1 00:00:00,000 --> 00:00:00,440 2 00:00:00,440 --> 00:00:02,930 In this video, I'm going to show you a technique called 3 00:00:02,930 --> 00:00:09,310 completing the square. 4 00:00:09,310 --> 00:00:14,490 And what's neat about this is that this will work for any 5 00:00:14,490 --> 00:00:18,210 quadratic equation, and it's actually the basis for the 6 00:00:18,210 --> 00:00:18,750 quadratic formula. 7 00:00:18,750 --> 00:00:21,990 And in the next video or the video after that I'll prove 8 00:00:21,989 --> 00:00:25,629 the quadratic formula using completing the square. 9 00:00:25,629 --> 00:00:28,449 But before we do that, we need to understand even 10 00:00:28,449 --> 00:00:29,469 what it's all about. 11 00:00:29,469 --> 00:00:32,070 And it really just builds off of what we did in the last 12 00:00:32,070 --> 00:00:33,880 video, where we solved quadratics 13 00:00:33,880 --> 00:00:36,130 using perfect squares. 14 00:00:36,130 --> 00:00:39,900 So let's say I have the quadratic equation x squared 15 00:00:39,899 --> 00:00:44,879 minus 4x is equal to 5. 16 00:00:44,880 --> 00:00:47,490 And I put this big space here for a reason. 17 00:00:47,490 --> 00:00:49,679 In the last video, we saw that these can be pretty 18 00:00:49,679 --> 00:00:53,200 straightforward to solve if the left-hand side is a 19 00:00:53,200 --> 00:00:56,500 perfect square. 20 00:00:56,500 --> 00:00:59,049 You see, completing the square is all about making the 21 00:00:59,049 --> 00:01:01,899 quadratic equation into a perfect square, engineering 22 00:01:01,899 --> 00:01:05,189 it, adding and subtracting from both sides so it becomes 23 00:01:05,189 --> 00:01:05,969 a perfect square. 24 00:01:05,969 --> 00:01:07,709 So how can we do that? 25 00:01:07,709 --> 00:01:10,129 Well, in order for this left-hand side to be a perfect 26 00:01:10,129 --> 00:01:12,989 square, there has to be some number here. 27 00:01:12,989 --> 00:01:17,509 There has to be some number here that if I have my number 28 00:01:17,510 --> 00:01:20,910 squared I get that number, and then if I have two times my 29 00:01:20,909 --> 00:01:22,890 number I get negative 4. 30 00:01:22,890 --> 00:01:24,750 Remember that, and I think it'll become 31 00:01:24,750 --> 00:01:27,700 clear with a few examples. 32 00:01:27,700 --> 00:01:35,230 I want x squared minus 4x plus something to be equal to x 33 00:01:35,230 --> 00:01:37,740 minus a squared. 34 00:01:37,739 --> 00:01:41,009 We don't know what a is just yet, but we 35 00:01:41,010 --> 00:01:42,109 know a couple of things. 36 00:01:42,109 --> 00:01:46,180 When I square things-- so this is going to be x squared minus 37 00:01:46,180 --> 00:01:49,330 2a plus a squared. 38 00:01:49,329 --> 00:01:53,640 So if you look at this pattern right here, that has to be-- 39 00:01:53,640 --> 00:01:59,879 sorry, x squared minus 2ax-- this right here has to be 2ax. 40 00:01:59,879 --> 00:02:03,530 And this right here would have to be a squared. 41 00:02:03,530 --> 00:02:07,689 So this number, a is going to be half of negative 4, a has 42 00:02:07,689 --> 00:02:10,370 to be negative 2, right? 43 00:02:10,370 --> 00:02:13,569 Because 2 times a is going to be negative 4. 44 00:02:13,569 --> 00:02:18,329 a is negative 2, and if a is negative 2, what is a squared? 45 00:02:18,330 --> 00:02:21,550 Well, then a squared is going to be positive 4. 46 00:02:21,550 --> 00:02:24,219 And this might look all complicated to you right now, 47 00:02:24,219 --> 00:02:25,909 but I'm showing you the rationale. 48 00:02:25,909 --> 00:02:29,079 You literally just look at this coefficient right here, 49 00:02:29,080 --> 00:02:32,670 and you say, OK, well what's half of that coefficient? 50 00:02:32,669 --> 00:02:35,919 Well, half of that coefficient is negative 2. 51 00:02:35,919 --> 00:02:40,229 So we could say a is equal to negative 2-- same idea there-- 52 00:02:40,229 --> 00:02:41,719 and then you square it. 53 00:02:41,719 --> 00:02:44,099 You square a, you get positive 4. 54 00:02:44,099 --> 00:02:46,539 So we add positive 4 here. 55 00:02:46,539 --> 00:02:47,629 Add a 4. 56 00:02:47,629 --> 00:02:50,990 Now, from the very first equation we ever did, you 57 00:02:50,990 --> 00:02:55,240 should know that you can never do something to just one side 58 00:02:55,240 --> 00:02:55,900 of the equation. 59 00:02:55,900 --> 00:02:58,700 You can't add 4 to just one side of the equation. 60 00:02:58,699 --> 00:03:02,709 If x squared minus 4x was equal to 5, then when I add 4 61 00:03:02,710 --> 00:03:04,719 it's not going to be equal to 5 anymore. 62 00:03:04,719 --> 00:03:07,949 It's going to be equal to 5 plus 4. 63 00:03:07,949 --> 00:03:11,429 We added 4 on the left-hand side because we wanted this to 64 00:03:11,430 --> 00:03:12,435 be a perfect square. 65 00:03:12,435 --> 00:03:15,210 But if you add something to the left-hand side, you've got 66 00:03:15,210 --> 00:03:17,320 to add it to the right-hand side. 67 00:03:17,319 --> 00:03:20,629 And now, we've gotten ourselves to a problem that's 68 00:03:20,629 --> 00:03:23,409 just like the problems we did in the last video. 69 00:03:23,409 --> 00:03:25,960 What is this left-hand side? 70 00:03:25,960 --> 00:03:27,000 Let me rewrite the whole thing. 71 00:03:27,000 --> 00:03:33,020 We have x squared minus 4x plus 4 is equal to 9 now. 72 00:03:33,020 --> 00:03:35,380 All we did is add 4 to both sides of the equation. 73 00:03:35,379 --> 00:03:39,069 But we added 4 on purpose so that this left-hand side 74 00:03:39,069 --> 00:03:41,079 becomes a perfect square. 75 00:03:41,080 --> 00:03:41,760 Now what is this? 76 00:03:41,759 --> 00:03:45,340 What number when I multiply it by itself is equal to 4 and 77 00:03:45,340 --> 00:03:47,770 when I add it to itself I'm equal to negative 2? 78 00:03:47,770 --> 00:03:49,000 Well, we already answered that question. 79 00:03:49,000 --> 00:03:50,039 It's negative 2. 80 00:03:50,039 --> 00:03:55,310 So we get x minus 2 times x minus 2 is equal to 9. 81 00:03:55,310 --> 00:03:59,349 Or we could have skipped this step and written x minus 2 82 00:03:59,349 --> 00:04:02,989 squared is equal to 9. 83 00:04:02,990 --> 00:04:07,280 And then you take the square root of both sides, you get x 84 00:04:07,280 --> 00:04:10,840 minus 2 is equal to plus or minus 3. 85 00:04:10,840 --> 00:04:16,870 Add 2 to both sides, you get x is equal to 2 plus or minus 3. 86 00:04:16,870 --> 00:04:22,439 That tells us that x could be equal to 2 plus 3, which is 5. 87 00:04:22,439 --> 00:04:28,959 Or x could be equal to 2 minus 3, which is negative 1. 88 00:04:28,959 --> 00:04:30,649 And we are done. 89 00:04:30,649 --> 00:04:31,839 Now I want to be very clear. 90 00:04:31,839 --> 00:04:34,299 You could have done this without completing the square. 91 00:04:34,300 --> 00:04:37,639 We could've started off with x squared minus 92 00:04:37,639 --> 00:04:39,849 4x is equal to 5. 93 00:04:39,850 --> 00:04:42,970 We could have subtracted 5 from both sides and gotten x 94 00:04:42,970 --> 00:04:47,160 squared minus 4x minus 5 is equal to 0. 95 00:04:47,160 --> 00:04:51,939 And you could say, hey, if I have a negative 5 times a 96 00:04:51,939 --> 00:04:56,189 positive 1, then their product is negative 5 and their sum is 97 00:04:56,189 --> 00:04:57,000 negative 4. 98 00:04:57,000 --> 00:05:00,800 So I could say this is x minus 5 times x plus 99 00:05:00,800 --> 00:05:02,480 1 is equal to 0. 100 00:05:02,480 --> 00:05:06,810 And then we would say that x is equal to 5 or x is equal to 101 00:05:06,810 --> 00:05:07,699 negative 1. 102 00:05:07,699 --> 00:05:10,349 And in this case, this actually probably would have 103 00:05:10,350 --> 00:05:13,450 been a faster way to do the problem. 104 00:05:13,449 --> 00:05:16,139 But the neat thing about the completing the square is it 105 00:05:16,139 --> 00:05:17,769 will always work. 106 00:05:17,769 --> 00:05:21,579 It'll always work no matter what the coefficients are or 107 00:05:21,579 --> 00:05:23,384 no matter how crazy the problem is. 108 00:05:23,384 --> 00:05:25,399 And let me prove it to you. 109 00:05:25,399 --> 00:05:28,439 Let's do one that traditionally would have been 110 00:05:28,439 --> 00:05:31,139 a pretty painful problem if we just tried to do it by 111 00:05:31,139 --> 00:05:36,199 factoring, especially if we did it using grouping or 112 00:05:36,199 --> 00:05:37,019 something like that. 113 00:05:37,019 --> 00:05:45,069 Let's say we had 10x squared minus 30x minus 114 00:05:45,069 --> 00:05:47,529 8 is equal to 0. 115 00:05:47,529 --> 00:05:50,059 Now, right from the get-go, you could say, hey look, we 116 00:05:50,060 --> 00:05:53,280 could maybe divide both sides by 2. 117 00:05:53,279 --> 00:05:54,799 That does simplify a little bit. 118 00:05:54,800 --> 00:05:56,449 Let's divide both sides by 2. 119 00:05:56,449 --> 00:06:02,149 So if you divide everything by 2, what do you get? 120 00:06:02,149 --> 00:06:11,989 We get 5x squared minus 15x minus 4 is equal to 0. 121 00:06:11,990 --> 00:06:14,540 But once again, now we have this crazy 5 in front of this 122 00:06:14,540 --> 00:06:16,810 coefficent and we would have to solve it by grouping which 123 00:06:16,810 --> 00:06:20,410 is a reasonably painful process. 124 00:06:20,410 --> 00:06:23,410 But we can now go straight to completing the square, and to 125 00:06:23,410 --> 00:06:27,500 do that I'm now going to divide by 5 to get a 1 leading 126 00:06:27,500 --> 00:06:28,870 coefficient here. 127 00:06:28,870 --> 00:06:31,660 And you're going to see why this is different than what 128 00:06:31,660 --> 00:06:33,010 we've traditionally done. 129 00:06:33,009 --> 00:06:35,730 So if I divide this whole thing by 5, I could have just 130 00:06:35,730 --> 00:06:38,050 divided by 10 from the get-go but I wanted to go to this the 131 00:06:38,050 --> 00:06:40,030 step first just to show you that this really 132 00:06:40,029 --> 00:06:41,799 didn't give us much. 133 00:06:41,800 --> 00:06:43,660 Let's divide everything by 5. 134 00:06:43,660 --> 00:06:52,693 So if you divide everything by 5, you get x squared minus 3x 135 00:06:52,692 --> 00:06:58,719 minus 4/5 is equal to 0. 136 00:06:58,720 --> 00:07:02,020 So, you might say, hey, why did we ever do that factoring 137 00:07:02,019 --> 00:07:02,629 by grouping? 138 00:07:02,629 --> 00:07:06,139 If we can just always divide by this leading coefficient, 139 00:07:06,139 --> 00:07:07,219 we can get rid of that. 140 00:07:07,220 --> 00:07:09,840 We can always turn this into a 1 or a negative 1 if we divide 141 00:07:09,839 --> 00:07:10,909 by the right number. 142 00:07:10,910 --> 00:07:14,410 But notice, by doing that we got this crazy 4/5 here. 143 00:07:14,410 --> 00:07:17,630 So this is super hard to do just using factoring. 144 00:07:17,629 --> 00:07:19,500 You'd have to say, what two numbers when I take the 145 00:07:19,500 --> 00:07:22,100 product is equal to negative 4/5? 146 00:07:22,100 --> 00:07:25,210 It's a fraction and when I take their sum, is equal to 147 00:07:25,209 --> 00:07:26,139 negative 3? 148 00:07:26,139 --> 00:07:29,310 This is a hard problem with factoring. 149 00:07:29,310 --> 00:07:36,860 This is hard using factoring. 150 00:07:36,860 --> 00:07:42,080 So, the best thing to do is to use completing the square. 151 00:07:42,079 --> 00:07:44,719 So let's think a little bit about how we can turn this 152 00:07:44,720 --> 00:07:45,950 into a perfect square. 153 00:07:45,949 --> 00:07:48,079 What I like to do-- and you'll see this done some ways and 154 00:07:48,079 --> 00:07:50,039 I'll show you both ways because you'll see teachers do 155 00:07:50,040 --> 00:07:53,879 it both ways-- I like to get the 4/5 on the other side. 156 00:07:53,879 --> 00:07:56,899 So let's add 4/5 to both sides of this equation. 157 00:07:56,899 --> 00:07:59,979 You don't have to do it this way, but I like to get the 4/5 158 00:07:59,980 --> 00:08:01,160 out of the way. 159 00:08:01,160 --> 00:08:04,010 And then what do we get if we add 4/5 to both 160 00:08:04,009 --> 00:08:05,250 sides of this equation? 161 00:08:05,250 --> 00:08:08,350 The left-hand hand side of the equation just becomes x 162 00:08:08,350 --> 00:08:11,800 squared minus 3x, no 4/5 there. 163 00:08:11,800 --> 00:08:13,660 I'm going to leave a little bit of space. 164 00:08:13,660 --> 00:08:17,790 And that's going to be equal to 4/5. 165 00:08:17,790 --> 00:08:19,990 Now, just like the last problem, we want to turn this 166 00:08:19,990 --> 00:08:23,350 left-hand side into the perfect square of a binomial. 167 00:08:23,350 --> 00:08:24,740 How do we do that? 168 00:08:24,740 --> 00:08:28,360 Well, we say, well, what number times 2 is equal to 169 00:08:28,360 --> 00:08:30,110 negative 3? 170 00:08:30,110 --> 00:08:32,308 So some number times 2 is negative 3. 171 00:08:32,308 --> 00:08:35,329 Or we essentially just take negative 3 and divide it by 2, 172 00:08:35,330 --> 00:08:37,370 which is negative 3/2. 173 00:08:37,370 --> 00:08:39,553 And then we square negative 3/2. 174 00:08:39,553 --> 00:08:44,839 So in the example, we'll say a is negative 3/2. 175 00:08:44,840 --> 00:08:48,379 And if we square negative 3/2, what do we get? 176 00:08:48,379 --> 00:08:54,100 We get positive 9/4. 177 00:08:54,100 --> 00:08:56,810 I just took half of this coefficient, squared it, got 178 00:08:56,809 --> 00:08:58,009 positive 9/4. 179 00:08:58,009 --> 00:09:00,720 The whole purpose of doing that is to turn this left-hand 180 00:09:00,720 --> 00:09:02,920 side into a perfect square. 181 00:09:02,919 --> 00:09:05,529 Now, anything you do to one side of the equation, you've 182 00:09:05,529 --> 00:09:06,600 got to do to the other side. 183 00:09:06,600 --> 00:09:11,029 So we added a 9/4 here, let's add a 9/4 over there. 184 00:09:11,029 --> 00:09:13,850 And what does our equation become? 185 00:09:13,850 --> 00:09:22,529 We get x squared minus 3x plus 9/4 is equal to-- let's see if 186 00:09:22,529 --> 00:09:24,459 we can get a common denominator. 187 00:09:24,460 --> 00:09:29,120 So, 4/5 is the same thing as 16/20. 188 00:09:29,120 --> 00:09:31,879 Just multiply the numerator and denominator by 4. 189 00:09:31,879 --> 00:09:33,820 Plus over 20. 190 00:09:33,820 --> 00:09:36,960 9/4 is the same thing if you multiply the 191 00:09:36,960 --> 00:09:42,150 numerator by 5 as 45/20. 192 00:09:42,149 --> 00:09:44,970 And so what is 16 plus 45? 193 00:09:44,970 --> 00:09:47,019 You see, this is kind of getting kind of hairy, but 194 00:09:47,019 --> 00:09:48,929 that's the fun, I guess, of 195 00:09:48,929 --> 00:09:50,379 completing the square sometimes. 196 00:09:50,379 --> 00:09:53,419 16 plus 45. 197 00:09:53,419 --> 00:09:55,779 See that's 55, 61. 198 00:09:55,779 --> 00:09:59,750 So this is equal to 61/20. 199 00:09:59,750 --> 00:10:02,679 So let me just rewrite it. 200 00:10:02,679 --> 00:10:09,479 x squared minus 3x plus 9/4 is equal to 61/20. 201 00:10:09,480 --> 00:10:11,029 Crazy number. 202 00:10:11,029 --> 00:10:13,629 Now this, at least on the left hand side, 203 00:10:13,629 --> 00:10:15,970 is a perfect square. 204 00:10:15,970 --> 00:10:21,610 This is the same thing as x minus 3/2 squared. 205 00:10:21,610 --> 00:10:24,200 And it was by design. 206 00:10:24,200 --> 00:10:27,590 Negative 3/2 times negative 3/2 is positive 9/4. 207 00:10:27,590 --> 00:10:32,790 Negative 3/2 plus negative 3/2 is equal to negative 3. 208 00:10:32,789 --> 00:10:37,959 So this squared is equal to 61/20. 209 00:10:37,960 --> 00:10:43,090 We can take the square root of both sides and we get x minus 210 00:10:43,090 --> 00:10:47,820 3/2 is equal to the positive or the negative 211 00:10:47,820 --> 00:10:53,320 square root of 61/20. 212 00:10:53,320 --> 00:10:57,640 And now, we can add 3/2 to both sides of this equation 213 00:10:57,639 --> 00:11:03,600 and you get x is equal to positive 3/2 plus or minus the 214 00:11:03,600 --> 00:11:07,300 square root of 61/20. 215 00:11:07,299 --> 00:11:09,289 And this is a crazy number and it's hopefully obvious you 216 00:11:09,289 --> 00:11:11,429 would not have been able to-- at least I would not have been 217 00:11:11,429 --> 00:11:15,250 able to-- get to this number just by factoring. 218 00:11:15,250 --> 00:11:17,259 And if you want their actual values, you can get your 219 00:11:17,259 --> 00:11:18,509 calculator out. 220 00:11:18,509 --> 00:11:20,620 221 00:11:20,620 --> 00:11:22,509 And then let me clear all of this. 222 00:11:22,509 --> 00:11:25,950 223 00:11:25,950 --> 00:11:28,759 And 3/2-- let's do the plus version first. So we want to 224 00:11:28,759 --> 00:11:33,710 do 3 divided by 2 plus the second square root. 225 00:11:33,710 --> 00:11:35,050 We want to pick that little yellow square root. 226 00:11:35,049 --> 00:11:46,479 So the square root of 61 divided by 20, which is 3.24. 227 00:11:46,480 --> 00:11:52,759 This crazy 3.2464, I'll just write 3.246. 228 00:11:52,759 --> 00:12:02,230 So this is approximately equal to 3.246, and that was just 229 00:12:02,230 --> 00:12:03,110 the positive version. 230 00:12:03,110 --> 00:12:06,710 Let's do the subtraction version. 231 00:12:06,710 --> 00:12:09,180 So we can actually put our entry-- if you do second and 232 00:12:09,179 --> 00:12:11,534 then entry, that we want that little yellow entry, that's 233 00:12:11,534 --> 00:12:12,464 why I pressed the second button. 234 00:12:12,465 --> 00:12:16,129 So I press enter, it puts in what we just put, we can just 235 00:12:16,129 --> 00:12:23,399 change the positive or the addition to a subtraction and 236 00:12:23,399 --> 00:12:27,970 you get negative 0.246. 237 00:12:27,970 --> 00:12:33,800 So you get negative 0.246. 238 00:12:33,799 --> 00:12:38,199 And you can actually verify that these satisfy our 239 00:12:38,200 --> 00:12:39,360 original equation. 240 00:12:39,360 --> 00:12:42,050 Our original equation was up here. 241 00:12:42,049 --> 00:12:43,839 Let me just verify for one of them. 242 00:12:43,840 --> 00:12:47,399 243 00:12:47,399 --> 00:12:50,129 So the second answer on your graphing calculator is the 244 00:12:50,129 --> 00:12:51,759 last answer you use. 245 00:12:51,759 --> 00:12:54,159 So if you use a variable answer, that's this number 246 00:12:54,159 --> 00:12:55,159 right here. 247 00:12:55,159 --> 00:13:00,089 So if I have my answer squared-- I'm using answer 248 00:13:00,090 --> 00:13:02,379 represents negative 0.24. 249 00:13:02,379 --> 00:13:11,975 Answer squared minus 3 times answer minus 4/5-- 4 divided 250 00:13:11,975 --> 00:13:16,029 by 5-- it equals--. 251 00:13:16,029 --> 00:13:18,490 And this just a little bit of explanation. 252 00:13:18,490 --> 00:13:21,860 This doesn't store the entire number, it goes up to some 253 00:13:21,860 --> 00:13:22,879 level of precision. 254 00:13:22,879 --> 00:13:24,909 It stores some number of digits. 255 00:13:24,909 --> 00:13:28,929 So when it calculated it using this stored number right here, 256 00:13:28,929 --> 00:13:32,239 it got 1 times 10 to the negative 14. 257 00:13:32,240 --> 00:13:34,980 So that is 0.0000. 258 00:13:34,980 --> 00:13:37,100 So that's 13 zeroes and then a 1. 259 00:13:37,100 --> 00:13:38,870 A decimal, then 13 zeroes and a 1. 260 00:13:38,870 --> 00:13:41,060 So this is pretty much 0. 261 00:13:41,059 --> 00:13:43,549 Or actually, if you got the exact answer right here, if 262 00:13:43,549 --> 00:13:46,479 you went through an infinite level of precision here, or 263 00:13:46,480 --> 00:13:49,050 maybe if you kept it in this radical form, you would get 264 00:13:49,049 --> 00:13:52,389 that it is indeed equal to 0. 265 00:13:52,389 --> 00:13:55,299 So hopefully you found that helpful, this whole notion of 266 00:13:55,299 --> 00:13:56,159 completing the square. 267 00:13:56,159 --> 00:13:58,669 Now we're going to extend it to the actual quadratic 268 00:13:58,669 --> 00:14:01,509 formula that we can use, we can essentially just plug 269 00:14:01,509 --> 00:14:03,610 things into to solve any quadratic equation. 270 00:14:03,610 --> 00:14:05,265