WEBVTT 00:00:05.220 --> 00:00:07.600 What are we getting an equation here? 00:00:13.220 --> 00:00:15.660 direction or solve with me. Solve 00:00:15.660 --> 00:00:17.740 find the solution set. 00:00:17.740 --> 00:00:18.380 That's great. 00:00:19.280 --> 00:00:20.520 Set of all solutions. 00:00:22.360 --> 00:00:24.580 What do we call this type of equation? 00:00:29.440 --> 00:00:32.100 This is called a linear equation. 00:00:32.100 --> 00:00:35.280 Why is this called a linear equation? 00:00:35.440 --> 00:00:38.940 The variable is only to the first power. 00:00:41.920 --> 00:00:42.680 I'll just tell you. 00:00:43.500 --> 00:00:44.840 This is what type of polynomial? 00:00:45.800 --> 00:00:46.740 Linear polynomial. 00:00:48.080 --> 00:00:50.880 A linear polynomial. Yes, it happens to be binomial. 00:00:52.200 --> 00:00:55.880 So this is a linear polynomial. 00:00:57.080 --> 00:00:57.860 And what is this? 00:01:00.000 --> 00:01:00.080 Constant. 00:01:02.820 --> 00:01:03.540 Polynomial. 00:01:04.000 --> 00:01:05.180 Definitely a monomial. 00:01:06.260 --> 00:01:06.280 Okay. 00:01:09.120 --> 00:01:10.840 The linear is the larger degree. 00:01:12.560 --> 00:01:14.220 Since it's the larger degree, 00:01:15.060 --> 00:01:18.100 the entire thing is called linear. Sorry, a linear equation. 00:01:19.120 --> 00:01:20.860 A linear equation. 00:01:23.320 --> 00:01:25.820 You could call it a linear polynomial equation. 00:01:26.760 --> 00:01:28.320 We simplify it. Linear equation. 00:01:29.680 --> 00:01:32.300 And we do a bunch of steps. 00:01:32.960 --> 00:01:33.780 What's the first step? 00:01:34.420 --> 00:01:36.720 Subtract seven. 00:01:38.660 --> 00:01:40.280 We get 3x 00:01:42.320 --> 00:01:43.440 equals two. 00:01:43.960 --> 00:01:46.680 And then we divide by three. 00:01:47.140 --> 00:01:47.800 Divide by three. 00:01:49.200 --> 00:01:50.520 Just two thirds. 00:01:51.280 --> 00:01:53.600 And then the solution is 00:01:54.800 --> 00:01:56.780 two thirds. 00:01:57.780 --> 00:01:59.580 And the solution set is 00:02:00.680 --> 00:02:02.720 the set containing two thirds. 00:02:05.000 --> 00:02:06.880 This process we went through, 00:02:07.360 --> 00:02:09.220 we learned in, I don't know, seventh grade? 00:02:09.580 --> 00:02:13.260 Is often known as isolating the variable. 00:02:23.820 --> 00:02:26.340 Once we got the variable all by itself, 00:02:26.700 --> 00:02:28.580 it's obvious what the solution is. 00:02:29.020 --> 00:02:30.340 It's the number on the other side. 00:02:31.420 --> 00:02:33.700 Because when you replace it for the x, 00:02:34.680 --> 00:02:35.900 makes the equation true. 00:02:37.760 --> 00:02:38.180 Okay. 00:02:41.660 --> 00:02:42.080 Next. 00:03:03.720 --> 00:03:05.240 What do we call this side of the equation? 00:03:07.280 --> 00:03:08.960 It's a quadratic equation. 00:03:10.100 --> 00:03:12.320 The left side is a quadratic polynomial, 00:03:12.400 --> 00:03:13.940 the right side is a constant polynomial, 00:03:14.300 --> 00:03:16.260 the larger the two degrees is two, 00:03:16.380 --> 00:03:17.240 the quadratic. 00:03:17.580 --> 00:03:20.760 We call the entire thing a quadratic equation. 00:03:23.480 --> 00:03:26.540 This is what we're talking about solving quadratic equations today. 00:03:31.120 --> 00:03:32.680 So, what are we going to do? 00:03:47.820 --> 00:03:50.280 We're going to isolate the variable. 00:03:51.800 --> 00:03:59.520 We want to get x equals how we got a problem. 00:03:59.760 --> 00:04:00.320 What's the problem? 00:04:04.280 --> 00:04:05.880 There's an x squared and there's an x. 00:04:06.420 --> 00:04:08.560 There's no way to combine those into one, right? 00:04:09.160 --> 00:04:10.540 Because they're different terms. 00:04:11.760 --> 00:04:12.140 You got an idea? 00:04:14.720 --> 00:04:16.440 Okay, we could factor out the x. 00:04:18.720 --> 00:04:20.120 Let's do that for a second. 00:04:23.020 --> 00:04:23.940 The x plus two. 00:04:35.880 --> 00:04:37.480 But there's no way to still... 00:04:37.480 --> 00:04:39.880 We still have two different x's. 00:04:44.100 --> 00:04:44.560 So now what? 00:04:45.880 --> 00:04:49.080 Subtract 15 and make it a quadratic trinomial. 00:04:49.420 --> 00:04:50.440 We want to factor them out. 00:04:51.980 --> 00:04:53.560 Okay, why would we do that? 00:04:54.040 --> 00:04:55.580 Because it will equal zero. 00:05:04.160 --> 00:05:06.260 Okay, how about this? 00:05:09.820 --> 00:05:13.940 Well, we're looking for a number and a number plus two 00:05:13.940 --> 00:05:15.340 more than that same number. 00:05:16.040 --> 00:05:17.500 It equals 15. 00:05:21.240 --> 00:05:23.440 We could start guessing and checking, 00:05:23.600 --> 00:05:24.520 but that's a terrible way. 00:05:25.820 --> 00:05:28.800 Okay, we have a very, very important property. 00:05:30.900 --> 00:05:33.500 Bet you even know what it is. 00:05:35.660 --> 00:05:38.540 It's called the zero product property. 00:05:38.540 --> 00:05:38.580 Okay. 00:05:48.800 --> 00:05:54.540 If a times b equals zero... 00:05:57.340 --> 00:06:01.540 If two numbers multiplied to be zero... 00:06:02.340 --> 00:06:03.600 What has to be true? 00:06:06.300 --> 00:06:07.600 Either a is zero... 00:06:11.860 --> 00:06:13.260 Or b equals zero. 00:06:14.840 --> 00:06:16.800 Or they both could be zero, correct? 00:06:18.180 --> 00:06:20.900 Okay, so that's still the zero product property. 00:06:25.620 --> 00:06:26.820 Why is it called that? 00:06:27.100 --> 00:06:29.340 Then why is it called the zero product property? 00:06:30.260 --> 00:06:31.000 Because it equals zero. 00:06:31.520 --> 00:06:33.840 Yeah, because we have a product that equals zero. 00:06:35.420 --> 00:06:36.900 This product equals zero. 00:06:37.760 --> 00:06:39.240 It's called the zero product property. 00:06:39.340 --> 00:06:43.180 Sometimes it's called the zero factor property. 00:06:44.540 --> 00:06:48.440 We have two factors that equal zero, 00:06:48.560 --> 00:06:50.560 but we're just going to call it the zero product property. 00:06:52.800 --> 00:06:53.700 Okay, so let's go back here. 00:06:55.420 --> 00:06:59.140 We have a product that equals 15. 00:07:02.500 --> 00:07:03.060 Okay. 00:07:08.380 --> 00:07:09.500 There's no... 00:07:09.500 --> 00:07:11.220 Well, let me give you this. 00:07:11.320 --> 00:07:13.720 What are some numbers that multiply to 15? 00:07:15.100 --> 00:07:16.500 Three and five. 00:07:17.340 --> 00:07:17.520 What else? 00:07:18.340 --> 00:07:18.740 15 and one. 00:07:20.080 --> 00:07:21.780 15 and one. What else? 00:07:23.660 --> 00:07:24.780 Negative three, negative five. 00:07:24.820 --> 00:07:26.940 Negative three, negative five. 00:07:27.260 --> 00:07:27.600 What else? 00:07:27.760 --> 00:07:29.040 Negative three, negative five. 00:07:30.220 --> 00:07:31.820 What else? 00:07:35.720 --> 00:07:38.260 How about 30 and one half? 00:07:40.400 --> 00:07:41.320 Negative 30, negative one half. 00:07:41.760 --> 00:07:44.500 How about 60 and a fourth? 00:07:45.840 --> 00:07:50.380 How many numbers multiply to be 15? 00:07:51.440 --> 00:07:52.740 More than a lot. 00:07:54.200 --> 00:07:54.600 An infinite. 00:07:54.600 --> 00:07:55.520 Okay. 00:07:56.820 --> 00:08:02.460 So we don't have a 15 product property, 00:08:04.200 --> 00:08:06.400 but we do have a zero product property. 00:08:07.380 --> 00:08:10.320 If we have zero, we've got only two options. 00:08:10.720 --> 00:08:12.900 Either the first number is zero or the second number is zero. 00:08:16.360 --> 00:08:18.640 So we are going to start with this. 00:08:19.860 --> 00:08:20.620 So we're going to go backwards. 00:08:21.300 --> 00:08:22.840 We have to have zero on one side. 00:08:22.840 --> 00:08:26.900 So we start by subtracting 15. 00:08:28.660 --> 00:08:30.240 Yes, people did say this before. 00:08:33.400 --> 00:08:36.740 And now we need to change this to a product. 00:08:40.100 --> 00:08:41.840 And that process is called... 00:08:44.040 --> 00:08:44.140 Factoring. 00:08:44.140 --> 00:08:44.580 Factoring. 00:08:44.820 --> 00:08:45.100 Thank you. 00:08:46.000 --> 00:08:49.620 We want to change our sum to a product that's called factoring, 00:08:50.260 --> 00:08:51.120 to write as a product. 00:08:51.120 --> 00:08:53.420 And so we're going to factor this. 00:08:56.820 --> 00:08:57.640 What's this factor to? 00:09:01.320 --> 00:09:03.240 Five and negative three. 00:09:03.480 --> 00:09:04.540 Five and negative three. 00:09:04.900 --> 00:09:05.340 Oops. 00:09:09.320 --> 00:09:09.940 Okay. 00:09:12.700 --> 00:09:15.860 Now we have a product that equals zero. 00:09:15.860 --> 00:09:17.980 Oh, so it's... 00:09:17.980 --> 00:09:25.380 By the zero product property, 00:09:28.860 --> 00:09:32.980 this number, x minus three, equals zero. 00:09:33.660 --> 00:09:38.960 Or this number, x plus five, equals zero. 00:09:42.080 --> 00:09:45.880 Now we don't have a quadratic equation. 00:09:46.180 --> 00:09:47.240 We have two what's that? 00:09:48.040 --> 00:09:48.920 Linear. 00:09:49.040 --> 00:09:50.180 Linear equations. 00:09:51.440 --> 00:09:55.780 And now all we have to do is isolate the variable. 00:09:59.480 --> 00:10:01.240 Add three. 00:10:04.080 --> 00:10:07.120 And the solution is three. 00:10:08.120 --> 00:10:10.800 And for x plus five, subtract five from both sides. 00:10:11.660 --> 00:10:13.780 Solution is negative five. 00:10:15.400 --> 00:10:15.420 Okay. 00:10:17.360 --> 00:10:19.200 And there it is. 00:10:20.120 --> 00:10:23.060 So once we turned it into two linear equations, 00:10:23.060 --> 00:10:24.780 we're able to just isolate the variable 00:10:26.260 --> 00:10:28.200 and get our solution set. 00:10:28.440 --> 00:10:31.540 And our solution set is negative five and three. 00:10:33.900 --> 00:10:34.580 Okay. 00:10:35.820 --> 00:10:38.620 So this is how we're going to solve quadratic equations. 00:10:40.320 --> 00:10:42.200 We need one side equals zero. 00:10:43.360 --> 00:10:44.700 Then we're going to factor. 00:10:46.540 --> 00:10:46.840 Okay. 00:10:47.920 --> 00:10:50.360 And then we use the zero product property, 00:10:50.420 --> 00:10:52.600 write two separate linear equations. 00:10:54.540 --> 00:10:58.500 And then we get our two solutions, if there are two solutions. 00:11:00.500 --> 00:11:01.000 Is that cool? 00:11:02.060 --> 00:11:02.340 Okay. 00:11:02.340 --> 00:11:04.080 We solved our first quadratic equation. 00:11:06.340 --> 00:11:09.260 This was the major point of why we spent so much time factoring. 00:11:10.640 --> 00:11:11.040 Okay. 00:11:11.460 --> 00:11:13.660 It allows us to solve quadratic equations, 00:11:13.900 --> 00:11:15.400 which before was impossible. 00:11:16.220 --> 00:11:17.100 You didn't know how to factor, 00:11:17.360 --> 00:11:19.980 and we didn't even know what factoring was before. 00:11:21.020 --> 00:11:22.320 Let's try another one. 00:11:42.260 --> 00:11:43.660 Okay. 00:12:04.880 --> 00:12:06.580 So what's the first step? 00:12:09.820 --> 00:12:11.260 We must have zero on one side. 00:12:11.440 --> 00:12:12.100 We don't care what side. 00:12:13.200 --> 00:12:14.360 We must have zero on one side. 00:12:15.020 --> 00:12:16.520 So we subtract seven X. 00:12:18.820 --> 00:12:20.080 And what order are we in writing? 00:12:22.460 --> 00:12:23.180 Descending order. 00:12:30.600 --> 00:12:32.100 And once we do that, what do we do? 00:12:33.000 --> 00:12:33.980 We need a factor. 00:12:40.560 --> 00:12:41.680 And what's this factor, two? 00:12:44.120 --> 00:12:46.700 Two numbers that multiply to be negative 30, 00:12:46.880 --> 00:12:47.840 that add to be negative seven. 00:12:49.760 --> 00:12:51.960 Negative 10, positive three. 00:12:54.140 --> 00:12:54.160 Okay. 00:12:54.400 --> 00:12:58.180 And then once we have a fact or a product that equals zero, 00:12:58.260 --> 00:12:58.880 what do we use? 00:13:00.780 --> 00:13:02.000 Zero product. 00:13:03.800 --> 00:13:04.880 Zero product. 00:13:10.480 --> 00:13:12.320 Either X minus 10. 00:13:14.280 --> 00:13:16.780 This number equals zero. 00:13:17.800 --> 00:13:19.240 Or X plus three. 00:13:21.480 --> 00:13:23.780 And now we got two linear equations. 00:13:24.780 --> 00:13:27.780 And we solve those. 00:13:47.120 --> 00:13:50.860 And we got our two solutions. 00:13:54.980 --> 00:13:55.440 Okay. 00:13:57.160 --> 00:14:00.980 Quadratics can have at most two equations, 00:14:01.340 --> 00:14:02.320 sorry, two solutions. 00:14:03.620 --> 00:14:05.240 It could have one solution. 00:14:05.860 --> 00:14:07.180 We'll give an example of that in a minute. 00:14:08.760 --> 00:14:10.340 Or it can have no solutions. 00:14:11.060 --> 00:14:12.320 And we'll give an example of that. 00:14:12.620 --> 00:14:14.360 But most of the time it's going to have two solutions. 00:14:16.640 --> 00:14:17.760 Let's try another one. 00:14:39.260 --> 00:14:40.220 Step one. 00:14:42.640 --> 00:14:44.080 We're not setting it to zero. 00:14:44.600 --> 00:14:46.620 We need to get one side to be zero. 00:14:48.200 --> 00:14:49.460 And we do that by? 00:14:50.400 --> 00:14:51.420 Subtracting 100. 00:14:58.460 --> 00:15:01.200 And now we need to factor that quadratic. 00:15:07.480 --> 00:15:11.700 Factors of negative 300 that add to be negative 40. 00:15:29.100 --> 00:15:30.540 You know what, I screwed up. 00:15:32.720 --> 00:15:34.560 Let's make this a negative 100. 00:15:34.860 --> 00:15:37.300 Sorry, I messed up the problem. 00:15:39.300 --> 00:15:40.660 So this becomes positive. 00:15:43.580 --> 00:15:45.020 So sorry, put that in there. 00:15:46.340 --> 00:15:48.200 Otherwise it would not be able to be factored. 00:15:49.200 --> 00:15:58.452 Now we're looking for factors of 300 that add to be negative 40: negative 10 and 00:15:58.452 --> 00:15:59.840 negative 30. 00:16:02.760 --> 00:16:05.100 Negative 10, thank you. 00:16:05.940 --> 00:16:06.560 Negative 30. 00:16:09.420 --> 00:16:14.160 Minus 10x minus 30x plus 100. 00:16:18.200 --> 00:16:19.940 And factor it by grouping. 00:16:50.360 --> 00:16:58.360 Okay, once we factor, we have a factor that equals, or a product that equals zero. 00:16:59.380 --> 00:16:59.780 What's next? 00:17:03.160 --> 00:17:10.040 So by the zero product property. 00:17:12.020 --> 00:17:17.514 And when you're solving this for homework, I want you to write that in every time 00:17:17.514 --> 00:17:18.260 you use it. 00:17:19.580 --> 00:17:21.200 You can abbreviate it if you want. 00:17:21.840 --> 00:17:23.180 What should we abbreviate it as? 00:17:24.300 --> 00:17:24.620 ZPP. 00:17:26.580 --> 00:17:34.120 Either x minus 10 equals zero, or 3x minus 10 equals zero. 00:17:44.940 --> 00:17:48.960 And we have our two solutions, 10 thirds and 10. 00:17:50.720 --> 00:17:54.160 How can we check to see if our solutions are actual solutions? 00:17:56.740 --> 00:17:59.080 We can plug it back into the original and? 00:18:02.600 --> 00:18:03.980 What does it mean to be a solution? 00:18:06.860 --> 00:18:09.940 A number that, when replaced for the variable, makes the equation true. 00:18:10.360 --> 00:18:12.420 If you plug it in, it makes the equation true. 00:18:12.940 --> 00:18:15.580 If you factor right, it's going to be the solutions. 00:18:17.880 --> 00:18:18.320 Okay. 00:18:35.720 --> 00:18:36.580 This is a binomial. 00:18:37.720 --> 00:18:38.000 What do I write? 00:18:38.260 --> 00:18:38.420 Binomial. 00:18:40.000 --> 00:18:42.020 The only one we know how to factor is what? 00:18:42.020 --> 00:18:45.040 Difference of squares. 00:18:46.160 --> 00:18:47.080 Is this the difference of squares? 00:18:50.700 --> 00:18:56.180 So if factors two, x minus seven, x plus seven. 00:19:05.840 --> 00:19:07.280 And now what? 00:19:09.380 --> 00:19:09.620 Zero. 00:19:10.620 --> 00:19:11.200 Zero. 00:19:14.780 --> 00:19:18.940 X minus seven equals zero, x plus seven equals zero. 00:19:19.780 --> 00:19:21.940 X equals seven, x equals negative seven. 00:19:29.120 --> 00:19:30.080 What? 00:19:33.060 --> 00:19:33.360 No. 00:19:44.960 --> 00:19:45.540 What? 00:19:52.280 --> 00:19:53.920 This is not the difference of squares. 00:19:54.080 --> 00:19:54.660 What is this? 00:19:55.980 --> 00:19:56.460 Sum of squares. 00:19:58.740 --> 00:20:00.640 Do sum of squares factor? 00:20:01.860 --> 00:20:01.940 No. 00:20:05.940 --> 00:20:06.800 Doesn't factor. 00:20:06.800 --> 00:20:09.340 That is not a factor. 00:20:13.980 --> 00:20:15.300 What does that mean? 00:20:18.780 --> 00:20:19.900 There's no solutions. 00:20:21.080 --> 00:20:22.060 There is a solution set. 00:20:22.240 --> 00:20:22.660 What is it? 00:20:23.660 --> 00:20:23.880 Empty set. 00:20:24.100 --> 00:20:24.660 The empty set. 00:20:25.160 --> 00:20:26.100 Either that or that. 00:20:27.300 --> 00:20:27.420 Okay. 00:20:28.600 --> 00:20:34.320 Just because a polynomial doesn't factor doesn't mean there's no solutions. 00:20:34.680 --> 00:20:35.400 We'll get to that later. 00:20:36.400 --> 00:20:38.460 But this one definitely does not factor. 00:20:39.640 --> 00:20:40.160 There's no solutions. 00:20:43.980 --> 00:20:45.840 You could also think this way. 00:20:46.280 --> 00:20:50.220 X squared equals negative twenty-five. 00:20:51.120 --> 00:20:55.003 There's no way or something squared which will always be positive to equal a 00:20:55.003 --> 00:20:55.820 negative number. 00:20:56.280 --> 00:20:57.740 There's definitely no solutions here. 00:21:18.100 --> 00:21:19.820 Is this the difference of squares? 00:21:22.760 --> 00:21:22.800 Yeah. 00:21:23.840 --> 00:21:24.340 It is. 00:21:30.400 --> 00:21:34.080 Two x, squared, is four x squared. 00:21:34.360 --> 00:21:35.260 What squared is 5? 00:21:35.820 --> 00:21:36.340 Square root of 5. 00:21:37.600 --> 00:21:40.400 Square root of 5. 00:21:51.840 --> 00:21:52.800 And now what? 00:21:55.960 --> 00:21:56.480 ZPP. 00:21:57.420 --> 00:21:58.980 We have a product that equals zero. 00:21:59.200 --> 00:22:01.340 We set each factor equal to zero. 00:22:30.860 --> 00:22:35.720 And we got negative the square root of 5 over 2. 00:22:36.460 --> 00:22:38.620 And the square root of 5 over 2. 00:22:42.280 --> 00:22:46.182 Sometimes when we have the same number but just opposite signs, well it's not the 00:22:46.182 --> 00:22:46.760 same number, 00:22:46.760 --> 00:22:48.040 but opposite numbers. 00:22:49.760 --> 00:22:51.100 We sometimes write it like this. 00:22:51.440 --> 00:22:54.380 Plus and minus the square root of 5 over 2. 00:22:58.440 --> 00:23:01.340 Like we could have done that for this one. 00:23:02.420 --> 00:23:05.980 We could have wrote plus and minus 7 which is two different numbers. 00:23:22.940 --> 00:23:24.420 Yeah, I think we got time. 00:23:32.340 --> 00:23:34.320 We had this equation before. 00:23:34.520 --> 00:23:34.660 Yes. 00:23:37.780 --> 00:23:38.280 Okay. 00:23:44.440 --> 00:23:48.360 I agree this is the equivalent equation. 00:23:48.940 --> 00:23:49.320 Yes. 00:23:55.280 --> 00:23:58.720 Can I just take the square root of both sides? 00:24:05.280 --> 00:24:05.420 Okay. 00:24:07.880 --> 00:24:08.340 Possibly. 00:24:10.820 --> 00:24:16.400 But then we only get one solution. 00:24:21.520 --> 00:24:24.240 So just taking the square root of both sides, you only get one solution. 00:24:27.740 --> 00:24:28.620 Okay. 00:24:36.280 --> 00:24:40.420 However, we know this can be factored, right? 00:24:41.820 --> 00:24:48.280 This can be factored to X minus 7 times X plus 7. 00:24:49.020 --> 00:25:02.900 And then by the ZPP, we get X minus 7 equals 0 and X plus 7 equals 0. 00:25:03.040 --> 00:25:06.000 And we got our two solutions, negative 7 and 7. 00:25:07.680 --> 00:25:08.480 Okay. 00:25:13.560 --> 00:25:19.540 So if X squared equals any number, let's call it A. 00:25:20.340 --> 00:25:22.620 A has to be greater than or equal to 0. 00:25:27.700 --> 00:25:32.540 X squared minus A equals 0 is equivalent. 00:25:41.700 --> 00:25:43.260 Is this the difference of squares? 00:25:44.960 --> 00:25:45.260 Yeah. 00:25:46.520 --> 00:25:54.431 X squared minus the square root of a squared factors to x minus the square root of 00:25:54.431 --> 00:25:58.000 a, times x plus the square root of a. 00:26:01.100 --> 00:26:04.200 And by the zero product property. 00:26:08.040 --> 00:26:18.440 X minus the square root of a equals zero, so x equals the square root of a. 00:26:19.280 --> 00:26:21.360 And X equals negative square root of A. 00:26:22.800 --> 00:26:29.000 And we get negative the square root of a. 00:26:34.420 --> 00:26:39.820 Which could be written as plus or minus the square root of A. 00:26:40.380 --> 00:26:43.580 And that number, number A does not matter. 00:26:44.120 --> 00:26:46.480 As long as a positive number. 00:26:47.440 --> 00:26:48.680 Why is the number of the negative number? 00:26:51.260 --> 00:26:52.060 What'd you say? 00:26:55.600 --> 00:26:59.780 This works always, except for when A is a negative number. 00:27:00.960 --> 00:27:01.200 It does. 00:27:02.680 --> 00:27:06.480 The square root of negative numbers don't exist. 00:27:07.560 --> 00:27:08.560 There's no real numbers. 00:27:10.020 --> 00:27:10.140 Okay. 00:27:10.900 --> 00:27:22.778 So if any time we have X squared equals a positive number, we know the two solutions 00:27:22.778 --> 00:27:29.000 are X equals the square root of that number. 00:27:30.220 --> 00:27:32.780 And X equals negative the square root of that number. 00:27:33.520 --> 00:27:34.980 We just proved it. 00:27:35.880 --> 00:27:36.400 Okay. 00:27:36.420 --> 00:27:38.320 We got these two equations. 00:27:40.720 --> 00:27:40.740 Okay. 00:27:41.400 --> 00:27:42.720 And we often write it like this. 00:27:42.960 --> 00:27:45.680 X equals plus or minus the square root of that number. 00:27:50.720 --> 00:27:51.240 Okay. 00:27:51.480 --> 00:27:55.700 So if this is true, then this is true. 00:27:55.820 --> 00:27:57.040 Let's write that down. 00:27:58.440 --> 00:28:08.832 If X squared equals a number, as long as positive, then X equals plus and minus the 00:28:08.832 --> 00:28:10.960 square root of A. 00:28:11.420 --> 00:28:12.640 Which is two equations. 00:28:12.940 --> 00:28:14.220 X equals the square root of A. 00:28:15.640 --> 00:28:17.980 And X equals the negative square root of A. 00:28:20.640 --> 00:28:21.280 Okay. 00:28:21.660 --> 00:28:24.000 This is called the square root principle. 00:28:36.860 --> 00:28:43.420 And it's going to allow us to solve quadratic equations without factoring. 00:28:44.000 --> 00:28:45.860 Because we just proved it always factors. 00:28:47.180 --> 00:28:52.420 So for example, if we have X squared equals 25. 00:28:57.520 --> 00:29:00.380 It looks like you're taking the square root of both sides, but we're not. 00:29:01.560 --> 00:29:03.240 It's by the square root principle. 00:29:15.960 --> 00:29:19.740 That X will equal plus or minus the square root of 25. 00:29:20.940 --> 00:29:21.500 Okay. 00:29:21.860 --> 00:29:23.800 We're not taking the square root of both sides. 00:29:25.200 --> 00:29:27.200 We're using the square root principle. 00:29:27.320 --> 00:29:31.878 Anytime something squared equals a number, whatever is squared equals plus or minus 00:29:31.878 --> 00:29:33.580 the square root of that number. 00:29:33.800 --> 00:29:35.220 Which is two separate equations. 00:29:36.500 --> 00:29:37.020 Okay. 00:29:38.520 --> 00:29:39.220 Plus or minus five. 00:29:40.140 --> 00:29:41.900 And that gives our two solutions. 00:29:46.220 --> 00:29:46.860 Okay. 00:29:49.300 --> 00:29:56.020 So, anytime you have X squared, let's just do another one. 00:30:01.280 --> 00:30:02.560 Equals 50. 00:30:06.660 --> 00:30:09.720 We're not taking the square root of both sides. 00:30:09.960 --> 00:30:11.380 We're going to use the what principle? 00:30:12.940 --> 00:30:14.160 Square root principle. 00:30:14.660 --> 00:30:17.400 Square root principle. 00:30:20.660 --> 00:30:25.600 Then X will equal plus or minus the square root of 50. 00:30:27.920 --> 00:30:29.680 And what does that simplify to? 00:30:31.180 --> 00:30:32.720 Five square root of two. 00:30:34.100 --> 00:30:34.860 Five square root of two. 00:30:37.000 --> 00:30:38.760 We have our two solutions. 00:30:41.380 --> 00:30:41.860 Okay. 00:30:42.620 --> 00:30:46.934 All the square root principle is saving time where we're not bringing this to the 00:30:46.934 --> 00:30:47.520 other side. 00:30:48.540 --> 00:30:49.700 And we're not factoring. 00:30:56.440 --> 00:30:58.880 And we use this as ZPP. 00:31:12.740 --> 00:31:13.380 Okay. 00:31:22.660 --> 00:31:24.520 We got our two solutions. 00:31:25.720 --> 00:31:26.400 Our two equations. 00:31:27.340 --> 00:31:27.360 Okay. 00:31:28.020 --> 00:31:34.086 So, because we know it's going to be the difference of squares, there's no linear 00:31:34.086 --> 00:31:34.460 term. 00:31:35.800 --> 00:31:36.440 Okay. 00:31:36.600 --> 00:31:39.920 It will factor and we'll end up with these two equations. 00:31:40.780 --> 00:31:41.180 Okay. 00:31:41.300 --> 00:31:42.960 And we write this all in one equation. 00:31:44.420 --> 00:31:44.560 Right. 00:31:44.700 --> 00:31:44.940 We could. 00:31:48.180 --> 00:31:48.580 Okay. 00:31:48.940 --> 00:31:50.020 So, that is the square root principle. 00:31:50.300 --> 00:31:54.120 We can only do it when we have something squared equals some positive number. 00:31:58.100 --> 00:31:58.500 Okay. 00:31:58.620 --> 00:31:59.840 And you go right to this equation. 00:32:01.360 --> 00:32:02.040 Does this make sense? 00:32:03.300 --> 00:32:03.600 Yes. 00:32:04.740 --> 00:32:08.100 This is going to be very important for the next thing we learn later. 00:32:12.980 --> 00:32:13.460 Okay. 00:32:13.780 --> 00:32:15.600 I think we could stop here. 00:32:16.560 --> 00:32:19.600 Most of the problems, we're not going to be able to use the square root principle. 00:32:20.180 --> 00:32:24.180 We just have to factor and use what property? 00:32:26.720 --> 00:32:27.880 Zero product property. 00:32:28.600 --> 00:32:30.520 And you need to write in there where you use it. 00:32:31.060 --> 00:32:33.360 You write two separate equations and then solve.