1 00:00:00,500 --> 00:00:02,300 Now let me show you ... 2 00:00:04,600 --> 00:00:06,700 So what if I do ... 3 00:00:06,950 --> 00:00:12,800 ... a x squared plus b x plus c. 4 00:00:13,000 --> 00:00:15,000 Is that scared? 5 00:00:15,700 --> 00:00:18,000 Can we complete square there? 6 00:00:18,300 --> 00:00:21,450 Oh, yes, we can. We factor a, right? 7 00:00:21,550 --> 00:00:28,000 We get x squared plus b over a x plus c over a. 8 00:00:28,750 --> 00:00:31,700 And then I have to take ... 9 00:00:32,400 --> 00:00:34,400 ... this part inside. 10 00:00:34,500 --> 00:00:36,250 And I have to look at the middle term. 11 00:00:36,350 --> 00:00:39,600 And I have to divide that number by two. 12 00:00:40,400 --> 00:00:42,700 And that will make a square for me. 13 00:00:42,800 --> 00:00:46,600 x plus b over two a ... 14 00:00:48,200 --> 00:00:49,200 ... squared ... 15 00:00:49,850 --> 00:00:52,400 ... is what the complete square should be. 16 00:00:52,800 --> 00:00:54,950 Well making that complete square ... 17 00:00:55,050 --> 00:00:57,000 ... I add it ... 18 00:00:57,450 --> 00:00:59,300 ... a square of this number ... 19 00:00:59,400 --> 00:01:02,000 ... that I have to subtract immediately. 20 00:01:02,100 --> 00:01:05,400 So that it doesn't change the total value. 21 00:01:06,050 --> 00:01:07,350 b over two a squared. 22 00:01:07,450 --> 00:01:11,650 So this difference is equal to these two terms together. 23 00:01:11,750 --> 00:01:14,500 And I still have that extra c over a. 24 00:01:18,800 --> 00:01:19,800 All right. 25 00:01:20,300 --> 00:01:23,000 And then what? And then I would ... 26 00:01:26,850 --> 00:01:28,750 Well I'm done with completing the square. 27 00:01:28,850 --> 00:01:31,600 So let me try to simplify a little. 28 00:01:32,650 --> 00:01:40,800 It is a times x plus b over two a squared ... 29 00:01:41,700 --> 00:01:45,400 If I multiply by a it's going to be ... 30 00:01:45,500 --> 00:01:49,400 ... minus b squared over ... 31 00:01:49,500 --> 00:01:50,500 ... four ... 32 00:01:51,800 --> 00:01:52,800 ... a ... 33 00:01:53,600 --> 00:01:56,250 ... plus c. 34 00:01:56,350 --> 00:02:01,800 And I'd also like to bring that c ... 35 00:02:01,900 --> 00:02:04,800 ... with this term to the common denominator. 36 00:02:05,150 --> 00:02:06,800 So c ... 37 00:02:08,100 --> 00:02:09,750 I have just plus c. 38 00:02:09,850 --> 00:02:15,400 So it will come to four a times c. 39 00:02:17,400 --> 00:02:18,600 Is that right? 40 00:02:18,700 --> 00:02:21,100 No. That's right. 41 00:02:22,100 --> 00:02:26,550 So it just may be common denominator for all the remaining terms. 42 00:02:31,000 --> 00:02:33,900 Isn't it reminding you of something? 43 00:02:34,000 --> 00:02:36,000 Minus b squared ... 44 00:02:36,650 --> 00:02:38,150 So er ... 45 00:02:40,100 --> 00:02:44,700 Isn't it reminding you of the quadratic equation? 46 00:02:44,800 --> 00:02:46,900 Right? If you want to solve that ... 47 00:02:47,000 --> 00:02:49,500 This is what I want to solve. And this is what you want to solve. 48 00:02:49,600 --> 00:02:52,900 And that is what you want to solve. Can you solve this? 49 00:02:56,700 --> 00:03:01,000 Well you have your ... You have to solve for this x. It appears only once. 50 00:03:02,100 --> 00:03:06,000 So all you have to do is you have to undo all the remaining stuff. 51 00:03:06,100 --> 00:03:07,100 So ... 52 00:03:07,300 --> 00:03:08,900 Let me do that. 53 00:03:09,000 --> 00:03:13,700 a times x plus b over two a squared. 54 00:03:13,800 --> 00:03:16,000 I will take this to the right. 55 00:03:16,500 --> 00:03:18,900 So that should equal to ... 56 00:03:19,450 --> 00:03:24,600 ... b squared minus four a c divided by four a. 57 00:03:26,550 --> 00:03:29,200 And then of course I have to divide by a. 58 00:03:29,800 --> 00:03:31,500 We divide by a ... 59 00:03:32,000 --> 00:03:34,400 ... making this four a squared. 60 00:03:34,500 --> 00:03:38,300 And then I will have to take a square root on both sides. 61 00:03:38,750 --> 00:03:41,200 Can I always take the square root? 62 00:03:42,650 --> 00:03:44,250 Of any number? 63 00:03:45,800 --> 00:03:48,500 What if that number is negative? 64 00:03:49,200 --> 00:03:51,000 That's a problem, right? 65 00:03:51,600 --> 00:03:55,500 So what can possibly make this number negative? 66 00:03:55,950 --> 00:03:58,600 Can four a squared make it negative? 67 00:03:58,700 --> 00:04:00,300 Never, right? 68 00:04:00,400 --> 00:04:02,800 So it is only that number. 69 00:04:03,700 --> 00:04:11,050 If b squared minus four a c is less than zero ... 70 00:04:11,150 --> 00:04:12,600 ... then what? 71 00:04:12,700 --> 00:04:15,800 Then I cannot take a square root. 72 00:04:15,900 --> 00:04:18,300 And then I cannot solve the equation. 73 00:04:18,700 --> 00:04:21,000 Is that of some importance? 74 00:04:21,450 --> 00:04:24,000 From what you learned in the school, right? 75 00:04:24,100 --> 00:04:26,750 So this is how you decide whether there are roots or not. 76 00:04:26,850 --> 00:04:29,250 And if this number is positive then what? 77 00:04:29,350 --> 00:04:32,900 Well then you solve fot x by taking the square root. 78 00:04:33,000 --> 00:04:37,300 So x equals negative that number. 79 00:04:37,400 --> 00:04:39,600 b over two a. 80 00:04:40,250 --> 00:04:44,200 And then of course we will take a square root. It should be plus-minus ... 81 00:04:45,300 --> 00:04:48,100 Because there are two possible way of taking the square root. 82 00:04:48,200 --> 00:04:51,700 b squared minus four a c over ... 83 00:04:52,500 --> 00:04:53,700 ... four a squared. 84 00:04:53,800 --> 00:04:57,100 And you notice that the square root of four a squared if two a. 85 00:04:57,200 --> 00:05:01,100 So the whole thing is common denominator two a ... 86 00:05:01,200 --> 00:05:05,000 ... and the numerator is minus b plus-minus square root of that ... 87 00:05:05,100 --> 00:05:08,800 ... our favorite number b squared minus four a c. 88 00:05:11,600 --> 00:05:14,900 Now my point is in doing this ... 89 00:05:15,000 --> 00:05:21,000 ... is ... Well, remember I did the Product Rule for some reason, right? 90 00:05:21,100 --> 00:05:26,600 I did this for the reason of showing you what is essential in the whole thing. 91 00:05:27,000 --> 00:05:32,100 Because probably some of you may thought that the most essential thing about it is the formula. 92 00:05:32,200 --> 00:05:36,200 And you memorized that formula and probably forgot it already several times. 93 00:05:36,300 --> 00:05:40,600 And no. The formula is not the most important thing. 94 00:05:41,400 --> 00:05:46,000 So what is it that makes the whole thing work? 95 00:05:47,300 --> 00:05:50,000 What is the most essential thing? 96 00:05:50,500 --> 00:05:52,600 Student: Completing the square. NB: Completing the square. 97 00:05:52,700 --> 00:05:55,900 One step that makes the whole thing work. 98 00:05:56,000 --> 00:06:00,500 So that's why that operation of completing square is so important. 99 00:06:00,600 --> 00:06:03,900 And unfortunately ... 100 00:06:04,000 --> 00:06:07,500 ... it is probably not emphasized in school ... 101 00:06:07,600 --> 00:06:12,900 ... that solving quadratic equation amounts to learning how to complete a square. 102 00:06:13,000 --> 00:06:14,750 Well at some point you have to learn it. 103 00:06:14,850 --> 00:06:17,600 So you've just learned it today. 104 00:06:17,700 --> 00:06:18,900 Congratulations. |

Video Lectures > Calculus III (2011 Summer) > Lecture 2011.06.15 Projection and Components of a Vector > Part 14 Solving Quadratic Equations >