WEBVTT 00:00:00.000 --> 00:00:00.890 00:00:00.890 --> 00:00:03.590 Welcome to part two of the presentation on 00:00:03.590 --> 00:00:05.660 quadratic equations. 00:00:05.660 --> 00:00:08.470 Well, I think I thoroughly confused you the last time 00:00:08.470 --> 00:00:11.170 around, so let me see if I can fix that a bit by doing 00:00:11.170 --> 00:00:12.770 several more examples. 00:00:12.770 --> 00:00:15.430 So let's just start with a review of what the 00:00:15.430 --> 00:00:16.380 quadratic equation is. 00:00:16.380 --> 00:00:19.650 The quadratic equation says, if I'm trying to solve the 00:00:19.650 --> 00:00:31.590 equation Ax squared plus Bx plus C equals 0, then the 00:00:31.590 --> 00:00:35.440 solution or the solutions because there's usually two 00:00:35.440 --> 00:00:38.970 times that it intersects the x-axis, or two solutions for 00:00:38.970 --> 00:00:47.610 this equation is x equals minus B plus or minus the square root 00:00:47.610 --> 00:00:56.390 of B squared minus 4 times A times C. 00:00:56.390 --> 00:01:00.270 And all of that over 2A. 00:01:00.270 --> 00:01:02.040 So let's do a problem and hopefully this should make 00:01:02.040 --> 00:01:02.690 a little more sense. 00:01:02.690 --> 00:01:04.620 That's a 2 on the bottom. 00:01:04.620 --> 00:01:13.890 So let's say I had the equation minus 9x squared minus 00:01:13.890 --> 00:01:19.950 9x plus 6 equals 0. 00:01:19.950 --> 00:01:22.230 So in this example what's A? 00:01:22.230 --> 00:01:25.410 Well, A is the coefficient on the x squared term. 00:01:25.410 --> 00:01:29.820 The x squared term is here, the coefficient is minus 9. 00:01:29.820 --> 00:01:30.620 So let's write that. 00:01:30.620 --> 00:01:34.120 A equals minus 9. 00:01:34.120 --> 00:01:35.400 What does B equal? 00:01:35.400 --> 00:01:39.180 B is the coefficient on the x term, so that's this term here. 00:01:39.180 --> 00:01:43.220 So B is also equal to minus 9. 00:01:43.220 --> 00:01:47.140 And C is the constant term, which in this example is 6. 00:01:47.140 --> 00:01:49.550 So C is equal to 6. 00:01:49.550 --> 00:01:52.070 Now we just substitute these values into the actual 00:01:52.070 --> 00:01:53.260 quadratic equation. 00:01:53.260 --> 00:01:59.600 So negative B, so it's negative times negative 9. 00:01:59.600 --> 00:02:00.780 That's B. 00:02:00.780 --> 00:02:08.110 Plus or minus the square root of B squared, so that's 81. 00:02:08.110 --> 00:02:08.390 Right? 00:02:08.390 --> 00:02:10.030 Negative 9 squared. 00:02:10.030 --> 00:02:14.720 Minus 4 times negative 9. 00:02:14.720 --> 00:02:16.140 That's A. 00:02:16.140 --> 00:02:19.480 Times C, which is 6. 00:02:19.480 --> 00:02:23.950 And all of that over 2 times negative 9, which 00:02:23.950 --> 00:02:25.630 is minus 18, right? 00:02:25.630 --> 00:02:26.720 2 times negative 9-- 2A. 00:02:26.720 --> 00:02:29.230 00:02:29.230 --> 00:02:33.760 Let's try to simplify this up here. 00:02:33.760 --> 00:02:37.930 Well, negative negative 9, that's positive 9. 00:02:37.930 --> 00:02:46.480 Plus or minus the square root of 81. 00:02:46.480 --> 00:02:47.900 Let's see. 00:02:47.900 --> 00:02:50.270 This is negative 4 times negative 9. 00:02:50.270 --> 00:02:53.470 Negative 4 times negative 9 is positive 36. 00:02:53.470 --> 00:02:58.310 And then positive 36 times 6 is-- let's see. 00:02:58.310 --> 00:03:01.330 30 times 6 is 180. 00:03:01.330 --> 00:03:07.890 And then 180 plus another 36 is 216. 00:03:07.890 --> 00:03:10.980 Plus 216, is that right? 00:03:10.980 --> 00:03:14.490 180 plus 36 is 216. 00:03:14.490 --> 00:03:16.840 All of that over 2A. 00:03:16.840 --> 00:03:19.570 2A we already said is minus 19. 00:03:19.570 --> 00:03:20.740 So we simplify that more. 00:03:20.740 --> 00:03:28.090 That's 9 plus or minus the square root 81 plus 216. 00:03:28.090 --> 00:03:30.400 That's 80 plus 217. 00:03:30.400 --> 00:03:38.040 That's 297. 00:03:38.040 --> 00:03:41.900 And all of that over minus 18. 00:03:41.900 --> 00:03:45.020 Now, this is actually-- the hardest part with the quadratic 00:03:45.020 --> 00:03:47.720 equation is oftentimes just simplifying this expression. 00:03:47.720 --> 00:03:50.860 We have to figure out if we can simplify this radical. 00:03:50.860 --> 00:03:53.090 Well, let's see. 00:03:53.090 --> 00:03:56.490 One way to figure out if a number is divisible by 9 is to 00:03:56.490 --> 00:03:58.320 actually add up the digits and see if the digits 00:03:58.320 --> 00:03:59.260 are divisible by 9. 00:03:59.260 --> 00:03:59.950 In this case, it is. 00:03:59.950 --> 00:04:02.510 2 plus 9 plus 7 is equal to 18. 00:04:02.510 --> 00:04:04.600 So let's see how many times 9 goes into that. 00:04:04.600 --> 00:04:07.150 I'll do it on the side here; I don't want to be too messy. 00:04:07.150 --> 00:04:09.450 9 goes into 2 97. 00:04:09.450 --> 00:04:13.630 00:04:13.630 --> 00:04:16.190 3 times 27. 00:04:16.190 --> 00:04:19.040 27-- it goes 33 times, right? 00:04:19.040 --> 00:04:24.290 So this is the same thing as 9 plus or minus the square root 00:04:24.290 --> 00:04:31.110 of 9 times 33 over minus 18. 00:04:31.110 --> 00:04:32.470 And 9 is a perfect square. 00:04:32.470 --> 00:04:34.650 That's why I actually wanted to see if 9 would work because 00:04:34.650 --> 00:04:36.390 that's the only way I could get it out of the radical, if 00:04:36.390 --> 00:04:37.390 it's a perfect square. 00:04:37.390 --> 00:04:40.410 As you learned in that exponent rules number one module. 00:04:40.410 --> 00:04:46.140 So this is equal to 9 plus or minus 3 times the square 00:04:46.140 --> 00:04:53.230 root of 33, and all of that over minus 18. 00:04:53.230 --> 00:04:54.570 We're almost done. 00:04:54.570 --> 00:04:57.840 We can actually simplify it because 9, 3, and minus 18 00:04:57.840 --> 00:05:00.650 are all divisible by 3. 00:05:00.650 --> 00:05:02.270 Let's divide everything by 3. 00:05:02.270 --> 00:05:14.370 3 plus or minus the square root of 33 over minus 6. 00:05:14.370 --> 00:05:15.610 And we are done. 00:05:15.610 --> 00:05:17.010 So as you see, the hardest thing with the quadratic 00:05:17.010 --> 00:05:20.110 equation is often just simplifying the expression. 00:05:20.110 --> 00:05:22.750 But what we've said, I know you might have lost track-- we did 00:05:22.750 --> 00:05:27.120 all this math-- is we said, this equation: minus 9x 00:05:27.120 --> 00:05:30.550 squared minus 9x plus 6. 00:05:30.550 --> 00:05:34.200 Now we found two x values that would satisfy this equation 00:05:34.200 --> 00:05:35.970 and make it equal to 0. 00:05:35.970 --> 00:05:39.830 One x value is x equals 3 plus the square root 00:05:39.830 --> 00:05:42.100 of 33 over minus 6. 00:05:42.100 --> 00:05:45.860 And the second value is 3 minus the square root 00:05:45.860 --> 00:05:50.160 of 33 over minus 6. 00:05:50.160 --> 00:05:52.250 And you might want to think about why we have 00:05:52.250 --> 00:05:53.370 that plus or minus. 00:05:53.370 --> 00:05:55.490 We have that plus or minus because a square root could 00:05:55.490 --> 00:05:59.550 actually be a positive or a negative number. 00:05:59.550 --> 00:06:02.180 Let's do another problem. 00:06:02.180 --> 00:06:05.890 Hopefully this one will be a little bit simpler. 00:06:05.890 --> 00:06:09.210 00:06:09.210 --> 00:06:16.780 Let's say I wanted to solve minus 8x squared 00:06:16.780 --> 00:06:21.000 plus 5x plus 9. 00:06:21.000 --> 00:06:23.150 Now I'm going to assume that you've memorized the quadratic 00:06:23.150 --> 00:06:25.310 equation because that's something you should do. 00:06:25.310 --> 00:06:26.630 Or you should write it down on a piece of paper. 00:06:26.630 --> 00:06:31.630 But the quadratic equation is negative B-- So b is 5, right? 00:06:31.630 --> 00:06:34.160 We're trying to solve that equal to 0, so negative B. 00:06:34.160 --> 00:06:39.790 So negative 5, plus or minus the square root of B squared- 00:06:39.790 --> 00:06:44.030 that's 5 squared, 25. 00:06:44.030 --> 00:06:50.470 Minus 4 times A, which is minus 8. 00:06:50.470 --> 00:06:53.820 Times C, which is 9. 00:06:53.820 --> 00:06:56.400 And all of that over 2 times A. 00:06:56.400 --> 00:07:00.320 Well, A is minus 8, so all of that is over minus 16. 00:07:00.320 --> 00:07:04.090 So let's simplify this expression up here. 00:07:04.090 --> 00:07:09.440 Well, that's equal to minus 5 plus or minus 00:07:09.440 --> 00:07:13.630 the square root of 25. 00:07:13.630 --> 00:07:14.620 Let's see. 00:07:14.620 --> 00:07:18.220 4 times 8 is 32 and the negatives cancel out, so 00:07:18.220 --> 00:07:21.520 that's positive 32 times 9. 00:07:21.520 --> 00:07:24.480 Positive 32 times 9, let's see. 00:07:24.480 --> 00:07:26.720 30 times 9 is 270. 00:07:26.720 --> 00:07:31.110 It's 288. 00:07:31.110 --> 00:07:31.570 I think. 00:07:31.570 --> 00:07:31.800 Right? 00:07:31.800 --> 00:07:36.130 00:07:36.130 --> 00:07:37.490 288. 00:07:37.490 --> 00:07:40.590 We have all of that over minus 16. 00:07:40.590 --> 00:07:42.560 Now simplify it more. 00:07:42.560 --> 00:07:47.760 Minus 5 plus or minus the square root-- 25 plus 00:07:47.760 --> 00:07:51.340 288 is 313 I believe. 00:07:51.340 --> 00:07:56.950 00:07:56.950 --> 00:08:00.230 And all of that over minus 16. 00:08:00.230 --> 00:08:03.430 And I think, I'm not 100% sure, although I'm pretty sure. 00:08:03.430 --> 00:08:04.570 I haven't checked it. 00:08:04.570 --> 00:08:10.370 That 313 can't be factored into a product of a perfect 00:08:10.370 --> 00:08:11.690 square and another number. 00:08:11.690 --> 00:08:13.670 In fact, it actually might be a prime number. 00:08:13.670 --> 00:08:15.600 That's something that you might want to check out. 00:08:15.600 --> 00:08:18.200 So if that is the case and we've got it in completely 00:08:18.200 --> 00:08:21.840 simplified form, and we say there are two solutions, two 00:08:21.840 --> 00:08:24.940 x values that will make this equation true. 00:08:24.940 --> 00:08:30.750 One of them is x is equal to minus 5 plus the square 00:08:30.750 --> 00:08:35.830 root of 313 over minus 16. 00:08:35.830 --> 00:08:44.110 And the other one is x is equal to minus 5 minus the square 00:08:44.110 --> 00:08:49.660 root of 313 over minus 16. 00:08:49.660 --> 00:08:51.760 Hopefully those two examples will give you a good 00:08:51.760 --> 00:08:53.940 sense of how to use the quadratic equation. 00:08:53.940 --> 00:08:55.860 I might add some more modules. 00:08:55.860 --> 00:08:58.230 And then, once you master this, I'll actually teach you how to 00:08:58.230 --> 00:09:00.370 solve quadratic equations when you actually get a negative 00:09:00.370 --> 00:09:01.910 number under the radical. 00:09:01.910 --> 00:09:03.140 Very interesting. 00:09:03.140 --> 00:09:06.760 Anyway, I hope you can do the module now and maybe I'll add a 00:09:06.760 --> 00:09:10.370 few more presentations because this isn't the easiest module. 00:09:10.370 --> 00:09:11.840 But I hope you have fun. 00:09:11.840 --> 00:09:13.140 Bye. 00:09:13.140 --> 00:09:13.482