WEBVTT 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 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 00:00:32.972 --> 00:00:33.540 we divide 00:00:33.540 --> 00:00:52.277 both sides by the leading coefficient. Alright, now we've got to complete the 00:00:52.277 --> 00:00:55.440 square. Okay, 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. 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 00:01:25.827 --> 00:01:30.040 could write it like this. 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 00:01:44.614 --> 00:01:47.800 squared over four a squared. 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 00:01:57.192 --> 00:02:02.460 normally write it just to the right over here. 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 00:02:15.228 --> 00:02:22.980 over two a. It is always this thing divided by two. 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 00:02:35.291 --> 00:02:45.880 common denominator. So the common denominator is going to be 4A squared. 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 00:03:01.190 --> 00:03:02.760 squared. 00:03:15.880 --> 00:03:26.041 Okay. Now we have something squared equals some number. Is it just a number? What do 00:03:26.041 --> 00:03:35.719 we use next? The square root principle. The something that is squared is plus or 00:03:35.719 --> 00:03:38.380 minus the square root. 00:03:53.040 --> 00:04:04.200 And what? Let's subtract B over 2A. And I put it on the left side. 00:04:26.740 --> 00:04:31.360 Then this denominator simplifies. What does that simplify to? 00:04:31.360 --> 00:04:56.360 Two a. Now we have a common denominator, so we can make one fraction. 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 00:05:24.018 --> 00:05:27.420 coefficients A, B, and C. 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. 00:05:35.852 --> 00:05:43.505 This is what's known as the quadratic formula. The very famous formula. It works for 00:05:43.505 --> 00:05:46.420 every single quadratic equation. 00:05:46.420 --> 00:05:54.080 Okay. So this is called the quadratic formula. 00:05:54.840 --> 00:06:02.886 In fact, we never have to use completing the square again. We did it for all 00:06:02.886 --> 00:06:11.673 quadratics. Instead of going through the whole thing, we plug the numbers a, b, and 00:06:11.673 --> 00:06:14.320 c into here and simplify. 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 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 00:06:28.616 --> 00:06:34.960 not quick and you cannot figure it out, we go to the quadratic formula. 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 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 00:06:52.996 --> 00:06:56.100 subtract 3x from both sides. 00:07:25.000 --> 00:07:26.400 Okay. 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 00:07:31.708 --> 00:07:32.420 over two a. 00:07:55.560 --> 00:08:07.637 We have negative b, the negative of negative three, plus or minus the square root of 00:08:07.637 --> 00:08:19.426 b squared: negative three squared minus four times a, which is one, times c, which 00:08:19.426 --> 00:08:25.320 is negative four, all over two times one. 00:08:27.720 --> 00:08:34.720 And now we simplify. 00:08:43.940 --> 00:08:49.101 Four times negative four is negative sixteen, so subtracting negative sixteen gives 00:08:49.101 --> 00:08:50.780 plus sixteen, all over two. 00:09:09.260 --> 00:09:20.178 And so again, this is two separate equations. Well, I'm going to write one more 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 00:09:31.373 --> 00:09:32.340 over 2. 00:09:59.600 --> 00:10:01.000 Okay. 00:10:01.000 --> 00:10:09.520 Our solutions are integers. That tells us it could have been solved by factoring. If 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 00:10:17.939 --> 00:10:22.300 quadratic formula or completing the square. 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 00:10:31.833 --> 00:10:34.840 formula. This is very important. 00:10:35.480 --> 00:10:39.420 One way to memorize it is by this song. 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 00:10:50.660 --> 00:10:52.060 over two a. 00:10:52.960 --> 00:11:01.980 Okay, now with your pen in hand on your paper, we're going to say. 00:11:05.240 --> 00:11:09.820 We are going to sing and write the formula as we go. 00:11:15.320 --> 00:11:17.200 Here we go. 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 00:11:30.636 --> 00:11:32.100 over two a. 00:11:33.100 --> 00:11:37.020 This is already in standard form. What is my a? 00:11:38.800 --> 00:11:39.920 Two. b? 00:11:41.480 --> 00:11:42.580 Negative five. c? 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 00:11:51.383 --> 00:11:53.760 numbers plugged in. So here we go. 00:12:33.100 --> 00:12:33.580 Okay. 00:13:22.040 --> 00:13:23.600 Can we simplify the square root of thirty-three? 00:13:30.200 --> 00:13:35.340 And we're done. That's it. These are this is my solution set. 00:13:39.500 --> 00:13:43.360 If we ever get this, we're going to learn this later. 00:13:44.920 --> 00:13:47.840 What was my original equation? Two x squared. 00:13:54.020 --> 00:13:56.660 It tells us this does factor. 00:14:00.120 --> 00:14:01.800 The two zeros. 00:14:03.620 --> 00:14:13.560 We subtract these numbers: five minus square root thirty-three, over four. 00:14:17.720 --> 00:14:21.840 Five plus square root thirty-three, over four. 00:14:25.140 --> 00:14:29.680 Except we're missing the leading coefficient. So we have the two out here. 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 00:14:36.682 --> 00:14:38.520 minus the other solution. 00:14:39.620 --> 00:14:43.876 It should make sense, right? You plug in a solution right here. That same number 00:14:43.876 --> 00:14:45.100 minus that number zero. 00:14:47.100 --> 00:14:50.940 And the other number plus. 00:14:51.640 --> 00:14:55.400 Five plus square root thirty-three, over four, minus that number. 00:14:56.360 --> 00:15:06.107 And that's it. So every polynomial does factor to exactly the leading coefficient, 00:15:06.107 --> 00:15:15.140 the A, times X minus one of the solutions times X minus the other solutions. 00:15:16.880 --> 00:15:19.340 OK, let's see some other. Let's try some more. 00:15:20.040 --> 00:15:25.980 This quadratic has a common factor in all three terms. 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. 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 00:15:40.579 --> 00:15:41.040 easier. 00:15:41.500 --> 00:15:49.099 Let's see what happens when we use quadratic formula when there is a common factor 00:15:49.099 --> 00:15:49.840 in here. 00:15:50.700 --> 00:15:53.220 Negative b. What is b? 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 00:16:08.957 --> 00:16:09.980 two a. 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 00:16:17.181 --> 00:16:19.820 plus or minus the square root of b 00:16:19.820 --> 00:16:21.780 Square minus four times A C. 00:16:25.820 --> 00:16:30.880 And now let's simplify. 00:16:36.440 --> 00:16:44.980 What is that plus sixteen? 00:17:16.860 --> 00:17:23.798 OK, this is the two solutions. However, we can simplify this. What is the square 00:17:23.798 --> 00:17:26.660 root of thirty two simplified to? 00:17:26.660 --> 00:17:27.740 Four square root two. 00:17:34.340 --> 00:17:39.260 Now both of these have a common factor of four. 00:17:41.080 --> 00:17:44.560 So we can either divide both of these by four. 00:17:47.900 --> 00:17:54.080 It would be negative four over four, plus or minus four square root two over four. 00:17:56.260 --> 00:18:02.960 We get negative one plus or minus square root two. 00:18:05.420 --> 00:18:10.500 The other option is you factor out a four. 00:18:15.540 --> 00:18:16.500 And those simplify. 00:18:18.440 --> 00:18:20.320 Negative one plus or minus square root two. 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 00:18:27.546 --> 00:18:30.120 definitely going to have to simplify. 00:18:31.180 --> 00:18:35.820 And there will be a common thing that factors out. 00:18:36.540 --> 00:18:38.180 OK, let's try another one. 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 00:19:13.580 --> 00:19:14.460 over two a. 00:19:21.420 --> 00:19:22.220 What is B? 00:19:23.220 --> 00:19:24.140 Negative ten. 00:19:26.000 --> 00:19:27.560 Negative ten. 00:19:27.880 --> 00:19:30.400 Negative ten squared. 00:20:19.400 --> 00:20:22.200 Negative ten. 00:20:28.120 --> 00:20:36.040 So our two solutions are ten plus zero over two and ten minus zero over two. 00:20:37.620 --> 00:20:40.360 OK, so we really only have one solution. 00:20:43.040 --> 00:20:48.121 Whenever we end up with just one solution or in other words, when this thing is 00:20:48.121 --> 00:20:51.080 zero, it's the only time you get one solution. 00:20:52.080 --> 00:20:57.720 It does tell us that this is a perfect squared trinomial. 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. 00:21:10.740 --> 00:21:20.160 Whenever the thing inside the radical of the quadratic formula is zero, 00:21:20.400 --> 00:21:21.500 We're only going to get one solution. 00:21:23.600 --> 00:21:27.640 And it tells us the quadratic is a perfect square trinomial. 00:21:32.040 --> 00:21:33.480 Now let's see what happens. 00:21:56.220 --> 00:21:57.620 OK. 00:22:28.820 --> 00:22:36.800 Sometimes b squared minus four a c is a negative number. 00:22:37.220 --> 00:22:38.720 So what does that tell us? 00:22:40.200 --> 00:22:41.840 This is not a real number. 00:22:47.800 --> 00:22:51.680 OK, so there are no real solutions. 00:22:57.060 --> 00:23:02.460 Does it have a solution set? Yes. Yes, and it is the empty set. 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. 00:23:10.180 --> 00:23:17.260 Later on, we will learn something very interesting, and we will not be done here. 00:23:17.880 --> 00:23:20.040 But right now we will be done. 00:23:21.480 --> 00:23:24.860 OK, so let's look one more time at this thing. 00:23:25.500 --> 00:23:35.340 OK, this radicand of the quadratic formula is called the discriminant. 00:23:36.140 --> 00:23:41.080 OK, so this number classifies our solutions. 00:23:43.160 --> 00:23:53.860 OK, if B squared minus four AC is greater than zero, what does that tell us? 00:23:56.240 --> 00:24:00.780 There's two real solutions. 00:24:10.020 --> 00:24:18.564 If the discriminant B squared minus four AC is equal to zero, what does that tell 00:24:18.564 --> 00:24:18.880 us? 00:24:20.060 --> 00:24:24.580 There's one real solution. 00:24:25.580 --> 00:24:42.340 It also tells us it can be factored: the quadratic is a perfect square trinomial. 00:24:51.880 --> 00:25:01.340 And if B squared minus four AC is less than zero, what does that tell us? 00:25:02.400 --> 00:25:05.340 There are no real solutions. 00:25:06.700 --> 00:25:10.600 OK, and then we tell this even more. 00:25:14.320 --> 00:25:21.040 If B squared minus four AC is a perfect square. 00:25:27.340 --> 00:25:29.020 What does that tell us? 00:25:30.900 --> 00:25:31.620 What's that? 00:25:31.620 --> 00:25:32.900 The solutions are integers? 00:25:33.780 --> 00:25:40.560 Not necessarily because we have to divide at least by two. 00:25:40.920 --> 00:25:46.220 It might not be an integer, but the square root of the perfect square. 00:25:48.840 --> 00:25:50.840 The square root will be an integer. 00:25:51.360 --> 00:25:52.280 Well, it could be. 00:25:53.320 --> 00:25:57.340 Yeah, but it tells us we're going to have two. 00:25:57.680 --> 00:26:00.060 So the square root will simplify since it's a perfect square. 00:26:00.060 --> 00:26:05.560 So we have two rational solutions. 00:26:10.380 --> 00:26:12.640 What's a rational number? 00:26:15.940 --> 00:26:19.580 Any number that can be written as the ratio of two integers. 00:26:21.580 --> 00:26:31.720 OK, if B squared minus four AC is not a perfect square. 00:26:38.020 --> 00:26:38.420 Then what? 00:26:41.540 --> 00:26:50.120 Then there are two irrational solutions. 00:26:54.580 --> 00:26:56.940 And your answer is going to have a square root. 00:26:58.940 --> 00:27:04.526 If we want to approximate the solution, we need a calculator to type in the square 00:27:04.526 --> 00:27:07.660 root, then add negative b and divide by two a. 00:27:08.860 --> 00:27:11.200 OK, so the discriminant tells a lot. 00:27:12.100 --> 00:27:13.160 If it's positive, two real. 00:27:13.860 --> 00:27:15.460 If it's zero, there's one real. 00:27:16.380 --> 00:27:19.200 And the quadratic is a perfect square trinomial. 00:27:19.800 --> 00:27:22.580 If the discriminant is negative, there's no real solutions. 00:27:23.800 --> 00:27:25.980 If it's a perfect square, there's two rational. 00:27:32.760 --> 00:27:34.940 And if it's not a perfect square, there's two irrational. 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 00:27:42.612 --> 00:27:43.440 factoring.