1 00:00:00,000 --> 00:00:00,720 2 00:00:00,720 --> 00:00:03,129 In the completing the square video I kept saying that all 3 00:00:03,129 --> 00:00:05,810 the quadratic equation is completing the square 4 00:00:05,809 --> 00:00:07,449 as kind of a short cut of completing square. 5 00:00:07,450 --> 00:00:10,220 And I was under the impression that I had done this 6 00:00:10,220 --> 00:00:12,120 proof already but now I realize that I haven't. 7 00:00:12,119 --> 00:00:15,750 So let me prove the quadratic equation to you, by 8 00:00:15,750 --> 00:00:16,690 completing the square. 9 00:00:16,690 --> 00:00:19,820 10 00:00:19,820 --> 00:00:23,210 So let's say I have a quadratic equation. 11 00:00:23,210 --> 00:00:25,870 I guess a quadratic equation is actually what you're trying to 12 00:00:25,870 --> 00:00:28,760 solve, and what a lot of people call the quadratic equation is 13 00:00:28,760 --> 00:00:29,960 actually the quadratic formula. 14 00:00:29,960 --> 00:00:33,100 But anyway I don't want to get caught up in terminology. 15 00:00:33,100 --> 00:00:36,030 But let's say that I have a quadratic equation that 16 00:00:36,030 --> 00:00:46,450 says ax squared plus bx plus c is equal to 0. 17 00:00:46,450 --> 00:00:48,280 And let's just complete the square here. 18 00:00:48,280 --> 00:00:49,480 So how do we do that? 19 00:00:49,479 --> 00:00:56,939 Well let's subtract c from both sides so we get ax squared plus 20 00:00:56,939 --> 00:01:00,739 the bx is equal to minus c. 21 00:01:00,740 --> 00:01:03,039 And just like I said in the completing the square video 22 00:01:03,039 --> 00:01:06,200 I don't like having this a coefficient here. 23 00:01:06,200 --> 00:01:08,355 I like just having one coefficient on my x squared 24 00:01:08,355 --> 00:01:10,980 term so let me divide everything by a. 25 00:01:10,980 --> 00:01:21,439 So I get x squared plus b/a x is equal to-- you have 26 00:01:21,439 --> 00:01:24,694 to divide both sides by a --minus c/a. 27 00:01:24,694 --> 00:01:27,939 28 00:01:27,939 --> 00:01:29,599 Now we are ready to complete the square. 29 00:01:29,599 --> 00:01:30,890 What was completing the square? 30 00:01:30,890 --> 00:01:34,790 Well it's somehow adding something to this expression so 31 00:01:34,790 --> 00:01:38,590 it has the form of something that is the square 32 00:01:38,590 --> 00:01:39,140 of an expression. 33 00:01:39,140 --> 00:01:39,939 What do i mean by that? 34 00:01:39,939 --> 00:01:43,399 Well, I'll do a little aside here. 35 00:01:43,400 --> 00:01:51,950 if I told you that x plus a squared, that equals 36 00:01:51,950 --> 00:01:57,500 x squared plus two ax plus a squared, right? 37 00:01:57,500 --> 00:02:01,329 So if I can add something here so that this left hand side 38 00:02:01,329 --> 00:02:05,810 this expression looks like this, then I could 39 00:02:05,810 --> 00:02:06,329 go the other way. 40 00:02:06,329 --> 00:02:09,689 I can say this is going to be x plus something squared. 41 00:02:09,689 --> 00:02:11,590 So what do I have to add on both sides? 42 00:02:11,590 --> 00:02:15,140 If you watched the completing the square video this should be 43 00:02:15,139 --> 00:02:17,729 hopefully intuitive for you. 44 00:02:17,729 --> 00:02:21,509 What you do is you say well this b/a, this corresponds to 45 00:02:21,509 --> 00:02:26,182 the 2a term, so a is going to be half of this, is going to 46 00:02:26,182 --> 00:02:28,009 be half of this coefficient. 47 00:02:28,009 --> 00:02:29,099 That would be the a. 48 00:02:29,099 --> 00:02:31,620 And then what I need to add is a squared. 49 00:02:31,620 --> 00:02:34,930 So I need to take half of this and then square it and 50 00:02:34,930 --> 00:02:36,110 then add it to both sides. 51 00:02:36,110 --> 00:02:38,865 Let me do that in a different color. 52 00:02:38,865 --> 00:02:40,810 Do it in this magenta. 53 00:02:40,810 --> 00:02:42,650 So I'm going to take half of this-- I'm just completing 54 00:02:42,650 --> 00:02:45,099 square, that's all I'm doing, no magic here --so 55 00:02:45,099 --> 00:02:47,449 plus half of this. 56 00:02:47,449 --> 00:02:50,229 Well half of that is b/2a right? 57 00:02:50,229 --> 00:02:52,129 You just multiply by 1/2. 58 00:02:52,129 --> 00:02:54,240 And I have to square it. 59 00:02:54,240 --> 00:02:55,893 Well if I did it to the left hand side of the equation, I 60 00:02:55,893 --> 00:02:57,659 have to do it to the right hand side. 61 00:02:57,659 --> 00:03:01,205 So plus b/2a squared. 62 00:03:01,205 --> 00:03:07,469 63 00:03:07,469 --> 00:03:10,669 And now I have this left hand side of the equation in the 64 00:03:10,669 --> 00:03:13,750 form that it is the square of an expression that is 65 00:03:13,750 --> 00:03:14,949 x plus something. 66 00:03:14,949 --> 00:03:15,879 And what is it? 67 00:03:15,879 --> 00:03:19,969 Well that's equal to-- let me switch colors again --what's 68 00:03:19,969 --> 00:03:21,729 the left hand side of this equation equal to? 69 00:03:21,729 --> 00:03:24,519 And you can just use this pattern and go to the left. 70 00:03:24,520 --> 00:03:28,730 It's x plus what? 71 00:03:28,729 --> 00:03:32,959 Well we said a, you can do one of two ways. a is 1/2 of this 72 00:03:32,960 --> 00:03:36,390 coefficient or a is the square root of this coefficient or 73 00:03:36,389 --> 00:03:38,309 since we didn't even square it we know that this 74 00:03:38,310 --> 00:03:40,969 is a. b/2a is a. 75 00:03:40,969 --> 00:03:49,060 So this is the same thing as x plus b over 2a everything 76 00:03:49,060 --> 00:03:55,979 squared, and then that equals-- let's see if we can simplify 77 00:03:55,979 --> 00:04:00,229 this or make this a little bit cleaner --that equals-- 78 00:04:00,229 --> 00:04:04,759 See, if I were to have a common denominator-- I'm just doing a 79 00:04:04,759 --> 00:04:07,599 little bit of algebra here --see, when I square this it's 80 00:04:07,599 --> 00:04:10,780 going to be 4a squared-- let me let me write this. 81 00:04:10,780 --> 00:04:15,740 This is equal to b squared over 4a squared. 82 00:04:15,740 --> 00:04:16,710 Right? 83 00:04:16,709 --> 00:04:19,859 And so if I have to add these two fractions, let me make 84 00:04:19,860 --> 00:04:29,550 this equal to 4a squared. 85 00:04:29,550 --> 00:04:30,329 Right? 86 00:04:30,329 --> 00:04:31,750 And if the denominator is 4a squared, what does 87 00:04:31,750 --> 00:04:34,360 the minus c/a become? 88 00:04:34,360 --> 00:04:40,280 I See if I multiply the denominator by 4a, I have to 89 00:04:40,279 --> 00:04:41,809 multiply the numerator by 4a. 90 00:04:41,810 --> 00:04:50,089 So this becomes minus 4ac, right? 91 00:04:50,089 --> 00:04:53,029 And then b squared over 4a squared, well that's 92 00:04:53,029 --> 00:04:54,809 just still b squared. 93 00:04:54,810 --> 00:04:56,519 I'm just doing a little bit of algrebra. 94 00:04:56,519 --> 00:04:57,519 Hopefully I'm not confusing you. 95 00:04:57,519 --> 00:04:59,469 I just expanded this. 96 00:04:59,470 --> 00:05:02,330 I just took the square of this, b squared over 4a squared. 97 00:05:02,329 --> 00:05:04,789 And then I added this to this, I got a common denominator. 98 00:05:04,790 --> 00:05:09,710 And minus c/a is the same thing as minus 4ac over 4a squared. 99 00:05:09,709 --> 00:05:11,569 And now we can take the square root of both 100 00:05:11,569 --> 00:05:13,240 sides of this equation. 101 00:05:13,240 --> 00:05:15,759 And this should hopefully start to look a little 102 00:05:15,759 --> 00:05:17,490 bit familiar to you now. 103 00:05:17,490 --> 00:05:19,290 So let's see, so we get x. 104 00:05:19,290 --> 00:05:21,080 So if we take the square root of both sides of this equation 105 00:05:21,079 --> 00:05:29,779 we get x plus b/2a is equal to the square root of this-- let's 106 00:05:29,779 --> 00:05:32,179 take the square root of the numerator and the demoninator. 107 00:05:32,180 --> 00:05:35,949 So the numerator is-- I'm going to put the b squared first, I'm 108 00:05:35,949 --> 00:05:38,110 just going to switch this order, it doesn't matter --the 109 00:05:38,110 --> 00:05:43,660 square root of b squared minus 4ac, right? 110 00:05:43,660 --> 00:05:46,439 That's just the numerator. 111 00:05:46,439 --> 00:05:48,240 I just the square root of it, and we have to get the square 112 00:05:48,240 --> 00:05:49,769 root of the denominator too. 113 00:05:49,769 --> 00:05:51,969 What's the square roof of 4a squared? 114 00:05:51,970 --> 00:05:54,020 Well it's just 2a, right? 115 00:05:54,019 --> 00:05:55,949 2a. 116 00:05:55,949 --> 00:05:56,800 And now what do we do? 117 00:05:56,800 --> 00:05:58,639 Oh, it's very important! 118 00:05:58,639 --> 00:06:00,399 When we're taking the square root, it's not just the 119 00:06:00,399 --> 00:06:01,069 positive square root. 120 00:06:01,069 --> 00:06:03,449 It's the positive or minus square root. 121 00:06:03,449 --> 00:06:06,599 We saw that couple of times when we did the-- and you could 122 00:06:06,600 --> 00:06:09,090 say it's a plus or minus here too, but if you look plus or 123 00:06:09,089 --> 00:06:10,799 minus on the top and a plus or minus on the bottom, you can 124 00:06:10,800 --> 00:06:12,290 just write it once on the top. 125 00:06:12,290 --> 00:06:14,930 I'll let you think about why you only have to write it once. 126 00:06:14,930 --> 00:06:17,560 If you had a negative an a plus, or negative and a plus 127 00:06:17,560 --> 00:06:19,250 sometimes cancel out, or a negative and a negative, 128 00:06:19,250 --> 00:06:20,790 that's the same thing as just having a plus on top. 129 00:06:20,790 --> 00:06:22,210 Anyway, I think you get that. 130 00:06:22,209 --> 00:06:26,139 And now we just have to subtract b/2a from both sides. 131 00:06:26,139 --> 00:06:33,680 and we get, we get-- and this is the exciting part --we get x 132 00:06:33,680 --> 00:06:42,850 is equal to minus be over to 2a plus or minus this thing, so 133 00:06:42,850 --> 00:06:51,790 minus b squared minus 4ac, all of that over 2a. 134 00:06:51,790 --> 00:06:53,850 And we already have a common denominator, so we can 135 00:06:53,850 --> 00:06:55,129 just add the fractions. 136 00:06:55,129 --> 00:06:58,879 So we got --and I'm going to do this in a vibrant bold-- I 137 00:06:58,879 --> 00:07:02,569 don't know maybe not so much bold, well green color --so we 138 00:07:02,569 --> 00:07:10,969 get x is equal to, numerator, negative b plus or minus square 139 00:07:10,970 --> 00:07:19,470 root of b squared minus 4ac, all of that over 2a. 140 00:07:19,470 --> 00:07:23,010 And that is the famous quadratic formula. 141 00:07:23,009 --> 00:07:25,480 So, there we go we proved it. 142 00:07:25,480 --> 00:07:28,410 And we proved it just from completing the square. 143 00:07:28,410 --> 00:07:31,570 I hope you found that vaguely interesting. 144 00:07:31,569 --> 00:07:33,449 See in the next Video.