1 00:00:05,640 --> 00:00:27,800 so first step is we're going to subtract C from both sides. Okay. And now, where are 2 00:00:27,800 --> 00:00:32,972 Where are we at? Divide by a. We need to turn the leading coefficient into one, so 3 00:00:32,972 --> 00:00:33,540 we divide 4 00:00:33,540 --> 00:00:52,277 both sides by the leading coefficient. Alright, now we've got to complete the 5 00:00:52,277 --> 00:00:55,440 square. Okay, 6 00:01:04,520 --> 00:01:11,840 We take this, which is no longer our b value, but it is like our new b value. 7 00:01:11,840 --> 00:01:25,827 We take that, divide by two, and square it. That becomes (b over two a) squared. We 8 00:01:25,827 --> 00:01:30,040 could write it like this. 9 00:01:35,740 --> 00:01:44,614 Or we can leave it like this, either one. This is what we add to both sides: b 10 00:01:44,614 --> 00:01:47,800 squared over four a squared. 11 00:01:47,800 --> 00:01:57,192 And I'm going to write on the left side here. You'll see why in a minute. Okay. We 12 00:01:57,192 --> 00:02:02,460 normally write it just to the right over here. 13 00:02:02,460 --> 00:02:15,228 If I put it here, you will see. Now this factors. What does this factor to? x plus b 14 00:02:15,228 --> 00:02:22,980 over two a. It is always this thing divided by two. 15 00:02:23,820 --> 00:02:35,291 So B over 2A. B over 2A. And then the right side, we're going to need to get a 16 00:02:35,291 --> 00:02:45,880 common denominator. So the common denominator is going to be 4A squared. 17 00:02:45,880 --> 00:03:01,190 So this needs a 4A over 4A. That's B squared over 4A squared minus 4AC over 4A 18 00:03:01,190 --> 00:03:02,760 squared. 19 00:03:15,880 --> 00:03:26,041 Okay. Now we have something squared equals some number. Is it just a number? What do 20 00:03:26,041 --> 00:03:35,719 we use next? The square root principle. The something that is squared is plus or 21 00:03:35,719 --> 00:03:38,380 minus the square root. 22 00:03:53,040 --> 00:04:04,200 And what? Let's subtract B over 2A. And I put it on the left side. 23 00:04:26,740 --> 00:04:31,360 Then this denominator simplifies. What does that simplify to? 24 00:04:31,360 --> 00:04:56,360 Two a. Now we have a common denominator, so we can make one fraction. 25 00:05:12,860 --> 00:05:24,018 And there it is. Okay. We just derived or proved that all we got to do is know the 26 00:05:24,018 --> 00:05:27,420 coefficients A, B, and C. 27 00:05:28,200 --> 00:05:35,852 And all we do is plug it into this formula. And this will give us our two solutions. 28 00:05:35,852 --> 00:05:43,505 This is what's known as the quadratic formula. The very famous formula. It works for 29 00:05:43,505 --> 00:05:46,420 every single quadratic equation. 30 00:05:46,420 --> 00:05:54,080 Okay. So this is called the quadratic formula. 31 00:05:54,840 --> 00:06:02,886 In fact, we never have to use completing the square again. We did it for all 32 00:06:02,886 --> 00:06:11,673 quadratics. Instead of going through the whole thing, we plug the numbers a, b, and 33 00:06:11,673 --> 00:06:14,320 c into here and simplify. 34 00:06:14,320 --> 00:06:21,379 But whenever you want to solve by factoring, you do not have to do this. If the 35 00:06:21,379 --> 00:06:28,616 quadratic can be solved by factoring, we usually do that if it is quick. If it is 36 00:06:28,616 --> 00:06:34,960 not quick and you cannot figure it out, we go to the quadratic formula. 37 00:06:35,480 --> 00:06:44,238 Okay. Before we use the quadratic formula, what do we have to do? Put it in the 38 00:06:44,238 --> 00:06:52,996 standard form. It's called Ax squared plus Bx plus C equals zero. So we have to 39 00:06:52,996 --> 00:06:56,100 subtract 3x from both sides. 40 00:07:25,000 --> 00:07:26,400 Okay. 41 00:07:26,400 --> 00:07:31,708 x equals negative b plus or minus the square root of b squared minus four a c, all 42 00:07:31,708 --> 00:07:32,420 over two a. 43 00:07:55,560 --> 00:08:07,637 We have negative b, the negative of negative three, plus or minus the square root of 44 00:08:07,637 --> 00:08:19,426 b squared: negative three squared minus four times a, which is one, times c, which 45 00:08:19,426 --> 00:08:25,320 is negative four, all over two times one. 46 00:08:27,720 --> 00:08:34,720 And now we simplify. 47 00:08:43,940 --> 00:08:49,101 Four times negative four is negative sixteen, so subtracting negative sixteen gives 48 00:08:49,101 --> 00:08:50,780 plus sixteen, all over two. 49 00:09:09,260 --> 00:09:20,178 And so again, this is two separate equations. Well, I'm going to write one more 50 00:09:20,178 --> 00:09:31,373 first. 3 plus or minus 5 over 2, which is x is 3 plus 5 over 2, or x is 3 minus 5 51 00:09:31,373 --> 00:09:32,340 over 2. 52 00:09:59,600 --> 00:10:01,000 Okay. 53 00:10:01,000 --> 00:10:09,520 Our solutions are integers. That tells us it could have been solved by factoring. If 54 00:10:09,520 --> 00:10:17,939 the thing inside the square root is not a perfect square, it could be solved by the 55 00:10:17,939 --> 00:10:22,300 quadratic formula or completing the square. 56 00:10:23,940 --> 00:10:31,833 This makes sense. Okay. That's the first one using a formula. We got to memorize the 57 00:10:31,833 --> 00:10:34,840 formula. This is very important. 58 00:10:35,480 --> 00:10:39,420 One way to memorize it is by this song. 59 00:10:40,220 --> 00:10:50,660 x equals negative b plus or minus the square root of b squared minus four a c, all 60 00:10:50,660 --> 00:10:52,060 over two a. 61 00:10:52,960 --> 00:11:01,980 Okay, now with your pen in hand on your paper, we're going to say. 62 00:11:05,240 --> 00:11:09,820 We are going to sing and write the formula as we go. 63 00:11:15,320 --> 00:11:17,200 Here we go. 64 00:11:19,720 --> 00:11:30,636 x equals negative b plus or minus the square root of b squared minus four a c, all 65 00:11:30,636 --> 00:11:32,100 over two a. 66 00:11:33,100 --> 00:11:37,020 This is already in standard form. What is my a? 67 00:11:38,800 --> 00:11:39,920 Two. b? 68 00:11:41,480 --> 00:11:42,580 Negative five. c? 69 00:11:45,860 --> 00:11:51,383 So first let's write it with the letters in there and then we'll write with the 70 00:11:51,383 --> 00:11:53,760 numbers plugged in. So here we go. 71 00:12:33,100 --> 00:12:33,580 Okay. 72 00:13:22,040 --> 00:13:23,600 Can we simplify the square root of thirty-three? 73 00:13:30,200 --> 00:13:35,340 And we're done. That's it. These are this is my solution set. 74 00:13:39,500 --> 00:13:43,360 If we ever get this, we're going to learn this later. 75 00:13:44,920 --> 00:13:47,840 What was my original equation? Two x squared. 76 00:13:54,020 --> 00:13:56,660 It tells us this does factor. 77 00:14:00,120 --> 00:14:01,800 The two zeros. 78 00:14:03,620 --> 00:14:13,560 We subtract these numbers: five minus square root thirty-three, over four. 79 00:14:17,720 --> 00:14:21,840 Five plus square root thirty-three, over four. 80 00:14:25,140 --> 00:14:29,680 Except we're missing the leading coefficient. So we have the two out here. 81 00:14:30,580 --> 00:14:36,682 So that's what this thing factors to two times X minus one of the solutions times X 82 00:14:36,682 --> 00:14:38,520 minus the other solution. 83 00:14:39,620 --> 00:14:43,876 It should make sense, right? You plug in a solution right here. That same number 84 00:14:43,876 --> 00:14:45,100 minus that number zero. 85 00:14:47,100 --> 00:14:50,940 And the other number plus. 86 00:14:51,640 --> 00:14:55,400 Five plus square root thirty-three, over four, minus that number. 87 00:14:56,360 --> 00:15:06,107 And that's it. So every polynomial does factor to exactly the leading coefficient, 88 00:15:06,107 --> 00:15:15,140 the A, times X minus one of the solutions times X minus the other solutions. 89 00:15:16,880 --> 00:15:19,340 OK, let's see some other. Let's try some more. 90 00:15:20,040 --> 00:15:25,980 This quadratic has a common factor in all three terms. 91 00:15:25,980 --> 00:15:33,120 What is the common factor? Two. But I want to show you what happens if we do this. 92 00:15:35,180 --> 00:15:40,579 We could divide both sides by two and make all the numbers smaller and it would be 93 00:15:40,579 --> 00:15:41,040 easier. 94 00:15:41,500 --> 00:15:49,099 Let's see what happens when we use quadratic formula when there is a common factor 95 00:15:49,099 --> 00:15:49,840 in here. 96 00:15:50,700 --> 00:15:53,220 Negative b. What is b? 97 00:15:55,140 --> 00:16:08,957 Negative four plus or minus the square root of b squared minus four a c, all over 98 00:16:08,957 --> 00:16:09,980 two a. 99 00:16:10,740 --> 00:16:17,181 Hopefully we go right to that. You do not even have to list a, b, and c: negative b 100 00:16:17,181 --> 00:16:19,820 plus or minus the square root of b 101 00:16:19,820 --> 00:16:21,780 Square minus four times A C. 102 00:16:25,820 --> 00:16:30,880 And now let's simplify. 103 00:16:36,440 --> 00:16:44,980 What is that plus sixteen? 104 00:17:16,860 --> 00:17:23,798 OK, this is the two solutions. However, we can simplify this. What is the square 105 00:17:23,798 --> 00:17:26,660 root of thirty two simplified to? 106 00:17:26,660 --> 00:17:27,740 Four square root two. 107 00:17:34,340 --> 00:17:39,260 Now both of these have a common factor of four. 108 00:17:41,080 --> 00:17:44,560 So we can either divide both of these by four. 109 00:17:47,900 --> 00:17:54,080 It would be negative four over four, plus or minus four square root two over four. 110 00:17:56,260 --> 00:18:02,960 We get negative one plus or minus square root two. 111 00:18:05,420 --> 00:18:10,500 The other option is you factor out a four. 112 00:18:15,540 --> 00:18:16,500 And those simplify. 113 00:18:18,440 --> 00:18:20,320 Negative one plus or minus square root two. 114 00:18:22,120 --> 00:18:27,546 So if there is a common factor, you can still leave it in there, except you're 115 00:18:27,546 --> 00:18:30,120 definitely going to have to simplify. 116 00:18:31,180 --> 00:18:35,820 And there will be a common thing that factors out. 117 00:18:36,540 --> 00:18:38,180 OK, let's try another one. 118 00:19:07,020 --> 00:19:13,580 x equals negative b plus or minus the square root of b squared minus four a c, all 119 00:19:13,580 --> 00:19:14,460 over two a. 120 00:19:21,420 --> 00:19:22,220 What is B? 121 00:19:23,220 --> 00:19:24,140 Negative ten. 122 00:19:26,000 --> 00:19:27,560 Negative ten. 123 00:19:27,880 --> 00:19:30,400 Negative ten squared. 124 00:20:19,400 --> 00:20:22,200 Negative ten. 125 00:20:28,120 --> 00:20:36,040 So our two solutions are ten plus zero over two and ten minus zero over two. 126 00:20:37,620 --> 00:20:40,360 OK, so we really only have one solution. 127 00:20:43,040 --> 00:20:48,121 Whenever we end up with just one solution or in other words, when this thing is 128 00:20:48,121 --> 00:20:51,080 zero, it's the only time you get one solution. 129 00:20:52,080 --> 00:20:57,720 It does tell us that this is a perfect squared trinomial. 130 00:20:59,740 --> 00:21:10,740 OK, this factors to X minus five times X minus five, which is X minus five squared. 131 00:21:10,740 --> 00:21:20,160 Whenever the thing inside the radical of the quadratic formula is zero, 132 00:21:20,400 --> 00:21:21,500 We're only going to get one solution. 133 00:21:23,600 --> 00:21:27,640 And it tells us the quadratic is a perfect square trinomial. 134 00:21:32,040 --> 00:21:33,480 Now let's see what happens. 135 00:21:56,220 --> 00:21:57,620 OK. 136 00:22:28,820 --> 00:22:36,800 Sometimes b squared minus four a c is a negative number. 137 00:22:37,220 --> 00:22:38,720 So what does that tell us? 138 00:22:40,200 --> 00:22:41,840 This is not a real number. 139 00:22:47,800 --> 00:22:51,680 OK, so there are no real solutions. 140 00:22:57,060 --> 00:23:02,460 Does it have a solution set? Yes. Yes, and it is the empty set. 141 00:23:03,820 --> 00:23:09,340 OK, so as soon as B squared minus four AC is a negative, there's no real solution. 142 00:23:10,180 --> 00:23:17,260 Later on, we will learn something very interesting, and we will not be done here. 143 00:23:17,880 --> 00:23:20,040 But right now we will be done. 144 00:23:21,480 --> 00:23:24,860 OK, so let's look one more time at this thing. 145 00:23:25,500 --> 00:23:35,340 OK, this radicand of the quadratic formula is called the discriminant. 146 00:23:36,140 --> 00:23:41,080 OK, so this number classifies our solutions. 147 00:23:43,160 --> 00:23:53,860 OK, if B squared minus four AC is greater than zero, what does that tell us? 148 00:23:56,240 --> 00:24:00,780 There's two real solutions. 149 00:24:10,020 --> 00:24:18,564 If the discriminant B squared minus four AC is equal to zero, what does that tell 150 00:24:18,564 --> 00:24:18,880 us? 151 00:24:20,060 --> 00:24:24,580 There's one real solution. 152 00:24:25,580 --> 00:24:42,340 It also tells us it can be factored: the quadratic is a perfect square trinomial. 153 00:24:51,880 --> 00:25:01,340 And if B squared minus four AC is less than zero, what does that tell us? 154 00:25:02,400 --> 00:25:05,340 There are no real solutions. 155 00:25:06,700 --> 00:25:10,600 OK, and then we tell this even more. 156 00:25:14,320 --> 00:25:21,040 If B squared minus four AC is a perfect square. 157 00:25:27,340 --> 00:25:29,020 What does that tell us? 158 00:25:30,900 --> 00:25:31,620 What's that? 159 00:25:31,620 --> 00:25:32,900 The solutions are integers? 160 00:25:33,780 --> 00:25:40,560 Not necessarily because we have to divide at least by two. 161 00:25:40,920 --> 00:25:46,220 It might not be an integer, but the square root of the perfect square. 162 00:25:48,840 --> 00:25:50,840 The square root will be an integer. 163 00:25:51,360 --> 00:25:52,280 Well, it could be. 164 00:25:53,320 --> 00:25:57,340 Yeah, but it tells us we're going to have two. 165 00:25:57,680 --> 00:26:00,060 So the square root will simplify since it's a perfect square. 166 00:26:00,060 --> 00:26:05,560 So we have two rational solutions. 167 00:26:10,380 --> 00:26:12,640 What's a rational number? 168 00:26:15,940 --> 00:26:19,580 Any number that can be written as the ratio of two integers. 169 00:26:21,580 --> 00:26:31,720 OK, if B squared minus four AC is not a perfect square. 170 00:26:38,020 --> 00:26:38,420 Then what? 171 00:26:41,540 --> 00:26:50,120 Then there are two irrational solutions. 172 00:26:54,580 --> 00:26:56,940 And your answer is going to have a square root. 173 00:26:58,940 --> 00:27:04,526 If we want to approximate the solution, we need a calculator to type in the square 174 00:27:04,526 --> 00:27:07,660 root, then add negative b and divide by two a. 175 00:27:08,860 --> 00:27:11,200 OK, so the discriminant tells a lot. 176 00:27:12,100 --> 00:27:13,160 If it's positive, two real. 177 00:27:13,860 --> 00:27:15,460 If it's zero, there's one real. 178 00:27:16,380 --> 00:27:19,200 And the quadratic is a perfect square trinomial. 179 00:27:19,800 --> 00:27:22,580 If the discriminant is negative, there's no real solutions. 180 00:27:23,800 --> 00:27:25,980 If it's a perfect square, there's two rational. 181 00:27:32,760 --> 00:27:34,940 And if it's not a perfect square, there's two irrational. 182 00:27:35,660 --> 00:27:42,612 If b squared minus four a c is a perfect square, it tells us we could have solved by 183 00:27:42,612 --> 00:27:43,440 factoring.