WEBVTT 00:00:01.156 --> 00:00:07.386 A:middle >> Welcome to the Cypress College math review on solving quadratic equations by factoring. 00:00:07.936 --> 00:00:12.356 A:middle In objective one, you will learn how 00:00:12.356 --> 00:00:17.876 A:middle to solve quadratic equations having two terms using factoring. 00:00:18.396 --> 00:00:22.186 A:middle Our strategy has four steps. 00:00:23.246 --> 00:00:28.296 A:middle First, gather all terms on one side of the equation if needed. 00:00:29.916 --> 00:00:42.436 A:middle Second, factor, third, apply the zero-product property and fourth, solve for the variable. 00:00:44.476 --> 00:00:46.096 A:middle So, let's get started. 00:00:46.696 --> 00:00:55.426 A:middle In this example, we have to solve 27y squared equals 36y. 00:00:58.216 --> 00:01:07.896 A:middle Remember our strategy, step one says to gather all terms on one side of the equation if needed. 00:01:08.516 --> 00:01:15.296 A:middle We definitely need to do that in this equation because we have terms on both sides. 00:01:15.866 --> 00:01:20.876 A:middle So, let's bring the 36y over to the left-hand side 00:01:21.416 --> 00:01:32.906 A:middle and we would have 27y squared minus 36y equals zero. 00:01:33.426 --> 00:01:39.126 A:middle Now it's important to remember when you're solving equations 00:01:39.666 --> 00:01:47.946 A:middle that every step should have a left-hand side, an equal sign, and a right-hand side. 00:01:48.466 --> 00:01:54.866 A:middle Let's move on to step two which says we should factor. 00:01:56.346 --> 00:02:03.326 A:middle Now between these two terms, we see that we have a greatest common factor of 9y 00:02:03.906 --> 00:02:14.496 A:middle and we would have left 3y minus 4 and then of course our equal sign, and we always have 00:02:14.496 --> 00:02:17.696 A:middle to have a right-hand side which, in this case, is zero. 00:02:19.556 --> 00:02:27.816 A:middle Now let's double check our factoring, 9y times 3y gives us 27y squared 00:02:28.746 --> 00:02:33.646 A:middle and 9y times a negative four gives us negative 36y. 00:02:34.176 --> 00:02:40.996 A:middle Step three says to apply the zero-product property. 00:02:41.566 --> 00:02:50.876 A:middle Now the zero-product property says that, if A times B is equal to zero, 00:02:52.086 --> 00:02:56.216 A:middle then A is equal to zero, or B is equal to zero. 00:02:57.626 --> 00:03:02.936 A:middle The only way that two numbers multiplied together can equal zero is 00:03:02.936 --> 00:03:05.196 A:middle if one of the numbers is zero. 00:03:06.776 --> 00:03:16.636 A:middle Now in our case the 9y is acting like the A and the 3y minus 4 is acting like the B. 00:03:17.126 --> 00:03:23.336 A:middle If these two factors multiplied together equals zero, 00:03:23.686 --> 00:03:27.206 A:middle then that means one of them has to equal zero. 00:03:27.896 --> 00:03:37.836 A:middle So, when we apply the zero-product property, we get that 9y would have to equal zero, 00:03:38.856 --> 00:03:44.966 A:middle or 3y minus 4 would have to equal zero. 00:03:47.796 --> 00:03:51.906 A:middle Step four now says to solve for the variable. 00:03:53.426 --> 00:04:01.706 A:middle So, in this first equation, if we divide both sides by 9, we would end up with y is equal 00:04:01.706 --> 00:04:05.466 A:middle to zero so that's our first solution. 00:04:06.056 --> 00:04:17.386 A:middle In the second equation, we would add four to both sides and get 3y is equal to four. 00:04:18.776 --> 00:04:27.206 A:middle Then we would divide both sides by 3 and get that y is equal to 4/3. 00:04:28.406 --> 00:04:31.146 A:middle And that is our second solution. 00:04:31.726 --> 00:04:41.376 A:middle I want to share with you now a tip about solving equations and help you avoid this common error. 00:04:42.876 --> 00:04:46.936 A:middle Never divide both sides by the variable. 00:04:48.096 --> 00:04:56.686 A:middle You always have to gather all terms to one side as we did here in the first step. 00:04:58.126 --> 00:05:00.226 A:middle Now here's why you can't do that. 00:05:01.496 --> 00:05:08.916 A:middle Suppose we started off with our equation and divided both sides by y. We would end 00:05:08.916 --> 00:05:16.226 A:middle up with 27y equals 36, then you would divide both sides by 27, 00:05:16.626 --> 00:05:22.156 A:middle you have 36/27 which reduces to 4/3. 00:05:23.816 --> 00:05:30.146 A:middle Now notice here, that while we got one of the solutions which is why equals 4/3, 00:05:30.936 --> 00:05:37.766 A:middle we are completely missing the second solution to this equation which is y equals zero. 00:05:38.926 --> 00:05:43.496 A:middle So, when you make the error of dividing by the variable in the first step, 00:05:44.076 --> 00:05:49.186 A:middle that leads to losing one of the solutions to the equation. 00:05:52.196 --> 00:05:57.306 A:middle Another way to think about it is, you know you can't divide by zero ever. 00:05:58.696 --> 00:06:02.466 A:middle And look at what one of the solutions to this equation is, zero. 00:06:03.256 --> 00:06:09.476 A:middle So, if you were to divide by y in that first step, there you are trying to divide 00:06:09.476 --> 00:06:13.276 A:middle by zero which you know is invalid. 00:06:14.956 --> 00:06:22.346 A:middle So, always gather all the terms onto one side of the equation and then factor 00:06:22.856 --> 00:06:24.936 A:middle and use the zero-product property. 00:06:36.036 --> 00:06:43.336 A:middle In this example, we have to solve 16a squared minus 25 is equal to zero. 00:06:45.376 --> 00:06:53.816 A:middle Now, in our strategy, step one is to gather all terms on one side of the equation if needed. 00:06:55.276 --> 00:07:01.156 A:middle Well that's already done for us, so let's move on to step two where we have to factor. 00:07:02.776 --> 00:07:07.256 A:middle We recognize that the left-hand side is the difference of squares 00:07:07.886 --> 00:07:17.206 A:middle so we can factor immediately into the quantity 4a plus 5 times the quantity 4a minus 5. 00:07:17.716 --> 00:07:23.246 A:middle We bring down the equal sign, we bring down the zero. 00:07:23.736 --> 00:07:29.536 A:middle Step three tells us to apply the zero-product property. 00:07:30.126 --> 00:07:38.716 A:middle Now we know that means we take each of the factors and set them equal to zero. 00:07:39.306 --> 00:07:50.256 A:middle So, we have 4a plus 5 is equal to zero or 4a minus 5 is equal to zero. 00:07:52.196 --> 00:07:55.786 A:middle Step four says to solve for the variable. 00:07:56.746 --> 00:08:05.766 A:middle In this first one, we would subtract 5 from both sides and have 4a is equal to negative 5, 00:08:06.336 --> 00:08:15.746 A:middle and then we would divide both sides by 4 and end up with A is equal to negative 5/4. 00:08:20.496 --> 00:08:29.606 A:middle In the second equation, we would add 5 to both sides, so we have 4a is equal to 5. 00:08:30.126 --> 00:08:39.386 A:middle Then we divide both sides by 4 and we end up with A is equal to 5/4. 00:08:39.946 --> 00:08:45.976 A:middle And these are the two solutions to this equation. 00:08:55.246 --> 00:08:57.566 A:middle It's time to check your understanding. 00:08:58.156 --> 00:09:00.856 A:middle So pause the video to try this one on your own, 00:09:01.256 --> 00:09:03.936 A:middle then restart when you are ready to check your answer. 00:09:11.536 --> 00:09:17.766 A:middle In this example, we have to solve 12h squared is equal to 21h. 00:09:19.636 --> 00:09:21.916 A:middle Let's refer to our strategy. 00:09:22.806 --> 00:09:27.816 A:middle Step one is to gather all terms on one side of the equation if needed, 00:09:28.656 --> 00:09:31.506 A:middle and we definitely need to do that in this one. 00:09:31.996 --> 00:09:34.866 A:middle So, we will bring everything to the left-hand side 00:09:35.446 --> 00:09:43.486 A:middle and have 12h squared minus 21h is equal to zero. 00:09:43.876 --> 00:09:49.636 A:middle Step two says to factor. 00:09:50.246 --> 00:09:57.656 A:middle Now, here we see that the greatest common factor is 3 times H, 00:09:58.616 --> 00:10:03.066 A:middle and what we would have left is 4h minus 7. 00:10:04.116 --> 00:10:08.996 A:middle Remember to bring down your equal sign and bring down the zero. 00:10:10.336 --> 00:10:17.026 A:middle Just a quick reminder that every step of every equation has to have a left-hand side, 00:10:17.916 --> 00:10:20.676 A:middle an equal sign, and a right-hand side. 00:10:21.246 --> 00:10:29.206 A:middle Now let's double check our factoring, 3h times 4h is 12h squared, 00:10:30.066 --> 00:10:34.836 A:middle and 3h times a negative 7 is negative 21h. 00:10:35.316 --> 00:10:41.936 A:middle Step three says to apply the zero-product property. 00:10:43.436 --> 00:10:49.826 A:middle So, we are going to take each of these factors and set it equal to zero. 00:10:51.086 --> 00:11:01.886 A:middle Then we would have 3h is equal to zero or 4h minus 7 is equal to zero. 00:11:02.396 --> 00:11:08.566 A:middle We move on to step four which says to solve for the variable. 00:11:09.686 --> 00:11:17.306 A:middle In this first equation, we will divide both sides by 3 and end up with H is equal 00:11:17.306 --> 00:11:20.556 A:middle to zero and that's our first solution. 00:11:21.136 --> 00:11:32.036 A:middle In the second equation, we would add 7 to both sides so that now we have 4h is equal to 7, 00:11:33.446 --> 00:11:39.446 A:middle and then we would divide both sides by 4 and end up with H is equal 00:11:39.446 --> 00:11:45.316 A:middle to 7/4, and that is our second solution. 00:11:47.656 --> 00:11:53.326 A:middle Another tip about solving equations is that, whenever you get your answers, 00:11:53.986 --> 00:11:58.076 A:middle you can always know whether you did the problem right or not, 00:11:58.876 --> 00:12:01.686 A:middle because you can always check your answers. 00:12:02.246 --> 00:12:04.616 A:middle So, for example, let's do that here. 00:12:05.276 --> 00:12:14.676 A:middle If we checked H is equal to zero, what we would do is plug that in to the original equation 00:12:15.266 --> 00:12:24.426 A:middle and double check that 12 times zero squared does equal 21 times zero, 00:12:24.926 --> 00:12:28.986 A:middle and because the left-hand side is zero 00:12:29.386 --> 00:12:33.886 A:middle and the right-hand side is zero, that means this one did work. 00:12:34.496 --> 00:12:43.376 A:middle We could also check the H equals 7/4 answer in the original. 00:12:44.456 --> 00:12:56.376 A:middle So, we would have to show that 12 times 7/4 squared is equal to 21 times 7/4. 00:12:56.806 --> 00:13:06.476 A:middle Well, on the left-hand side, we have 12 times 49/16. 00:13:07.826 --> 00:13:18.016 A:middle On the right-hand side, we know that 21 times 7 would be 147/4. 00:13:18.636 --> 00:13:27.276 A:middle On the left-hand side, the 12 and 16 have a common factor of 4, so let's reduce that. 00:13:28.486 --> 00:13:40.966 A:middle Four goes into 12 three times and 4 goes into 16 four times and 3 times 49 is 147. 00:13:41.616 --> 00:13:46.296 A:middle The denominator is 4 so, those two worked out. 00:13:46.826 --> 00:13:52.516 A:middle So, in this way, you can always check your answers 00:13:52.826 --> 00:13:55.806 A:middle and know whether you did the problem correctly or not. 00:14:03.106 --> 00:14:06.426 A:middle Below are some practice problems you can try on your own. 00:14:07.286 --> 00:14:13.136 A:middle You may either pause the video to work on them now or write them down to work on them later. 00:14:14.216 --> 00:14:16.976 A:middle After a few seconds, the answers will be revealed. 00:14:33.286 --> 00:14:39.186 A:middle In objective two, you will learn how to solve quadratic equations using foil factoring. 00:14:39.706 --> 00:14:42.776 A:middle Let's look at our strategy. 00:14:44.206 --> 00:14:49.226 A:middle Step one will be to gather all terms on one side of the equation if needed. 00:14:50.656 --> 00:14:57.986 A:middle Step two will be to factor using either trial and error or the AC method. 00:14:58.516 --> 00:15:07.116 A:middle Step three, apply the zero-product property, and step four, solve for the variable. 00:15:13.356 --> 00:15:19.236 A:middle In this example, we have to solve 2x squared equals 7x plus 15. 00:15:19.816 --> 00:15:26.346 A:middle If we consult our strategy, step one is to gather all terms 00:15:26.346 --> 00:15:28.836 A:middle on one side of the equation if needed. 00:15:29.436 --> 00:15:33.906 A:middle You can see we have terms on both sides of the equation. 00:15:35.176 --> 00:15:40.616 A:middle Now, because the coefficient of the x squared is positive on the left-hand side, 00:15:41.266 --> 00:15:47.086 A:middle I'm going to bring the right-hand side terms over to the left-hand side to keep 00:15:47.086 --> 00:15:49.136 A:middle that leading coefficient positive. 00:15:49.626 --> 00:16:00.676 A:middle Then this will become 2x squared minus 7x minus 15 is equal to zero. 00:16:01.116 --> 00:16:11.066 A:middle Step two says to factor and we will be using trial and error method in this example. 00:16:11.606 --> 00:16:23.836 A:middle So, the factors of 2 are 1 times 2 and the factors of 15 are 1 times 15 or 3 times 5. 00:16:24.386 --> 00:16:32.686 A:middle So, the only way we're going to get this 2x squared is to take X times 2x. 00:16:34.796 --> 00:16:38.306 A:middle Then we have some choices for the constant term, 00:16:38.856 --> 00:16:42.266 A:middle typically we will choose the factors that are closer together. 00:16:42.896 --> 00:16:45.726 A:middle So, we'll say 3 and 5. 00:16:47.616 --> 00:16:51.566 A:middle Now we know one of them will be negative and one will be positive 00:16:51.566 --> 00:16:53.776 A:middle because the constant term is negative. 00:16:54.336 --> 00:17:00.326 A:middle But we will wait until we do our cross-product terms to pick which one. 00:17:02.216 --> 00:17:09.896 A:middle The first cross-product is 6x and the second cross-product is 5x. 00:17:11.066 --> 00:17:13.276 A:middle Now because the middle term is negative, 00:17:13.956 --> 00:17:19.996 A:middle we will always pick the larger magnitude cross-product to be the one that's negative. 00:17:20.736 --> 00:17:25.186 A:middle So, I would put the negative here which means the 3 is negative. 00:17:25.876 --> 00:17:34.996 A:middle Now, when we add up these cross-products, we get negative X which does not equal the negative 7x 00:17:34.996 --> 00:17:39.086 A:middle that we need, so that tells us we have to try something else. 00:17:40.416 --> 00:17:48.906 A:middle So, if we keep the X and the 2x going, maybe we will try switching the 3 and the 5. 00:17:49.226 --> 00:17:52.786 A:middle So, we'll put the 5 up here and the 3 down here. 00:17:54.056 --> 00:18:03.136 A:middle The first cross-product is 10x, the second cross-product is 3x and, remember, 00:18:03.136 --> 00:18:07.666 A:middle we said we'll pick the larger magnitude cross-product and make it negative. 00:18:07.936 --> 00:18:11.926 A:middle So that's the 10x which means the 5 would have to be negative. 00:18:13.056 --> 00:18:18.246 A:middle Now when we add these two together, we get negative 7x 00:18:18.896 --> 00:18:22.586 A:middle which is exactly what we needed for the middle term. 00:18:23.166 --> 00:18:28.596 A:middle Now, that tells us that our factors are going 00:18:28.596 --> 00:18:37.956 A:middle to be the quantity X minus 5 times the quantity 2x plus 3. 00:18:38.476 --> 00:18:43.816 A:middle And, of course, that still is equal to zero. 00:18:44.296 --> 00:18:53.186 A:middle Step three says to apply the zero-product property, so we will take each 00:18:53.186 --> 00:18:56.936 A:middle of these factors and set them equal to zero. 00:18:57.776 --> 00:19:06.806 A:middle So, X minus 5 could be zero or 2x plus 3 could be zero. 00:19:10.856 --> 00:19:14.396 A:middle Step four is to solve for the variable. 00:19:15.736 --> 00:19:22.896 A:middle In this first equation, we add 5 to both sides and get that X is equal 00:19:22.896 --> 00:19:26.516 A:middle to 5, so that our first solution. 00:19:27.056 --> 00:19:37.936 A:middle In the second equation, we would subtract 3 from both sides and get 2x is equal to negative 3. 00:19:39.336 --> 00:19:48.026 A:middle Then we would divide both sides by 2 and end up with X is equal to negative three halves. 00:19:49.036 --> 00:19:51.366 A:middle And that is our second solution. 00:19:51.916 --> 00:19:59.266 A:middle And, if you have time, remember it's always a good idea to check your answers 00:19:59.266 --> 00:20:02.096 A:middle by plugging them back into the original equation. 00:20:02.626 --> 00:20:07.056 A:middle For the purposes of this lesson though, we are going to move on. 00:20:14.376 --> 00:20:21.706 A:middle In this example, we have to solve 6n squared minus 25n is equal to negative 4. 00:20:23.856 --> 00:20:29.336 A:middle From our strategy, we know that step one is to gather all terms 00:20:29.336 --> 00:20:31.396 A:middle on one side of the equation if needed. 00:20:31.926 --> 00:20:41.266 A:middle So, we will choose to move this negative 4 over to the left-hand side by adding 4 to both sides. 00:20:42.116 --> 00:20:52.026 A:middle So, we'll end up with 6n squared minus 25n plus 4 is equal to zero. 00:20:52.416 --> 00:21:02.376 A:middle Step two says to factor and, in this case, we'll be using the AC method. 00:21:02.896 --> 00:21:12.226 A:middle To do that, we make the little cross grid here, we know that the AC term goes 00:21:12.226 --> 00:21:17.786 A:middle in this upper area meaning the leading coefficient of 6 has 00:21:17.786 --> 00:21:23.676 A:middle to multiply the constant term of 4, and we get a positive 24. 00:21:25.776 --> 00:21:34.146 A:middle Whatever the B term is, the middle term, that coefficient which is negative 25 goes down here. 00:21:34.806 --> 00:21:45.806 A:middle So, we need two numbers that multiply to be a positive 24, but add to be a negative 25. 00:21:46.316 --> 00:21:53.116 A:middle And those two numbers would be negative 1 and negative 24. 00:21:53.726 --> 00:22:00.926 A:middle Now, these two numbers tell us how to break up the middle term. 00:22:02.756 --> 00:22:15.776 A:middle So, it will become negative N minus 24n, the 6n squared comes straight down. 00:22:16.336 --> 00:22:26.896 A:middle And the constant term of 4 comes straight down, equals zero. 00:22:28.856 --> 00:22:30.916 A:middle Now we factor 2 by 2. 00:22:31.956 --> 00:22:41.636 A:middle From these first two terms, we can factor out an N and we would have left 6n minus 1. 00:22:43.046 --> 00:22:43.986 A:middle Let's check it. 00:22:44.726 --> 00:22:57.076 A:middle N times 6n is 6n squared, N times a negative 1 is negative N. Whatever this math symbol is, 00:22:57.076 --> 00:23:00.296 A:middle you bring it straight down, so that's a minus there. 00:23:00.866 --> 00:23:06.076 A:middle From these two terms, we can factor out a 4. 00:23:07.086 --> 00:23:10.986 A:middle Now, you want to be careful because, remember, it's really negative 4 00:23:11.976 --> 00:23:17.046 A:middle so then what we would have left is 6n minus 1. 00:23:18.316 --> 00:23:24.986 A:middle Let's check our factoring, negative 4 times 6n is negative 24n, 00:23:25.746 --> 00:23:29.586 A:middle negative 4 times negative 1 is positive 4. 00:23:30.156 --> 00:23:37.046 A:middle We have to bring down the equal sign and bring down the right-hand side which is zero. 00:23:37.586 --> 00:23:46.466 A:middle Now from these two terms, we can factor out 6n minus 1 00:23:47.816 --> 00:23:56.026 A:middle and what we have left is N minus 4 equals zero. 00:23:56.556 --> 00:24:04.096 A:middle Step three says to apply the zero-product property, 00:24:05.286 --> 00:24:11.926 A:middle so we will do that in this next step here, where we take each 00:24:11.926 --> 00:24:13.976 A:middle of these factors and set it to zero. 00:24:14.176 --> 00:24:25.056 A:middle So, we have 6n minus 1 is equal to zero or N minus 4 is equal to zero. 00:24:28.116 --> 00:24:31.836 A:middle Step four says to solve for the variable. 00:24:33.236 --> 00:24:41.166 A:middle In this first equation, we would add one to both sides and get 6n is equal to 1, 00:24:41.736 --> 00:24:51.626 A:middle and then we would divide by 6 and get that N is equal to 1/6. 00:24:53.016 --> 00:24:54.956 A:middle That's our first solution. 00:24:55.496 --> 00:25:05.636 A:middle In the second equation, we would add 4 to both sides and get N is equal 00:25:05.636 --> 00:25:10.226 A:middle to 4, and that is our second solution. 00:25:18.336 --> 00:25:23.936 A:middle It's time to check your understanding, so pause the video to try this one on your own, 00:25:24.576 --> 00:25:26.966 A:middle then restart when you are ready to check your answer. 00:25:35.326 --> 00:25:45.016 A:middle In this example, we have to solve 2 is equal to 10k squared plus K. In our strategy, 00:25:45.456 --> 00:25:50.846 A:middle step one is to gather all terms on one side of the equation if needed. 00:25:52.086 --> 00:25:58.426 A:middle Now the only thing we need to move here is this 2, so we will subtract 2 from both sides 00:25:58.426 --> 00:26:08.216 A:middle of the equation and get zero is equal to 10k squared plus K minus 2. 00:26:08.716 --> 00:26:17.566 A:middle Step two is to factor using trial and error or the AC method. 00:26:17.566 --> 00:26:23.566 A:middle And I think we'll go ahead and use trial and error in this example. 00:26:24.106 --> 00:26:33.786 A:middle When we look at the factors of 10, we know it could be 1 times 10 or 2 times 5 and, 00:26:34.396 --> 00:26:39.346 A:middle for the factors of 2, it's going to be 1 times 2. 00:26:39.906 --> 00:26:46.616 A:middle Now in deciding which of these to try first, we typically will start 00:26:46.616 --> 00:26:48.706 A:middle with the factors that are closer together. 00:26:49.616 --> 00:26:53.956 A:middle So, let's do 2k and 5k. 00:26:54.566 --> 00:27:05.426 A:middle Now the only way to get 2 is with a 1 times a 2 because that 2 is negative, 00:27:05.846 --> 00:27:11.866 A:middle we know that one of these will be negative and one will be positive, but we will wait to decide 00:27:12.276 --> 00:27:14.456 A:middle until after we do our cross-products. 00:27:14.996 --> 00:27:24.726 A:middle The first cross-product is 5k, the second cross-product is 4k. 00:27:25.326 --> 00:27:29.176 A:middle Now, because the middle term is positive, 00:27:29.626 --> 00:27:35.426 A:middle we are going to want the larger magnitude cross-product to be positive. 00:27:35.946 --> 00:27:42.686 A:middle So, that means we'll have the 5k be positive, we'll want to make the 4k negative 00:27:42.686 --> 00:27:45.226 A:middle which means that 2 has to be negative. 00:27:46.516 --> 00:27:49.776 A:middle Now, when we add these up, we get k 00:27:50.926 --> 00:27:55.646 A:middle which already is exactly what we needed for our middle term. 00:27:56.206 --> 00:28:02.106 A:middle So, then we've got zero is equal to, 00:28:02.956 --> 00:28:16.806 A:middle now our factors will be the quantity 2k plus 1 times the quantity 5k minus 2. 00:28:17.276 --> 00:28:29.586 A:middle Step three is to apply the zero-product property which tells us that we have 00:28:29.586 --> 00:28:33.376 A:middle to set each of these factors equal to zero. 00:28:34.476 --> 00:28:46.276 A:middle So, we would have 2k plus 1 has to equal zero or 5k minus 2 has to equal zero. 00:28:49.756 --> 00:28:53.576 A:middle Step four is to solve for the variable. 00:28:54.136 --> 00:29:05.156 A:middle In this first equation, we would subtract 1 from both sides and get 2k is equal to negative 1, 00:29:05.736 --> 00:29:21.766 A:middle and then we would divide by 2 so we'd get K is equal to negative ½. In the second equation, 00:29:22.406 --> 00:29:31.826 A:middle we would add 2 to both sides, so we have 5k is equal to 2 and then we would divide 00:29:31.826 --> 00:29:37.256 A:middle by 5 to get that K is equal to 2/5. 00:29:42.776 --> 00:29:45.116 A:middle And these are our two solutions. 00:29:51.516 --> 00:29:55.156 A:middle Below, are some practice problems you can try on your own. 00:29:56.016 --> 00:30:01.876 A:middle You may either pause the video to work on them now, or write them down to work on them later. 00:30:02.956 --> 00:30:05.976 A:middle After a few seconds, the answers will be revealed. 00:30:20.436 --> 00:30:23.536 A:middle In objective three, you will learn how 00:30:23.536 --> 00:30:27.766 A:middle to solve quadratic equations with the greatest common factor. 00:30:28.296 --> 00:30:33.686 A:middle Our strategy has five steps. 00:30:35.026 --> 00:30:40.646 A:middle Step one is to gather all terms on one side of the equation if needed. 00:30:41.966 --> 00:30:48.206 A:middle Step two, factor out the GCF, the greatest common factor. 00:30:48.736 --> 00:30:57.806 A:middle Step three, factor the quadratic using trial and error or AC method. 00:30:59.366 --> 00:31:07.436 A:middle Step four, apply the zero-product property, and step five, solve for the variable. 00:31:09.116 --> 00:31:10.676 A:middle So, let's get started. 00:31:11.266 --> 00:31:22.046 A:middle In this example, we have to solve 30x minus 16x squared is equal to 24x cubed. 00:31:22.656 --> 00:31:31.986 A:middle In our strategy, step one is to gather all terms on one side of the equation if needed. 00:31:33.656 --> 00:31:38.286 A:middle Because we have terms on both sides, we will definitely need to do that, 00:31:39.386 --> 00:31:45.696 A:middle and I'm going to choose to take the left-hand side terms and move them over to the right, 00:31:46.556 --> 00:31:51.406 A:middle because my leading coefficient of 24 is already positive there. 00:31:51.946 --> 00:32:01.026 A:middle So, the left-hand side will become zero, the right-hand side we keep that 24x cubed. 00:32:02.376 --> 00:32:08.916 A:middle Now, over here, we are subtracting 16x squared, so that means we'll have 00:32:08.916 --> 00:32:12.446 A:middle to add 16x squared to both sides. 00:32:13.046 --> 00:32:20.546 A:middle And then, here, we will have to subtract 30x from both sides. 00:32:24.696 --> 00:32:32.156 A:middle Step two says to factor out the GCF, which we know is the greatest common factor. 00:32:33.636 --> 00:32:38.876 A:middle So, we bring down the zero on the left-hand side, bring down the equal sign and, 00:32:39.566 --> 00:32:45.036 A:middle among these three terms, our GCF is going to be 2x 00:32:46.076 --> 00:32:54.666 A:middle and we would have left 12x squared plus 8x minus 15. 00:32:56.936 --> 00:33:08.856 A:middle Always double check that 2x times 12x squared is 24x cubed, 2x times 8x is 16x squared, 00:33:09.576 --> 00:33:14.626 A:middle and 2x times a negative 15 is negative 30x. 00:33:15.186 --> 00:33:25.876 A:middle Now step three says to factor the quadratic using trial and error or AC method, and I think, 00:33:25.876 --> 00:33:29.556 A:middle for this example, we'll go ahead and use trial and error. 00:33:30.066 --> 00:33:37.516 A:middle Let's take a look at the leading coefficient of 12, and we know it could be 1 times 12, 00:33:38.406 --> 00:33:43.206 A:middle it could be 2 times 6, or 3 times 4. 00:33:43.726 --> 00:33:52.936 A:middle For the constant term of 15, it could be 1 times 15, or 3 times 5. 00:33:53.506 --> 00:34:00.816 A:middle Now, typically, we will try starting out with the factors that are closest together. 00:34:01.736 --> 00:34:08.046 A:middle So, maybe for the X squared term, we'll do 3x and 4x, 00:34:08.656 --> 00:34:15.226 A:middle and for the constant term, we'll do 3 and 5. 00:34:15.886 --> 00:34:22.476 A:middle Now because the 15 is negative, we know that one of these will be positive 00:34:22.886 --> 00:34:27.496 A:middle and one will be negative, but let's do the cross-products first. 00:34:28.716 --> 00:34:37.246 A:middle The first cross-product is 12x, the second cross-product is 15x. 00:34:39.296 --> 00:34:43.656 A:middle Because this middle term is positive, we know that means we want 00:34:43.656 --> 00:34:47.686 A:middle to make the larger magnitude cross-product positive, 00:34:48.116 --> 00:34:51.026 A:middle which means make the smaller one negative. 00:34:51.026 --> 00:34:55.916 A:middle If this is negative, then that means we have to make that 3 negative. 00:34:56.426 --> 00:35:04.776 A:middle Now when we add these two, we get 3x but, unfortunately, 00:35:04.776 --> 00:35:08.986 A:middle that does not equal the 8x that we need. 00:35:10.316 --> 00:35:19.736 A:middle So, maybe we'll try switching the 3 and the 5 so, 3x and 4x will stay the same. 00:35:21.016 --> 00:35:24.986 A:middle And then we'll put the 5 here and the 3 here. 00:35:26.556 --> 00:35:34.866 A:middle The first cross-product would be 20x, the second cross-product would be 9x. 00:35:35.706 --> 00:35:41.106 A:middle We know we're keeping the bigger one positive and taking the smaller one, making it negative. 00:35:42.316 --> 00:35:52.496 A:middle Now, when we add these, we get 11x, but that does not equal the 8x that we need. 00:35:52.726 --> 00:35:57.326 A:middle So, maybe we need to try switching these first ones. 00:35:57.976 --> 00:36:00.336 A:middle So, maybe we'll try 2 and 6. 00:36:01.206 --> 00:36:11.706 A:middle We've got 2x and 6x and we'll try putting the 3 and the 5. 00:36:12.386 --> 00:36:22.056 A:middle Now, the first cross-product is 18x, the second cross-product is 10x. 00:36:23.436 --> 00:36:27.556 A:middle We keep the larger one positive, make the smaller one negative, 00:36:27.706 --> 00:36:30.216 A:middle which means we have to make that 5 negative. 00:36:31.496 --> 00:36:35.696 A:middle Now when we add these two up, we get 8x 00:36:36.396 --> 00:36:40.616 A:middle which does match exactly what we need in the middle term. 00:36:41.186 --> 00:36:50.966 A:middle So, this tells us that our equation will become zero equals, 00:36:51.616 --> 00:36:54.666 A:middle take this 2x and bring it straight down. 00:36:55.276 --> 00:37:00.546 A:middle And now, this quadratic is going to factor 00:37:01.176 --> 00:37:15.846 A:middle into the quantity 2x plus 3 times the quantity 6x minus 5. 00:37:21.856 --> 00:37:26.926 A:middle Step four is to apply the zero-product property. 00:37:28.416 --> 00:37:34.306 A:middle Now this time, we have three factors being multiplied together and getting zero. 00:37:34.816 --> 00:37:38.346 A:middle So, that means, when we apply the zero-product property, 00:37:38.786 --> 00:37:42.316 A:middle we have to take all three and set them to zero. 00:37:43.606 --> 00:37:52.916 A:middle So, we would have 2x is equal to zero, or 2x plus 3 is equal 00:37:52.916 --> 00:38:00.106 A:middle to zero, or 6x minus 5 is equal to zero. 00:38:02.476 --> 00:38:06.126 A:middle Step five says to solve for the variable. 00:38:07.576 --> 00:38:09.436 A:middle So, let's take these one at a time. 00:38:10.506 --> 00:38:18.746 A:middle In this first equation, we would divide by 2 and get that X is equal to zero. 00:38:22.836 --> 00:38:32.406 A:middle In the second equation, we would subtract 3 so now we have 2x is equal to negative 3. 00:38:32.976 --> 00:38:41.976 A:middle And then we would divide both sides by 2 and get that X is equal to negative three halves. 00:38:49.506 --> 00:38:58.056 A:middle In this third equation, we would add 5 to both sides, so we have 6x is equal to 5, 00:38:59.216 --> 00:39:07.786 A:middle then we would divide both sides by 6 and we would get that X is equal to 5/6. 00:39:12.996 --> 00:39:16.776 A:middle And these are the three solutions to this equation. 00:39:26.476 --> 00:39:37.226 A:middle In this example, we have to solve 120y cubed minus 44y squared minus 8y is equal to zero. 00:39:39.416 --> 00:39:45.226 A:middle Referring to our strategy, step one says to gather all terms 00:39:45.226 --> 00:39:47.776 A:middle on one side of the equation if needed. 00:39:49.096 --> 00:39:56.386 A:middle Well, that's already been done for us, so we can move on to step two which says to factor 00:39:56.386 --> 00:39:59.966 A:middle out the GCF, the greatest common factor. 00:40:00.466 --> 00:40:09.256 A:middle Now for these three terms, we can see that we would have a greatest common factor of 4y. 00:40:10.466 --> 00:40:26.476 A:middle And what we would have left is 30y squared minus 11y minus 2, close the bracket, 00:40:27.076 --> 00:40:29.986 A:middle bring down the equal sign, and the zero. 00:40:30.476 --> 00:40:39.476 A:middle Step three says to factor the quadratic using trial and error or AC method. 00:40:40.636 --> 00:40:42.246 A:middle Here's our quadratic. 00:40:43.126 --> 00:40:47.856 A:middle I think we'll use the AC method in this one, so we practice it as well. 00:40:49.296 --> 00:40:53.436 A:middle So, we make our little X grid, we know 00:40:53.436 --> 00:41:02.426 A:middle that the AC term would be 30 times a negative 2, so that would be negative 60. 00:41:02.946 --> 00:41:13.886 A:middle And the B term is negative 11, so we need two numbers that will multiply 00:41:13.886 --> 00:41:18.266 A:middle to be negative 60, but add to be negative 11. 00:41:18.736 --> 00:41:26.906 A:middle Now the best way to not miss any possibilities is to just go through the factors in order. 00:41:27.736 --> 00:41:35.986 A:middle So, if we try 1 and negative 60, that adds to negative 59 instead of negative 11. 00:41:37.006 --> 00:41:44.086 A:middle Two and negative 30 adds to negative 28, again not what we need. 00:41:45.416 --> 00:41:57.956 A:middle Three and negative 20 adds to negative 17, and we can see that 4 and negative 15 would multiply 00:41:57.956 --> 00:42:02.556 A:middle to be negative 60, but add to be negative 11. 00:42:04.676 --> 00:42:18.226 A:middle Remember that these two terms tell us how to break up the B term into plus 4y and minus 15y. 00:42:18.816 --> 00:42:24.486 A:middle The 30y squared comes straight down. 00:42:30.076 --> 00:42:33.806 A:middle And the minus 2 comes straight down. 00:42:36.576 --> 00:42:41.386 A:middle We also have to bring down that GCF of 4y, 00:42:42.496 --> 00:42:47.466 A:middle and then we have to put our equal sign and the right-hand side of zero. 00:42:49.616 --> 00:42:59.046 A:middle Then we would bring down the 4y and we would look to factor these two by two. 00:43:00.396 --> 00:43:05.586 A:middle So, from these first two, we see that we can factor out a 2y 00:43:06.416 --> 00:43:10.926 A:middle and we would have left 15y plus 2. 00:43:13.696 --> 00:43:19.716 A:middle Whatever math symbol is here, you bring it directly down, so that's a minus sign. 00:43:20.356 --> 00:43:28.186 A:middle Now from these two terms, there's nothing that can be factored out other than 1 and, of course, 00:43:28.246 --> 00:43:41.416 A:middle it's a minus 1, so then that means we would have left 15y plus 2, close the bracket, equals zero. 00:43:42.856 --> 00:43:44.856 A:middle So, let's double check this whole step. 00:43:45.826 --> 00:44:02.386 A:middle 2y times 15y is 30y squared, 2y times 2 is plus 4y negative 1 times 15y is negative 15y 00:44:03.226 --> 00:44:09.456 A:middle and negative 1 times 2 is negative 2. 00:44:09.686 --> 00:44:18.196 A:middle Now from these two terms, we see that we can factor out 15y plus 2 00:44:18.736 --> 00:44:25.746 A:middle and we would have left the quantity 2y minus 1. 00:44:26.296 --> 00:44:32.296 A:middle We have to bring down that GCF of 4y. 00:44:33.186 --> 00:44:36.136 A:middle We have to bring down our equal sign and the zero. 00:44:38.336 --> 00:44:43.266 A:middle Step four says to apply the zero-product property. 00:44:43.796 --> 00:44:53.356 A:middle We have three factors, and we have to take each of them and set it equal to zero. 00:44:54.366 --> 00:45:04.096 A:middle So, we will have 4y is equal to zero, or 15y plus 2 is equal 00:45:04.096 --> 00:45:11.446 A:middle to zero, or 2y minus 1 is equal to zero. 00:45:14.356 --> 00:45:17.946 A:middle Step five says to solve for the variable. 00:45:18.496 --> 00:45:24.506 A:middle In this first equation, we would divide both sides by 4 00:45:25.176 --> 00:45:32.586 A:middle and end up with Y equals zero, there's our first solution. 00:45:33.106 --> 00:45:39.466 A:middle In the second equation, we would subtract 2 from both sides, 00:45:39.996 --> 00:45:45.326 A:middle so now we have 15y equals negative 2. 00:45:46.746 --> 00:45:55.966 A:middle Then we would divide both sides by 15 and end up with Y is equal to negative 2/15. 00:46:01.736 --> 00:46:10.466 A:middle And, in this third equation, we would add 1 to both sides, so that we have 2y equals 1, 00:46:11.936 --> 00:46:17.426 A:middle then we would divide by 2 and end up with Y is equal to ½. 00:46:24.096 --> 00:46:27.916 A:middle And these are the three solutions to this equation. 00:46:36.916 --> 00:46:42.346 A:middle It's time to check your understanding, so pause the video to try this one on your own, 00:46:42.906 --> 00:46:45.886 A:middle then restart when you are ready to check your answer. 00:46:51.616 --> 00:47:01.686 A:middle In this example, we have to solve negative 8p cubed plus 18p squared is equal to 7p. 00:47:02.236 --> 00:47:10.876 A:middle In our strategy, step one is to gather all terms on one side of the equation if needed. 00:47:12.146 --> 00:47:18.246 A:middle We definitely need to do that here and, because this leading coefficient is negative 00:47:18.246 --> 00:47:23.566 A:middle on the left-hand side, I'm going to choose to move all the terms 00:47:23.566 --> 00:47:26.536 A:middle from the left-hand side over to the right-hand side. 00:47:27.036 --> 00:47:37.126 A:middle So, then that means this left-hand side will become zero and the right-hand side we'll have 00:47:37.126 --> 00:47:43.676 A:middle to add 8p cubed to both sides so it becomes positive 8p cubed. 00:47:45.066 --> 00:47:57.026 A:middle We would subtract 18p squared and we bring down the positive 7p. 00:47:57.516 --> 00:48:08.646 A:middle Step two says to factor out the GCF, so bring down the zero and the equal sign 00:48:09.536 --> 00:48:14.886 A:middle from these three terms, we see that our GCF is P 00:48:15.826 --> 00:48:24.246 A:middle and what we would have left is 8p squared minus 18p plus 7. 00:48:28.716 --> 00:48:36.236 A:middle Step three says to factor the quadratic using trial and error or AC method. 00:48:38.286 --> 00:48:42.266 A:middle Let's use trial and error for factoring this quadratic. 00:48:42.726 --> 00:48:52.046 A:middle For the factors of 8, we could do 1 times 8 or 2 times 4. 00:48:52.636 --> 00:48:58.036 A:middle And for the factors of 7, there's only 1 times 7. 00:49:00.556 --> 00:49:04.366 A:middle Typically, in trial and error, we choose to start 00:49:04.416 --> 00:49:06.756 A:middle with the two factors that are closer together. 00:49:07.796 --> 00:49:12.346 A:middle So we will try 2p times 4p. 00:49:12.916 --> 00:49:19.796 A:middle And then, for the constant term, we know it's going to be 1 and 7. 00:49:20.946 --> 00:49:26.256 A:middle Now, because the constant term is positive but the middle term is negative, 00:49:26.746 --> 00:49:31.426 A:middle we know that both the 1 and the 7 will have to be negative. 00:49:34.236 --> 00:49:37.096 A:middle So, let's check our cross-products now. 00:49:38.006 --> 00:49:48.846 A:middle The first cross-product is negative 4p, the second cross-product is negative 14p and, 00:49:49.436 --> 00:49:54.226 A:middle when we add those up, we get negative 18p 00:49:55.076 --> 00:49:59.386 A:middle which already is exactly what we need for the middle term. 00:50:01.796 --> 00:50:11.676 A:middle So, let's bring our equation down, zero equals, we have to bring that GCF of P down. 00:50:12.206 --> 00:50:18.196 A:middle And now what we discovered was this quadratic is going to factor 00:50:18.786 --> 00:50:32.456 A:middle into the quantity 2p minus 1 times the quantity 4p minus 7. 00:50:35.636 --> 00:50:40.846 A:middle Now step four tells us to apply the zero-product property. 00:50:45.696 --> 00:50:52.616 A:middle And we know that that means we have to take each of these three factors and set it to zero, 00:50:53.836 --> 00:51:08.486 A:middle so we have P equals zero or 2p minus 1 equals zero, or 4p minus 7 equals zero. 00:51:11.656 --> 00:51:15.286 A:middle Step five says to solve for the variable. 00:51:15.886 --> 00:51:25.466 A:middle In this first equation, we're already done solving, so that solution is P equals zero. 00:51:25.976 --> 00:51:35.736 A:middle In the second equation, we would add 1 to both sides, so now we have 2p equals 1. 00:51:36.916 --> 00:51:49.636 A:middle Then we would divide by 2 and get P is equal to ½. In the third equation, 00:51:50.796 --> 00:51:57.576 A:middle we would add 7 to both sides and have 4p is equal to 7. 00:51:59.356 --> 00:52:06.816 A:middle Then we would divide by 4 and have that P is equal to 7/4. 00:52:12.456 --> 00:52:16.246 A:middle And these are the three solutions to this equation. 00:52:22.696 --> 00:52:26.326 A:middle Below, are some practice problems you can try on your own. 00:52:27.186 --> 00:52:33.036 A:middle You may either pause the video to work on them now or write them down to work on them later. 00:52:34.116 --> 00:52:36.976 A:middle After a few seconds, the answers will be revealed.