewhere? At what value of X 00:01:17.677 --> 00:01:22.249 does that minimum value occur? We could use calculus if we knew 00:01:22.249 --> 00:01:26.440 calculus, but sometimes we don't know calculus. We might not have 00:01:26.440 --> 00:01:27.583 reached it yet. 00:01:28.420 --> 00:01:31.907 At other occasions it might be rather like using a sledgehammer 00:01:31.907 --> 00:01:36.662 to crack or not, so let's have a look at how we can deal with 00:01:36.662 --> 00:01:37.930 this kind of function. 00:01:38.720 --> 00:01:42.240 What we're going to do is a process known as 00:01:42.380 --> 00:01:50.110 "completing ... the ... square" 00:01:51.970 --> 00:01:55.267 OK, "completing the square", what does that mean? 00:01:55.447 --> 00:02:00.137 Well, let's have a look at something that is a "complete square". 00:02:00.297 --> 00:02:05.045 That is, an exact square. 00:02:06.110 --> 00:02:12.662 So that's a complete and exact square. If we multiply out the 00:02:12.662 --> 00:02:20.306 brackets, x plus a times by x plus a, what we end up with 00:02:20.306 --> 00:02:21.944 is x squared... 00:02:22.690 --> 00:02:25.070 that's x times by x... 00:02:26.320 --> 00:02:33.488 a times by x, and of course x times by a, so that gives 00:02:33.488 --> 00:02:38.336 us 2ax, and then finally a times by a... 00:02:38.386 --> 00:02:40.656 and that gives us a squared. 00:02:40.656 --> 00:02:46.288 So this expression is a complete square, a complete 00:02:46.288 --> 00:02:51.408 and exact square. Because it's "x plus a" all squared. 00:02:52.410 --> 00:02:57.369 Similarly, we can have "x minus a" all squared. 00:02:58.339 --> 00:03:00.638 And if we multiply out, these brackets 00:03:00.778 --> 00:03:06.798 we will end up with the same result, except, we will have 00:03:06.798 --> 00:03:12.562 minus 2ax plus a squared. And again this is a complete 00:03:12.562 --> 00:03:17.278 square an exact square because it's equal to x minus a... 00:03:17.278 --> 00:03:19.374 all squared. 00:03:20.790 --> 00:03:23.880 So,... we go back to this. 00:03:24.450 --> 00:03:28.861 Expression here x squared, plus 5x, minus two and what we're 00:03:28.861 --> 00:03:33.673 going to do is complete the square. In other words we're going to 00:03:33.673 --> 00:03:38.886 try and make it look like this. We're going to try and complete it. 00:03:38.886 --> 00:03:44.099 Make it up so it's a full square. In order to do that, 00:03:44.099 --> 00:03:48.109 what we're going to do is compare that expression directly 00:03:48.109 --> 00:03:49.312 with that one. 00:03:50.080 --> 00:03:55.965 And we've chosen this expression here because that's a plus sign 00:03:55.965 --> 00:04:01.850 plus 5x, and that's a plus sign there plus 2ax. 00:04:01.850 --> 00:04:05.480 So. 00:04:05.500 --> 00:04:11.790 x squared, plus 5x, minus 2. 00:04:12.520 --> 00:04:22.142 And we have x squared plus 2ax plus a squared 00:04:22.172 --> 00:04:24.324 These two match up 00:04:24.734 --> 00:04:29.852 Somehow we've got to match these two up. 00:04:29.852 --> 00:04:31.620 Well,... the x's are the same. 00:04:32.260 --> 00:04:38.995 So the 5 and the 2a have got to be the same and that would 00:04:38.995 --> 00:04:44.383 suggest to us that a has got to be 5 / 2. 00:04:44.980 --> 00:04:50.836 So that x squared plus 5x minus 2... 00:04:52.510 --> 00:04:59.350 becomes x squared plus 5x... 00:04:59.626 --> 00:05:02.150 now... plus a squared and 00:05:02.690 --> 00:05:06.310 now we decided that 5 was to be equal to 2a 00:05:06.310 --> 00:05:12.520 and so a was equal to 5 over 2. 00:05:13.090 --> 00:05:20.214 So to complete the square, we've got to add on 5 over 2... 00:05:20.214 --> 00:05:21.858 and square it. 00:05:24.400 --> 00:05:26.200 But that isn't equal to that. 00:05:28.360 --> 00:05:32.650 It's equal, this is equal to that, but not to that 00:05:32.650 --> 00:05:36.550 well. Clearly we need to put the minus two on. 00:05:38.600 --> 00:05:42.846 But then it's still not equal, because here we've added on 00:05:42.846 --> 00:05:47.864 something extra 5 over 2 [squared]. So we've got to take off that five 00:05:47.864 --> 00:05:51.724 over 2 all squared. We've got to take that away. 00:05:53.410 --> 00:05:56.278 Now let's look at this bit. 00:05:57.140 --> 00:06:02.207 This is an exact square. It's that expression there. 00:06:03.060 --> 00:06:10.560 No, this began life as x plus a all squared, so this 00:06:10.560 --> 00:06:18.060 bit has got to be the same, x plus (5 over 2) all 00:06:18.060 --> 00:06:25.080 squared. And now we can play with this. We've got minus 2 00:06:25.080 --> 00:06:31.956 minus 25 over 4. We can combine that so we have 00:06:31.956 --> 00:06:35.394 x plus (5 over 2) all squared... 00:06:36.160 --> 00:06:37.390 minus... 00:06:38.600 --> 00:06:43.242 Now we're taking away two, so in terms of quarters, that's 00:06:43.242 --> 00:06:47.462 8 quarters were taking away, and we're taking away 25 00:06:47.462 --> 00:06:50.838 quarters as well, so altogether, that's 33 00:06:50.838 --> 00:06:52.948 quarters that we're taking away. 00:06:54.400 --> 00:06:59.068 Now let's have a look at this expression... x squared plus 5x minus 2. 00:06:59.068 --> 00:07:03.347 Remember what we were asking was "what's its minimum value?" 00:07:03.347 --> 00:07:06.815 Its graph looked like that. We were interested in... 00:07:06.815 --> 00:07:08.004 "where's this point?" 00:07:08.134 --> 00:07:09.803 "where is the lowest point?" 00:07:10.173 --> 00:07:11.734 "what's the x-coordinate?" 00:07:11.905 --> 00:07:13.874 "and what's the y-coordinate?" 00:07:14.529 --> 00:07:16.993 Let's have a look at this expression here. 00:07:17.114 --> 00:07:21.904 This is a square. A square is always 00:07:21.904 --> 00:07:26.720 positive unless it's equal to 0, so its lowest value 00:07:26.930 --> 00:07:30.639 that this expression [can take] is 0. 00:07:30.780 --> 00:07:34.546 So the lowest value of the whole expression... 00:07:34.586 --> 00:07:38.686 is that "minus 33 over 4". 00:07:38.928 --> 00:07:43.070 So therefore we can say that the minimum value... 00:07:48.592 --> 00:07:55.431 of x squared, plus 5x, minus 2 equals... minus 33 over 4. 00:07:55.948 --> 00:07:58.556 And we need to be able to say when 00:07:58.876 --> 00:08:00.614 "what's the x-value there?" 00:08:00.864 --> 00:08:04.927 well, it occurs when this bracket is at its lowest value. 00:08:05.194 --> 00:08:11.142 When this bracket is at 0. In other words, when x equals... 00:08:11.142 --> 00:08:14.458 minus 5 over 2. 00:08:16.550 --> 00:08:23.380 So we found the minimum value and exactly when it happens. 00:08:25.850 --> 00:08:31.900 Let's take a second example. Our quadratic function this time, f of x, 00:08:31.900 --> 00:08:37.400 is x squared, minus 6 x, minus 12. 00:08:38.210 --> 00:08:45.254 We've got a minus sign in here, so let's line this up with the 00:08:45.254 --> 00:08:50.537 complete square: x squared, minus 2ax, plus a squared. 00:08:51.210 --> 00:08:56.010 The x squared terms are the same, and we want these two to be the 00:08:56.010 --> 00:09:00.490 same as well. That clearly means that 2a has got to be the same 00:09:00.490 --> 00:09:03.050 as 6, so a has got to be 3. 00:09:03.610 --> 00:09:09.790 So f(x) is equal to x squared, minus 6x, 00:09:09.790 --> 00:09:11.559 plus... 00:09:11.559 --> 00:09:17.338 a squared (which is 3 squared), minus 12, and now we added on 00:09:17.338 --> 00:09:22.462 3 squared. So we've got to take the 3 squared away in order 00:09:22.462 --> 00:09:27.586 to make it equal. To keep the value of the original expression 00:09:27.586 --> 00:09:29.294 that we started with. 00:09:29.930 --> 00:09:37.766 We can now identify this as being (x minus 3) all squared. 00:09:39.060 --> 00:09:41.166 And these numbers at the end... 00:09:41.750 --> 00:09:47.915 minus 12 minus 9, altogether gives us minus 21. 00:09:49.020 --> 00:09:54.610 Again, we can say does it have a maximum value or a minimum value? 00:09:54.610 --> 00:09:57.730 Well what we know that we began with a positive x squared term, 00:09:57.730 --> 00:10:01.820 so the shape of the graph has got to be like that. So we 00:10:01.820 --> 00:10:03.740 know that we're looking for a 00:10:03.740 --> 00:10:08.223 minimum value. We know that that minimum value will occur when 00:10:08.223 --> 00:10:13.607 this bit is 0 because it's a square, it's least value is 00:10:13.607 --> 00:10:21.341 going to be 0, so therefore we can say the minimum value. 00:10:22.090 --> 00:10:25.996 of our quadratic function f of x 00:10:25.996 --> 00:10:31.860 is minus 21, [occurring] when... 00:10:32.870 --> 00:10:38.953 this bit is 0. In other words, when x equals 3. 00:10:40.650 --> 00:10:46.182 The two examples we've taken so far have both had a positive x squared 00:10:46.182 --> 00:10:51.714 and a unit coefficient of x squared, in other words 1 x squared. 00:10:51.714 --> 00:10:57.145 We'll now look at an example where we've got a number here 00:10:57.145 --> 00:10:59.910 in front of the x squared. 00:10:59.920 --> 00:11:04.156 So the example that will take. 00:11:05.980 --> 00:11:11.290 f of x equals 2x squared, 00:11:11.570 --> 00:11:16.550 minus 6x, plus one. 00:11:17.260 --> 00:11:22.852 Our first step is to take out that 2 as a factor. 00:11:23.800 --> 00:11:27.870 2, brackets x squared, 00:11:28.482 --> 00:11:31.020 minus 3x,... 00:11:31.630 --> 00:11:37.368 we've got to take the 2 out of this as well, so 00:11:37.368 --> 00:11:38.700 that's a half. 00:11:39.370 --> 00:11:45.086 And now we do the same as we've done before with this bracket here. 00:11:45.766 --> 00:11:52.654 We line this one up with x squared, minus 2ax, plus a squared. 00:11:53.380 --> 00:11:59.988 When making these two terms the same 3 has to be the same as 2a, 00:11:59.988 --> 00:12:05.652 and so 3 over 2 has to be equal to a. 00:12:06.770 --> 00:12:13.742 So our function f of x is going to be equal to 2 times... 00:12:14.530 --> 00:12:18.438 x squared, minus 3x,... 00:12:19.170 --> 00:12:27.267 now we want plus a squared, so that's plus (3 over 2) all squared 00:12:28.257 --> 00:12:31.743 Plus the half that was there originally and now 00:12:31.743 --> 00:12:35.910 we've added on this, so we've got to take it away,... 00:12:36.144 --> 00:12:42.418 (3 over 2) all squared. And finally we opened a bracket, so we must 00:12:42.418 --> 00:12:44.553 close it at the end. 00:12:45.740 --> 00:12:53.024 Equals... 2, bracket,... now this is going to be our complete square 00:12:53.024 --> 00:12:59.404 (x minus 3 over 2) all squared. 00:12:59.930 --> 00:13:05.371 And then here we've got some calculation to do. We've plus a half, 00:13:05.651 --> 00:13:12.551 take away (3 over 2) squared, so that's plus 1/2 00:13:12.551 --> 00:13:14.955 take away 9 over 4. 00:13:16.620 --> 00:13:21.230 The front bit is going to stay the same 00:13:24.508 --> 00:13:27.230 And now we can juggle with these fractions. At the end, 00:13:27.460 --> 00:13:32.017 we've got plus 1/2 take away 9 quarters or 1/2 is 2 quarters, 00:13:32.017 --> 00:13:36.547 so if we're taking away, nine quarters must be 00:13:36.987 --> 00:13:40.095 ultimately taking away 7 quarters. 00:13:40.355 --> 00:13:44.927 So again, what's the minimum value of this function? 00:13:44.927 --> 00:13:49.039 It had a positive 2 in front of the x squared, so again, it 00:13:49.039 --> 00:13:52.396 looks like that. And again, we're asking the question, 00:13:52.396 --> 00:13:56.499 "what's this point down here?" What's the lowest point and that 00:13:56.499 --> 00:13:59.483 lowest point must occur when this is 0. 00:14:00.340 --> 00:14:02.350 So the min ... 00:14:02.890 --> 00:14:10.828 value of f of x must be equal to... now that's going to be 0 00:14:10.828 --> 00:14:18.268 But we're still multiplying by the 2, so it's 2 times 00:14:18.268 --> 00:14:24.724 minus 7 over 4. That's minus 14 over 4, which reduces to 00:14:24.724 --> 00:14:27.414 minus 7 over 2. When? 00:14:28.100 --> 00:14:34.756 And that will happen when this is zero. In other words, when x 00:14:34.756 --> 00:14:36.804 equals 3 over 2. 00:14:38.170 --> 00:14:43.955 So a minimum value of minus 7 over 2 when x equals 3 over 2. 00:14:43.955 --> 00:14:49.295 Let's take one final example and this time when the 00:14:49.295 --> 00:14:51.520 coefficient of x squared is 00:14:51.520 --> 00:14:53.986 actually negative. 00:14:54.106 --> 00:14:57.991 So for this will take our quadratic function to be f of x 00:14:58.600 --> 00:15:02.084 equals... 3 plus 00:15:02.084 --> 00:15:08.510 8x minus 2(x squared). 00:15:09.520 --> 00:15:16.002 We operate in just the same way as we did before. We take out 00:15:16.002 --> 00:15:21.095 the factor that is multiplying the x squared and on this 00:15:21.095 --> 00:15:24.336 occasion it's minus 2. The "- 2" comes out. 00:15:24.336 --> 00:15:31.805 Times by x squared, we take a minus 2 out of the 8x, 00:15:31.805 --> 00:15:39.080 that leaves us minus 4x and the minus 2 out of the 3 is a 00:15:39.080 --> 00:15:42.960 factor which gives us minus three over 2. 00:15:44.760 --> 00:15:50.870 We line this one up with X squared minus 2X plus A squared. 00:15:50.870 --> 00:15:57.920 Those two are the same. We want these two to be the same. 2A is 00:15:57.920 --> 00:16:03.560 equal to four, so a has got to be equal to two. 00:16:04.120 --> 00:16:10.324 So our F of X is going to be minus 2. 00:16:11.280 --> 00:16:17.110 X squared minus 4X plus A squared, so that's +2 squared 00:16:17.110 --> 00:16:23.470 minus the original 3 over 2, but we've added on a 2 00:16:23.470 --> 00:16:30.360 squared, so we need to take it away again to keep the balance 00:16:30.360 --> 00:16:32.480 to keep the equality. 00:16:33.820 --> 00:16:40.398 Minus two, this is now our complete square, so that's X 00:16:40.398 --> 00:16:47.574 minus two all squared. And here we've got minus three over 2 00:16:47.574 --> 00:16:54.444 - 4. Well, let's have it all over too. So minus four is minus 00:16:54.444 --> 00:16:59.141 8 over 2, so altogether we've got minus 11 over 2. 00:17:00.910 --> 00:17:06.188 And we can look at this. We can see that when this is 00:17:06.188 --> 00:17:10.654 zero, we've got. In this case a maximum value, because this 00:17:10.654 --> 00:17:15.526 is a negative X squared term. So we know that we're looking 00:17:15.526 --> 00:17:20.398 for a graph like that. So it's this point that we're looking 00:17:20.398 --> 00:17:24.052 for the maximum point, and so therefore maximum value. 00:17:26.300 --> 00:17:28.308 Solve F of X. 00:17:28.810 --> 00:17:33.828 Will occur when this square term is equal to 0 'cause the square 00:17:33.828 --> 00:17:36.530 term can never be less than 0. 00:17:37.420 --> 00:17:40.577 And so we have minus two times. 00:17:41.080 --> 00:17:46.778 Minus 11. And altogether that gives us plus 11. 00:17:47.300 --> 00:17:51.236 And it will occur when this. 00:17:51.750 --> 00:17:58.933 Is equal to 0. In other words, when X equals 2.