WEBVTT 00:00:00.870 --> 00:00:03.610 Welcome to the video on completing the square. 00:00:03.610 --> 00:00:04.440 What's completing the square? 00:00:04.440 --> 00:00:06.740 Well, it's a way to solve a quadratic equation. 00:00:06.740 --> 00:00:09.700 And so before I--actually, let me just write down a quadratic equation, and 00:00:09.700 --> 00:00:11.570 then I will show you how to complete the square. 00:00:11.570 --> 00:00:13.460 And then we'll do another example, and then maybe we'll talk 00:00:13.460 --> 00:00:16.650 a little bit about why it's called completing the square. 00:00:16.650 --> 00:00:27.770 So let's say I have this equation: x squared plus 16x 00:00:27.770 --> 00:00:32.600 minus 57 is equal to 0. 00:00:32.600 --> 00:00:36.130 So what are the tools in our toolkit right now that we 00:00:36.130 --> 00:00:36.970 could use to solve this? 00:00:36.970 --> 00:00:38.570 Well, we could try to factor it out. 00:00:38.570 --> 00:00:41.770 We could say, what two numbers add up to 16, and then when you 00:00:41.770 --> 00:00:44.060 multiply them they're minus 57? 00:00:44.060 --> 00:00:45.450 And you'd have to think about it a little bit. 00:00:45.450 --> 00:00:47.360 And you might get whole numbers, but you're not even 00:00:47.360 --> 00:00:49.050 sure if there are two whole numbers that work 00:00:49.050 --> 00:00:49.540 out like that. 00:00:49.540 --> 00:00:50.630 This problem there are. 00:00:50.630 --> 00:00:53.510 But, you know, sometimes the solution is a decimal number 00:00:53.510 --> 00:00:54.190 and you don't know it. 00:00:54.190 --> 00:00:58.150 So the only time you can really factor is if you're sure that 00:00:58.150 --> 00:01:01.000 you could factor this into kind of integer expressions. 00:01:01.000 --> 00:01:03.620 You know, x plus some integer or x minus some integer 00:01:03.620 --> 00:01:05.920 times, you know, x plus some other integer. 00:01:05.920 --> 00:01:06.990 Or likewise. 00:01:06.990 --> 00:01:09.240 The other option is to do the quadratic equation. 00:01:09.240 --> 00:01:11.420 And what we're going to see is actually the quadratic equation 00:01:11.420 --> 00:01:15.510 is just essentially a shortcut to completing the square. 00:01:15.510 --> 00:01:18.410 The quadratic equation is actually proven using 00:01:18.410 --> 00:01:19.420 completing the square. 00:01:19.420 --> 00:01:21.420 So what is completing the square? 00:01:21.420 --> 00:01:23.340 So what do we do? 00:01:23.340 --> 00:01:27.080 Well, before we move into this video let's see what happens 00:01:27.080 --> 00:01:30.930 if I square an expression. 00:01:30.930 --> 00:01:33.220 Let me do it in this down here. 00:01:33.220 --> 00:01:40.250 What is x plus a, squared? 00:01:40.250 --> 00:01:50.940 Well that equals x squared plus 2ax plus a squared. 00:01:50.940 --> 00:01:51.680 Right? 00:01:51.680 --> 00:01:55.420 So if you ever see something in this form, you know that it's 00:01:55.420 --> 00:01:57.740 x plus something squared. 00:01:57.740 --> 00:02:01.040 So wouldn't it be neat if we could manipulate this equation 00:02:01.040 --> 00:02:05.900 so we can write it as x plus a squared equals something, 00:02:05.900 --> 00:02:08.140 and then we could just take the square root? 00:02:08.140 --> 00:02:11.580 And what we're going to do is, actually, do just that. 00:02:11.580 --> 00:02:13.090 And that is completing the square. 00:02:13.090 --> 00:02:15.010 So let me show you an example. 00:02:15.010 --> 00:02:16.515 I think an example will make it a little clearer. 00:02:16.515 --> 00:02:17.620 Let me box this away. 00:02:17.620 --> 00:02:19.310 This is what you need to remember. 00:02:19.310 --> 00:02:22.130 This is the whole rationale behind competing the squares-- 00:02:22.130 --> 00:02:25.650 to get an equation into this form, onto one side of the 00:02:25.650 --> 00:02:27.940 equation, and just have a number on the other side, so 00:02:27.940 --> 00:02:31.210 you could take the square root of both sides. 00:02:31.210 --> 00:02:32.000 So let's see. 00:02:32.000 --> 00:02:33.970 First of all, let's just check to make sure this isn't 00:02:33.970 --> 00:02:35.020 a perfect square. 00:02:35.020 --> 00:02:39.700 If this were, this coefficient would be equivalent to the 2a. 00:02:39.700 --> 00:02:40.470 Right? 00:02:40.470 --> 00:02:44.440 So a would be 8, and then this would be 64. 00:02:44.440 --> 00:02:48.270 This is clearly not 64, so this right here is not 00:02:48.270 --> 00:02:50.840 a squared expression. 00:02:50.840 --> 00:02:51.680 So what can we do? 00:02:51.680 --> 00:02:55.990 Well let me get rid of the 57 by adding 57 to both 00:02:55.990 --> 00:02:57.200 sides of this equation. 00:02:57.200 --> 00:03:07.550 So I would get x squared plus 16x is equal to 57. 00:03:07.550 --> 00:03:11.470 All I did is I added 57 to both sides of this equation. 00:03:11.470 --> 00:03:16.300 Now, what could I add here so that this, the left-hand side 00:03:16.300 --> 00:03:21.480 of this equation, becomes a square of some expression 00:03:21.480 --> 00:03:24.820 like x plus a? 00:03:24.820 --> 00:03:28.790 If you just follow this pattern down here, we have x squared 00:03:28.790 --> 00:03:37.880 plus 2ax-- so you could view this right here as 2ax. 00:03:37.880 --> 00:03:39.090 Right? 00:03:39.090 --> 00:03:40.900 That's 2ax. 00:03:40.900 --> 00:03:43.520 And then we need to add an a squared to it. 00:03:43.520 --> 00:03:44.040 Right? 00:03:44.040 --> 00:03:46.300 Plus a squared. 00:03:46.300 --> 00:03:48.010 And then we would have the form here. 00:03:48.010 --> 00:03:50.510 But we know from basic algebra that anything you do to one 00:03:50.510 --> 00:03:52.080 side of an equation you have to do to another. 00:03:52.080 --> 00:03:54.230 So we added an a squared here, so let's add an a 00:03:54.230 --> 00:03:56.840 squared here as well. 00:03:56.840 --> 00:04:01.350 And now you could essentially rewrite this as a square 00:04:01.350 --> 00:04:02.260 of some expression. 00:04:02.260 --> 00:04:04.210 But before that we have to figure out what a was? 00:04:04.210 --> 00:04:05.520 Well how do we do that? 00:04:05.520 --> 00:04:06.740 Well, what is a? 00:04:06.740 --> 00:04:10.720 If this expression right here is 2ax, what is a? 00:04:10.720 --> 00:04:15.380 Well 2a is going to equal 16, so a is equal to 8. 00:04:15.380 --> 00:04:18.020 And you could usually do that just by inspection; 00:04:18.020 --> 00:04:18.630 do it in your head. 00:04:18.630 --> 00:04:20.930 But if you wanted to see it done algebraically you could 00:04:20.930 --> 00:04:25.690 actually write 2ax is equal to 16x. 00:04:25.690 --> 00:04:29.090 And then divide both sides by 2x, and you get a is 00:04:29.090 --> 00:04:31.430 equal to 16x over 2x. 00:04:31.430 --> 00:04:36.950 And assuming that x doesn't equal 0 this evaluates to 8. 00:04:36.950 --> 00:04:38.130 So a is 8. 00:04:38.130 --> 00:04:42.430 So if a is 8 we could rewrite that expression-- I'll switch 00:04:42.430 --> 00:04:49.030 colors arbitrarily-- as x squared plus 16x 00:04:49.030 --> 00:04:50.470 plus a squared. 00:04:50.470 --> 00:04:54.180 Well, it's 64, because a is 8. 00:04:54.180 --> 00:04:59.170 Is equal to 57 plus 64. 00:04:59.170 --> 00:05:00.720 Right? 00:05:00.720 --> 00:05:04.600 I went through a fairly tedious explanation here, but all we've 00:05:04.600 --> 00:05:08.890 really done to get from there to there is we added 57 to both 00:05:08.890 --> 00:05:10.870 sides of this equation to kind of get it on the right-hand 00:05:10.870 --> 00:05:14.320 side, and then we added 64 to both sides of this equation. 00:05:14.320 --> 00:05:16.830 And why did I add 64 to both sides of this equation? 00:05:16.830 --> 00:05:21.070 So that the left-hand side expression takes this form. 00:05:21.070 --> 00:05:23.200 Now that the left-hand side expression takes this form 00:05:23.200 --> 00:05:26.030 I can rewrite it as what? 00:05:26.030 --> 00:05:27.170 x plus a, squared. 00:05:27.170 --> 00:05:28.620 I can rewrite it in this form. 00:05:28.620 --> 00:05:35.550 And we know that a is 8, so it becomes x plus 8, squared, 00:05:35.550 --> 00:05:39.730 is equal to-- and what's 57 plus 64? 00:05:39.730 --> 00:05:43.090 It's 121. 00:05:43.090 --> 00:05:47.270 Now we have what looks like a fairly straightforward-- it's 00:05:47.270 --> 00:05:48.960 still a quadratic equation, actually, because if you 00:05:48.960 --> 00:05:50.350 were to expand this side you'd get a quadratic. 00:05:50.350 --> 00:05:53.065 But we can solve this without using the quadratic equation 00:05:53.065 --> 00:05:54.610 or without having to factor. 00:05:54.610 --> 00:05:57.390 We can just take the square root of both sides of this. 00:05:57.390 --> 00:06:00.550 And if we take the square root of both sides what do we get? 00:06:00.550 --> 00:06:03.610 We get-- once again, arbitrarily switching colors-- 00:06:03.610 --> 00:06:09.230 that x plus 8 is equal to, and remember this, the plus or 00:06:09.230 --> 00:06:12.880 minus square root of 121. 00:06:12.880 --> 00:06:14.590 And what's the square root of 121? 00:06:14.590 --> 00:06:15.960 Well it's 11, right? 00:06:15.960 --> 00:06:17.630 So then we come here. 00:06:17.630 --> 00:06:18.800 Let me box this away. 00:06:18.800 --> 00:06:20.620 This was just an aside. 00:06:20.620 --> 00:06:26.830 So we get x plus 8 is equal to plus or minus 11. 00:06:26.830 --> 00:06:30.420 And so x is equal to-- subtract 8 from both sides-- minus 00:06:30.420 --> 00:06:33.860 8 plus or minus 11. 00:06:33.860 --> 00:06:41.590 And so x could equal-- so minus 8 plus 11 is 3. 00:06:41.590 --> 00:06:41.970 Right? 00:06:44.800 --> 00:06:48.160 Let me make sure I did that right. 00:06:48.160 --> 00:06:53.310 x is equal to minus 8 plus or minus 11. 00:06:53.310 --> 00:06:54.140 Yes. 00:06:54.140 --> 00:06:55.350 That's right. 00:06:55.350 --> 00:06:59.270 So x could be equal to 3. 00:06:59.270 --> 00:07:02.960 And then if I took minus 8 minus 11, x could 00:07:02.960 --> 00:07:10.416 also equal minus 19. 00:07:10.416 --> 00:07:11.350 All right. 00:07:11.350 --> 00:07:13.200 And let's see if that makes sense. 00:07:13.200 --> 00:07:18.680 So in theory this should be able to be factored as x 00:07:18.680 --> 00:07:23.770 minus 3 times x plus 19 is equal to 0. 00:07:23.770 --> 00:07:24.030 Right? 00:07:24.030 --> 00:07:26.160 Because these are the two solutions of this equation. 00:07:26.160 --> 00:07:28.190 And that works out, right? 00:07:28.190 --> 00:07:31.340 Minus 3 times 19 is minus 57. 00:07:31.340 --> 00:07:36.920 And minus 3 plus 19 is plus 16x. 00:07:36.920 --> 00:07:39.120 We could have just immediately factored it this way, but if 00:07:39.120 --> 00:07:41.030 that wasn't obvious to us-- because, you know, at least 00:07:41.030 --> 00:07:43.600 19 is kind of a strange number-- we could do it 00:07:43.600 --> 00:07:46.800 completing the square. 00:07:46.800 --> 00:07:47.690 And so why is it called completing the square? 00:07:47.690 --> 00:07:49.920 Because you get it in this form and then you have to add this 00:07:49.920 --> 00:07:52.950 64 here to kind of complete the square-- to turn this 00:07:52.950 --> 00:07:56.020 left-hand expression into a squared expression. 00:07:56.020 --> 00:07:56.770 Let's do one more. 00:07:56.770 --> 00:07:59.920 And I'll do less explanation and more just chugging through 00:07:59.920 --> 00:08:02.105 the problem, and that actually might make it seem simpler. 00:08:04.800 --> 00:08:07.080 But this is going to be a hairier problem. 00:08:07.080 --> 00:08:19.930 So let's say I have 6x squared minus 7x minus 3 is equal to 0. 00:08:19.930 --> 00:08:22.980 You could try to factor it, but personally I don't 00:08:22.980 --> 00:08:25.260 enjoy factoring things when I have a coefficient. 00:08:25.260 --> 00:08:27.590 And you can say, oh well why don't we divide both sides 00:08:27.590 --> 00:08:28.970 of this equation by 6? 00:08:28.970 --> 00:08:30.960 But then you'd get a fraction here and a fraction here. 00:08:30.960 --> 00:08:33.580 And that's even worse to factor just by inspection. 00:08:33.580 --> 00:08:35.190 You could do the quadratic equation. 00:08:35.190 --> 00:08:37.310 And maybe I'll show you in a future video, the quadratic 00:08:37.310 --> 00:08:39.500 equation-- and I think I've already done one where I proved 00:08:39.500 --> 00:08:40.630 the quadratic equation. 00:08:40.630 --> 00:08:42.380 But the quadratic equation is essentially 00:08:42.380 --> 00:08:43.170 completing the square. 00:08:43.170 --> 00:08:44.090 It's kind of a shortcut. 00:08:44.090 --> 00:08:46.280 It's just kind of remembering the formula. 00:08:46.280 --> 00:08:48.320 But let's complete the square here, because that's what the 00:08:48.320 --> 00:08:50.640 point of this video was. 00:08:50.640 --> 00:08:54.650 So let's add the 3 to both sides of that equation. 00:08:54.650 --> 00:08:56.300 We could do-- well, let's add the 3 first. 00:08:56.300 --> 00:09:04.820 So you get 6 x squared minus 7x is equal to 3. 00:09:04.820 --> 00:09:06.770 I added 3 to both sides. 00:09:06.770 --> 00:09:09.470 And some teachers will leave the minus 3 here, and then try 00:09:09.470 --> 00:09:11.050 to figure out what to add to it and all of that. 00:09:11.050 --> 00:09:13.170 But I like to get it out of the way so that I can figure out 00:09:13.170 --> 00:09:16.080 very clearly what number I should put here. 00:09:16.080 --> 00:09:18.230 But I also don't like the 6 here. 00:09:18.230 --> 00:09:19.550 It just complicates things. 00:09:19.550 --> 00:09:25.990 I like to have it x plus a squared, not some square root 00:09:25.990 --> 00:09:27.450 coefficient on the x term. 00:09:27.450 --> 00:09:31.530 So let's divide both sides of this equation by 6, and you get 00:09:31.530 --> 00:09:39.730 x squared minus 7/6 x is equal to-- 3 divided by 6 00:09:39.730 --> 00:09:41.566 is equal to 1/2. 00:09:41.566 --> 00:09:43.190 And we could have made that our first step. 00:09:43.190 --> 00:09:46.450 We could have divided by 6 right at that first step. 00:09:46.450 --> 00:09:49.250 Anyway, now let's try to complete the square. 00:09:49.250 --> 00:09:51.800 So we have x squared-- I'm just going to open up some space-- 00:09:51.800 --> 00:09:59.530 minus 7/6 x plus something is going to be equal to 1/2. 00:09:59.530 --> 00:10:02.400 And so we have to add something here so that this left-hand 00:10:02.400 --> 00:10:05.290 expression becomes a squared expression. 00:10:05.290 --> 00:10:06.620 So how do we do that? 00:10:06.620 --> 00:10:10.770 Well essentially we look at this coefficient, and keep 00:10:10.770 --> 00:10:14.610 in mind this is not just 7/6 it's minus 7/6. 00:10:14.610 --> 00:10:17.460 You take 1/2 of it, and then you square it. 00:10:17.460 --> 00:10:18.610 Right? 00:10:18.610 --> 00:10:19.690 Let me do it. 00:10:19.690 --> 00:10:25.290 x plus a, squared, is equal to x squared plus 00:10:25.290 --> 00:10:28.820 2ax plus a squared. 00:10:28.820 --> 00:10:29.070 Right? 00:10:29.070 --> 00:10:30.750 This is what you have to remember all the time. 00:10:30.750 --> 00:10:33.560 That's all completing the square is based off of. 00:10:33.560 --> 00:10:34.980 So what did I say just now? 00:10:34.980 --> 00:10:37.260 Well, this term is going to be 1/2 of this 00:10:37.260 --> 00:10:39.190 coefficient squared. 00:10:39.190 --> 00:10:40.190 And how do we know that? 00:10:40.190 --> 00:10:43.880 Because a is going to be 1/2 of this coefficient if you just 00:10:43.880 --> 00:10:45.850 do a little bit of pattern matching. 00:10:45.850 --> 00:10:48.760 So what's 1/2 of this coefficient? 00:10:48.760 --> 00:10:54.050 1/2 of minus 7/6 is minus 7/12. 00:10:54.050 --> 00:10:56.640 So if you want you could write a equals minus 00:10:56.640 --> 00:10:58.770 7/12 for our example. 00:10:58.770 --> 00:11:00.770 And I just multiplied this by 1/2. 00:11:00.770 --> 00:11:01.980 Right? 00:11:01.980 --> 00:11:03.660 So what do I add to both sides? 00:11:03.660 --> 00:11:06.030 I add a squared. 00:11:06.030 --> 00:11:08.930 So what's 7/12 squared? 00:11:08.930 --> 00:11:13.220 Well that's going to be 49/144. 00:11:13.220 --> 00:11:15.000 If I did it to the left-hand side I have to do it to 00:11:15.000 --> 00:11:16.630 the right-hand side. 00:11:16.630 --> 00:11:22.240 Plus 49/144. 00:11:22.240 --> 00:11:26.120 And now how can I simplify this left-hand side? 00:11:26.120 --> 00:11:26.880 What's our next step? 00:11:26.880 --> 00:11:28.470 Well we now know it is a perfect square. 00:11:28.470 --> 00:11:31.550 In fact, we know what a is. a is minus 7/12. 00:11:31.550 --> 00:11:35.200 And so we know that this left-hand side of this equation 00:11:35.200 --> 00:11:43.390 is x minus a-- or x plus a, but a is a negative number. 00:11:43.390 --> 00:11:47.980 So x plus a, and a is negative, squared. 00:11:47.980 --> 00:11:50.350 And if you want you can multiply this out and confirm 00:11:50.350 --> 00:11:53.130 that it truly equals this. 00:11:53.130 --> 00:11:55.920 And that is going to be equal to-- let's get a common 00:11:55.920 --> 00:11:58.360 denominator, 144. 00:11:58.360 --> 00:12:04.070 So 72 plus 49 equals 121. 00:12:04.070 --> 00:12:06.300 121/144. 00:12:06.300 --> 00:12:09.210 So we have x minus 7/12, all of that squared 00:12:09.210 --> 00:12:13.180 is equal to 121/144. 00:12:13.180 --> 00:12:14.300 So what do we do now? 00:12:14.300 --> 00:12:15.570 Well now we just take the square root of both 00:12:15.570 --> 00:12:17.700 sides of this equation. 00:12:17.700 --> 00:12:20.140 And I'm trying to free up some space. 00:12:20.140 --> 00:12:22.215 Switch to green. 00:12:22.215 --> 00:12:25.320 Let me partition this off. 00:12:25.320 --> 00:12:33.310 And we get x minus 7/12 is equal to the plus or minus 00:12:33.310 --> 00:12:33.940 square root of that. 00:12:33.940 --> 00:12:38.120 So plus or minus 11/12. 00:12:38.120 --> 00:12:38.390 Right? 00:12:38.390 --> 00:12:39.660 Square root of 121 is 11. 00:12:39.660 --> 00:12:42.420 Square root of 144 is 12. 00:12:42.420 --> 00:12:44.480 So then we could add 7/12 to both sides of this equation, 00:12:44.480 --> 00:12:53.100 and we get x is equal to 7/12 plus or minus 11/12. 00:12:53.100 --> 00:12:58.660 Well that equals 7 plus or minus 11/12. 00:12:58.660 --> 00:13:00.050 So what are the two options? 00:13:00.050 --> 00:13:03.930 7 plus 11 is 18, over 12. 00:13:03.930 --> 00:13:08.210 So x could equal 18/12, is 3/2. 00:13:08.210 --> 00:13:11.010 Or, what's 7 minus 11? 00:13:11.010 --> 00:13:12.760 That's minus 4/12. 00:13:12.760 --> 00:13:15.370 So it's minus 1/3. 00:13:15.370 --> 00:13:16.630 There you have it. 00:13:16.630 --> 00:13:17.940 That is completing the square. 00:13:17.940 --> 00:13:20.220 Hopefully you found that reasonably insightful. 00:13:20.220 --> 00:13:23.340 And if you want to prove the quadratic equation, all you 00:13:23.340 --> 00:13:27.320 have to do is instead of having numbers here, write a x squared 00:13:27.320 --> 00:13:29.820 plus bx plus c equals 0. 00:13:29.820 --> 00:13:34.130 And then complete the square using the a, b, and c's 00:13:34.130 --> 00:13:35.060 instead of numbers. 00:13:35.060 --> 00:13:37.180 And you will end up with the quadratic equation 00:13:37.180 --> 00:13:38.110 by this point. 00:13:38.110 --> 00:13:39.510 And I think I did that in a video. 00:13:39.510 --> 00:13:41.600 Let me know if I didn't and I'll do it for you. 00:13:41.600 --> 00:13:44.540 Anyway, I'll see you in the next video.