0:00:00.440,0:00:02.930 In this video, I'm going to show[br]you a technique called 0:00:02.930,0:00:09.310 completing the square. 0:00:09.310,0:00:14.490 And what's neat about this is[br]that this will work for any 0:00:14.490,0:00:18.210 quadratic equation, and it's[br]actually the basis for the 0:00:18.210,0:00:18.750 quadratic formula. 0:00:18.750,0:00:21.990 And in the next video or the[br]video after that I'll prove 0:00:21.990,0:00:25.630 the quadratic formula using[br]completing the square. 0:00:25.630,0:00:28.450 But before we do that, we[br]need to understand even 0:00:28.450,0:00:29.470 what it's all about. 0:00:29.470,0:00:32.070 And it really just builds off[br]of what we did in the last 0:00:32.070,0:00:33.880 video, where we solved[br]quadratics 0:00:33.880,0:00:36.130 using perfect squares. 0:00:36.130,0:00:39.900 So let's say I have the[br]quadratic equation x squared 0:00:39.900,0:00:44.880 minus 4x is equal to 5. 0:00:44.880,0:00:47.490 And I put this big space[br]here for a reason. 0:00:47.490,0:00:49.680 In the last video, we saw[br]that these can be pretty 0:00:49.680,0:00:53.200 straightforward to solve if[br]the left-hand side is a 0:00:53.200,0:00:56.500 perfect square. 0:00:56.500,0:00:59.050 You see, completing the square[br]is all about making the 0:00:59.050,0:01:01.900 quadratic equation into a[br]perfect square, engineering 0:01:01.900,0:01:05.190 it, adding and subtracting from[br]both sides so it becomes 0:01:05.190,0:01:05.970 a perfect square. 0:01:05.970,0:01:07.710 So how can we do that? 0:01:07.710,0:01:10.130 Well, in order for this[br]left-hand side to be a perfect 0:01:10.130,0:01:12.990 square, there has to be[br]some number here. 0:01:12.990,0:01:17.510 There has to be some number here[br]that if I have my number 0:01:17.510,0:01:20.910 squared I get that number, and[br]then if I have two times my 0:01:20.910,0:01:22.890 number I get negative 4. 0:01:22.890,0:01:24.750 Remember that, and I[br]think it'll become 0:01:24.750,0:01:27.700 clear with a few examples. 0:01:27.700,0:01:35.230 I want x squared minus 4x plus[br]something to be equal to x 0:01:35.230,0:01:37.740 minus a squared. 0:01:37.740,0:01:41.010 We don't know what a[br]is just yet, but we 0:01:41.010,0:01:42.110 know a couple of things. 0:01:42.110,0:01:46.180 When I square things-- so this[br]is going to be x squared minus 0:01:46.180,0:01:49.330 2a plus a squared. 0:01:49.330,0:01:53.640 So if you look at this pattern[br]right here, that has to be-- 0:01:53.640,0:01:59.880 sorry, x squared minus 2ax--[br]this right here has to be 2ax. 0:01:59.880,0:02:03.530 And this right here would[br]have to be a squared. 0:02:03.530,0:02:07.690 So this number, a is going to[br]be half of negative 4, a has 0:02:07.690,0:02:10.370 to be negative 2, right? 0:02:10.370,0:02:13.570 Because 2 times a is going[br]to be negative 4. 0:02:13.570,0:02:18.330 a is negative 2, and if a is[br]negative 2, what is a squared? 0:02:18.330,0:02:21.550 Well, then a squared is going[br]to be positive 4. 0:02:21.550,0:02:24.220 And this might look all[br]complicated to you right now, 0:02:24.220,0:02:25.910 but I'm showing you[br]the rationale. 0:02:25.910,0:02:29.080 You literally just look at this[br]coefficient right here, 0:02:29.080,0:02:32.670 and you say, OK, well what's[br]half of that coefficient? 0:02:32.670,0:02:35.920 Well, half of that coefficient[br]is negative 2. 0:02:35.920,0:02:40.230 So we could say a is equal to[br]negative 2-- same idea there-- 0:02:40.230,0:02:41.720 and then you square it. 0:02:41.720,0:02:44.100 You square a, you[br]get positive 4. 0:02:44.100,0:02:46.540 So we add positive 4 here. 0:02:46.540,0:02:47.630 Add a 4. 0:02:47.630,0:02:50.990 Now, from the very first[br]equation we ever did, you 0:02:50.990,0:02:55.240 should know that you can never[br]do something to just one side 0:02:55.240,0:02:55.900 of the equation. 0:02:55.900,0:02:58.700 You can't add 4 to just one[br]side of the equation. 0:02:58.700,0:03:02.710 If x squared minus 4x was equal[br]to 5, then when I add 4 0:03:02.710,0:03:04.720 it's not going to be[br]equal to 5 anymore. 0:03:04.720,0:03:07.950 It's going to be equal[br]to 5 plus 4. 0:03:07.950,0:03:11.430 We added 4 on the left-hand side[br]because we wanted this to 0:03:11.430,0:03:12.435 be a perfect square. 0:03:12.435,0:03:15.210 But if you add something to the[br]left-hand side, you've got 0:03:15.210,0:03:17.320 to add it to the right-hand[br]side. 0:03:17.320,0:03:20.630 And now, we've gotten ourselves[br]to a problem that's 0:03:20.630,0:03:23.410 just like the problems we[br]did in the last video. 0:03:23.410,0:03:25.960 What is this left-hand side? 0:03:25.960,0:03:27.000 Let me rewrite the[br]whole thing. 0:03:27.000,0:03:33.020 We have x squared minus 4x[br]plus 4 is equal to 9 now. 0:03:33.020,0:03:35.380 All we did is add 4 to both[br]sides of the equation. 0:03:35.380,0:03:39.070 But we added 4 on purpose so[br]that this left-hand side 0:03:39.070,0:03:41.080 becomes a perfect square. 0:03:41.080,0:03:41.760 Now what is this? 0:03:41.760,0:03:45.340 What number when I multiply it[br]by itself is equal to 4 and 0:03:45.340,0:03:47.770 when I add it to itself I'm[br]equal to negative 2? 0:03:47.770,0:03:49.000 Well, we already answered[br]that question. 0:03:49.000,0:03:50.040 It's negative 2. 0:03:50.040,0:03:55.310 So we get x minus 2 times[br]x minus 2 is equal to 9. 0:03:55.310,0:03:59.350 Or we could have skipped this[br]step and written x minus 2 0:03:59.350,0:04:02.990 squared is equal to 9. 0:04:02.990,0:04:07.280 And then you take the square[br]root of both sides, you get x 0:04:07.280,0:04:10.840 minus 2 is equal to[br]plus or minus 3. 0:04:10.840,0:04:16.870 Add 2 to both sides, you get x[br]is equal to 2 plus or minus 3. 0:04:16.870,0:04:22.440 That tells us that x could be[br]equal to 2 plus 3, which is 5. 0:04:22.440,0:04:28.960 Or x could be equal to 2 minus[br]3, which is negative 1. 0:04:28.960,0:04:30.650 And we are done. 0:04:30.650,0:04:31.840 Now I want to be very clear. 0:04:31.840,0:04:34.300 You could have done this without[br]completing the square. 0:04:34.300,0:04:37.640 We could've started off[br]with x squared minus 0:04:37.640,0:04:39.850 4x is equal to 5. 0:04:39.850,0:04:42.970 We could have subtracted 5 from[br]both sides and gotten x 0:04:42.970,0:04:47.160 squared minus 4x minus[br]5 is equal to 0. 0:04:47.160,0:04:51.940 And you could say, hey, if I[br]have a negative 5 times a 0:04:51.940,0:04:56.190 positive 1, then their product[br]is negative 5 and their sum is 0:04:56.190,0:04:57.000 negative 4. 0:04:57.000,0:05:00.800 So I could say this is x[br]minus 5 times x plus 0:05:00.800,0:05:02.480 1 is equal to 0. 0:05:02.480,0:05:06.810 And then we would say that x is[br]equal to 5 or x is equal to 0:05:06.810,0:05:07.700 negative 1. 0:05:07.700,0:05:10.350 And in this case, this actually[br]probably would have 0:05:10.350,0:05:13.450 been a faster way to[br]do the problem. 0:05:13.450,0:05:16.140 But the neat thing about the[br]completing the square is it 0:05:16.140,0:05:17.770 will always work. 0:05:17.770,0:05:21.580 It'll always work no matter what[br]the coefficients are or 0:05:21.580,0:05:23.385 no matter how crazy[br]the problem is. 0:05:23.385,0:05:25.400 And let me prove it to you. 0:05:25.400,0:05:28.440 Let's do one that traditionally[br]would have been 0:05:28.440,0:05:31.140 a pretty painful problem if[br]we just tried to do it by 0:05:31.140,0:05:36.200 factoring, especially if we[br]did it using grouping or 0:05:36.200,0:05:37.020 something like that. 0:05:37.020,0:05:45.070 Let's say we had 10x squared[br]minus 30x minus 0:05:45.070,0:05:47.530 8 is equal to 0. 0:05:47.530,0:05:50.060 Now, right from the get-go, you[br]could say, hey look, we 0:05:50.060,0:05:53.280 could maybe divide[br]both sides by 2. 0:05:53.280,0:05:54.800 That does simplify[br]a little bit. 0:05:54.800,0:05:56.450 Let's divide both sides by 2. 0:05:56.450,0:06:02.150 So if you divide everything[br]by 2, what do you get? 0:06:02.150,0:06:11.990 We get 5x squared minus 15x[br]minus 4 is equal to 0. 0:06:11.990,0:06:14.540 But once again, now we have this[br]crazy 5 in front of this 0:06:14.540,0:06:16.810 coefficent and we would have to[br]solve it by grouping which 0:06:16.810,0:06:20.410 is a reasonably painful[br]process. 0:06:20.410,0:06:23.410 But we can now go straight to[br]completing the square, and to 0:06:23.410,0:06:27.500 do that I'm now going to divide[br]by 5 to get a 1 leading 0:06:27.500,0:06:28.870 coefficient here. 0:06:28.870,0:06:31.660 And you're going to see why this[br]is different than what 0:06:31.660,0:06:33.010 we've traditionally done. 0:06:33.010,0:06:35.730 So if I divide this whole thing[br]by 5, I could have just 0:06:35.730,0:06:38.050 divided by 10 from the get-go[br]but I wanted to go to this the 0:06:38.050,0:06:40.030 step first just to show[br]you that this really 0:06:40.030,0:06:41.800 didn't give us much. 0:06:41.800,0:06:43.660 Let's divide everything by 5. 0:06:43.660,0:06:52.693 So if you divide everything by[br]5, you get x squared minus 3x 0:06:52.693,0:06:58.720 minus 4/5 is equal to 0. 0:06:58.720,0:07:02.020 So, you might say, hey, why did[br]we ever do that factoring 0:07:02.020,0:07:02.630 by grouping? 0:07:02.630,0:07:06.140 If we can just always divide by[br]this leading coefficient, 0:07:06.140,0:07:07.220 we can get rid of that. 0:07:07.220,0:07:09.840 We can always turn this into a 1[br]or a negative 1 if we divide 0:07:09.840,0:07:10.910 by the right number. 0:07:10.910,0:07:14.410 But notice, by doing that we[br]got this crazy 4/5 here. 0:07:14.410,0:07:17.630 So this is super hard to do[br]just using factoring. 0:07:17.630,0:07:19.500 You'd have to say, what two[br]numbers when I take the 0:07:19.500,0:07:22.100 product is equal to[br]negative 4/5? 0:07:22.100,0:07:25.210 It's a fraction and when I take[br]their sum, is equal to 0:07:25.210,0:07:26.140 negative 3? 0:07:26.140,0:07:29.310 This is a hard problem[br]with factoring. 0:07:29.310,0:07:36.860 This is hard using factoring. 0:07:36.860,0:07:42.080 So, the best thing to do is to[br]use completing the square. 0:07:42.080,0:07:44.720 So let's think a little bit[br]about how we can turn this 0:07:44.720,0:07:45.950 into a perfect square. 0:07:45.950,0:07:48.080 What I like to do-- and you'll[br]see this done some ways and 0:07:48.080,0:07:50.040 I'll show you both ways because[br]you'll see teachers do 0:07:50.040,0:07:53.880 it both ways-- I like to get[br]the 4/5 on the other side. 0:07:53.880,0:07:56.900 So let's add 4/5 to both[br]sides of this equation. 0:07:56.900,0:07:59.980 You don't have to do it this[br]way, but I like to get the 4/5 0:07:59.980,0:08:01.160 out of the way. 0:08:01.160,0:08:04.010 And then what do we get[br]if we add 4/5 to both 0:08:04.010,0:08:05.250 sides of this equation? 0:08:05.250,0:08:08.350 The left-hand hand side of the[br]equation just becomes x 0:08:08.350,0:08:11.800 squared minus 3x,[br]no 4/5 there. 0:08:11.800,0:08:13.660 I'm going to leave a little[br]bit of space. 0:08:13.660,0:08:17.790 And that's going to[br]be equal to 4/5. 0:08:17.790,0:08:19.990 Now, just like the last problem,[br]we want to turn this 0:08:19.990,0:08:23.350 left-hand side into the perfect[br]square of a binomial. 0:08:23.350,0:08:24.740 How do we do that? 0:08:24.740,0:08:28.360 Well, we say, well, what number[br]times 2 is equal to 0:08:28.360,0:08:30.110 negative 3? 0:08:30.110,0:08:32.309 So some number times[br]2 is negative 3. 0:08:32.309,0:08:35.330 Or we essentially just take[br]negative 3 and divide it by 2, 0:08:35.330,0:08:37.370 which is negative 3/2. 0:08:37.370,0:08:39.554 And then we square[br]negative 3/2. 0:08:39.554,0:08:44.840 So in the example, we'll[br]say a is negative 3/2. 0:08:44.840,0:08:48.380 And if we square negative[br]3/2, what do we get? 0:08:48.380,0:08:54.100 We get positive 9/4. 0:08:54.100,0:08:56.810 I just took half of this[br]coefficient, squared it, got 0:08:56.810,0:08:58.010 positive 9/4. 0:08:58.010,0:09:00.720 The whole purpose of doing that[br]is to turn this left-hand 0:09:00.720,0:09:02.920 side into a perfect square. 0:09:02.920,0:09:05.530 Now, anything you do to one side[br]of the equation, you've 0:09:05.530,0:09:06.600 got to do to the other side. 0:09:06.600,0:09:11.030 So we added a 9/4 here, let's[br]add a 9/4 over there. 0:09:11.030,0:09:13.850 And what does our[br]equation become? 0:09:13.850,0:09:22.530 We get x squared minus 3x plus[br]9/4 is equal to-- let's see if 0:09:22.530,0:09:24.460 we can get a common[br]denominator. 0:09:24.460,0:09:29.120 So, 4/5 is the same[br]thing as 16/20. 0:09:29.120,0:09:31.880 Just multiply the numerator[br]and denominator by 4. 0:09:31.880,0:09:33.820 Plus over 20. 0:09:33.820,0:09:36.960 9/4 is the same thing[br]if you multiply the 0:09:36.960,0:09:42.150 numerator by 5 as 45/20. 0:09:42.150,0:09:44.970 And so what is 16 plus 45? 0:09:44.970,0:09:47.020 You see, this is kind of getting[br]kind of hairy, but 0:09:47.020,0:09:48.930 that's the fun, I guess, of 0:09:48.930,0:09:50.380 completing the square sometimes. 0:09:50.380,0:09:53.420 16 plus 45. 0:09:53.420,0:09:55.780 See that's 55, 61. 0:09:55.780,0:09:59.750 So this is equal to 61/20. 0:09:59.750,0:10:02.680 So let me just rewrite it. 0:10:02.680,0:10:09.480 x squared minus 3x plus[br]9/4 is equal to 61/20. 0:10:09.480,0:10:11.030 Crazy number. 0:10:11.030,0:10:13.630 Now this, at least on[br]the left hand side, 0:10:13.630,0:10:15.970 is a perfect square. 0:10:15.970,0:10:21.610 This is the same thing as[br]x minus 3/2 squared. 0:10:21.610,0:10:24.200 And it was by design. 0:10:24.200,0:10:27.590 Negative 3/2 times negative[br]3/2 is positive 9/4. 0:10:27.590,0:10:32.790 Negative 3/2 plus negative 3/2[br]is equal to negative 3. 0:10:32.790,0:10:37.960 So this squared is[br]equal to 61/20. 0:10:37.960,0:10:43.090 We can take the square root of[br]both sides and we get x minus 0:10:43.090,0:10:47.820 3/2 is equal to the positive[br]or the negative 0:10:47.820,0:10:53.320 square root of 61/20. 0:10:53.320,0:10:57.640 And now, we can add 3/2 to both[br]sides of this equation 0:10:57.640,0:11:03.600 and you get x is equal to[br]positive 3/2 plus or minus the 0:11:03.600,0:11:07.300 square root of 61/20. 0:11:07.300,0:11:09.290 And this is a crazy number and[br]it's hopefully obvious you 0:11:09.290,0:11:11.430 would not have been able to-- at[br]least I would not have been 0:11:11.430,0:11:15.250 able to-- get to this number[br]just by factoring. 0:11:15.250,0:11:17.260 And if you want their actual[br]values, you can get your 0:11:17.260,0:11:18.510 calculator out. 0:11:20.620,0:11:22.510 And then let me clear[br]all of this. 0:11:25.950,0:11:28.760 And 3/2-- let's do the plus[br]version first. So we want to 0:11:28.760,0:11:33.710 do 3 divided by 2 plus the[br]second square root. 0:11:33.710,0:11:35.050 We want to pick that little[br]yellow square root. 0:11:35.050,0:11:46.480 So the square root of 61 divided[br]by 20, which is 3.24. 0:11:46.480,0:11:52.760 This crazy 3.2464, I'll[br]just write 3.246. 0:11:52.760,0:12:02.230 So this is approximately equal[br]to 3.246, and that was just 0:12:02.230,0:12:03.110 the positive version. 0:12:03.110,0:12:06.710 Let's do the subtraction[br]version. 0:12:06.710,0:12:09.180 So we can actually put our[br]entry-- if you do second and 0:12:09.180,0:12:11.535 then entry, that we want that[br]little yellow entry, that's 0:12:11.535,0:12:12.465 why I pressed the[br]second button. 0:12:12.465,0:12:16.130 So I press enter, it puts in[br]what we just put, we can just 0:12:16.130,0:12:23.400 change the positive or the[br]addition to a subtraction and 0:12:23.400,0:12:27.970 you get negative 0.246. 0:12:27.970,0:12:33.800 So you get negative 0.246. 0:12:33.800,0:12:38.200 And you can actually verify[br]that these satisfy our 0:12:38.200,0:12:39.360 original equation. 0:12:39.360,0:12:42.050 Our original equation[br]was up here. 0:12:42.050,0:12:43.840 Let me just verify[br]for one of them. 0:12:47.400,0:12:50.130 So the second answer on your[br]graphing calculator is the 0:12:50.130,0:12:51.760 last answer you use. 0:12:51.760,0:12:54.160 So if you use a variable answer,[br]that's this number 0:12:54.160,0:12:55.160 right here. 0:12:55.160,0:13:00.090 So if I have my answer squared--[br]I'm using answer 0:13:00.090,0:13:02.380 represents negative 0.24. 0:13:02.380,0:13:11.975 Answer squared minus 3 times[br]answer minus 4/5-- 4 divided 0:13:11.975,0:13:16.030 by 5-- it equals--. 0:13:16.030,0:13:18.490 And this just a little[br]bit of explanation. 0:13:18.490,0:13:21.860 This doesn't store the entire[br]number, it goes up to some 0:13:21.860,0:13:22.880 level of precision. 0:13:22.880,0:13:24.910 It stores some number[br]of digits. 0:13:24.910,0:13:28.930 So when it calculated it using[br]this stored number right here, 0:13:28.930,0:13:32.240 it got 1 times 10 to[br]the negative 14. 0:13:32.240,0:13:34.980 So that is 0.0000. 0:13:34.980,0:13:37.100 So that's 13 zeroes[br]and then a 1. 0:13:37.100,0:13:38.870 A decimal, then 13[br]zeroes and a 1. 0:13:38.870,0:13:41.060 So this is pretty much 0. 0:13:41.060,0:13:43.550 Or actually, if you got the[br]exact answer right here, if 0:13:43.550,0:13:46.480 you went through an infinite[br]level of precision here, or 0:13:46.480,0:13:49.050 maybe if you kept it in this[br]radical form, you would get 0:13:49.050,0:13:52.390 that it is indeed equal to 0. 0:13:52.390,0:13:55.300 So hopefully you found that[br]helpful, this whole notion of 0:13:55.300,0:13:56.160 completing the square. 0:13:56.160,0:13:58.670 Now we're going to extend it[br]to the actual quadratic 0:13:58.670,0:14:01.510 formula that we can use, we[br]can essentially just plug 0:14:01.510,0:14:03.610 things into to solve any[br]quadratic equation.