WEBVTT mathematics/algebra-2/eaton 00:00:00.000 --> 00:00:02.500 Welcome to Educator.com. 00:00:02.500 --> 00:00:07.000 In a previous lesson, we talked about solving quadratic equations by graphing. 00:00:07.000 --> 00:00:14.700 And we are going to go on to discuss another technique, an algebraic technique, which is solving quadratic equations by factoring. 00:00:14.700 --> 00:00:21.700 So, first I am starting out with just a brief review of the different factoring techniques. 00:00:21.700 --> 00:00:24.800 And each of these is covered in-depth in the Algebra I series. 00:00:24.800 --> 00:00:29.300 So, if you are unsure on any of these, make sure you review that before you go on, 00:00:29.300 --> 00:00:36.200 because we are just using these techniques as a tool, now, to actually solve quadratic equations. 00:00:36.200 --> 00:00:47.900 OK, first, remember the greatest common factor: the greatest common factor of two or more numbers is the product of their common prime factors. 00:00:47.900 --> 00:01:06.800 So, the GCF of two or more numbers is the product of their common prime factors. 00:01:06.800 --> 00:01:14.500 And, in factoring, often, when you are factoring an equation, if there is a GCF, you want to factor that out first, 00:01:14.500 --> 00:01:17.100 because then you are working with something less complicated. 00:01:17.100 --> 00:01:26.800 For example, if you had 4x² + 16x + 36, you will recognize that there is a GCF of 4. 00:01:26.800 --> 00:01:37.400 So, I am going to factor that out and get 4 times x squared, plus 4x, plus 9. 00:01:37.400 --> 00:01:42.800 Factoring out the 4 gives me x² + 4x + 9; and this is much easier to work with. 00:01:42.800 --> 00:01:48.800 So, remember to factor out the GCF first, if there is one. 00:01:48.800 --> 00:02:01.000 Recall the difference of two squares: you will recognize this, because it is going to be in the form a² - b². 00:02:01.000 --> 00:02:13.700 So, an example would be x² - 4; and that is equal to (x + 2) (x - 2). 00:02:13.700 --> 00:02:19.000 So, this is the difference of two squares; here is the plus and the minus. 00:02:19.000 --> 00:02:24.600 And so, when you are factoring, if you recognize this, you can quickly factor it as the difference of two squares, 00:02:24.600 --> 00:02:30.100 knowing that it fits into this formula, a² - b². 00:02:30.100 --> 00:02:44.400 Perfect square trinomials: perfect square trinomials would be the result of squaring a binomial. 00:02:44.400 --> 00:02:54.700 For example, if you have x² + 6x + 9, this is actually equal to (x + 3)². 00:02:54.700 --> 00:03:00.300 So, if you take a binomial and square it, you will get a perfect square trinomial. 00:03:00.300 --> 00:03:05.500 Take a binomial; multiply it times itself; that is what a perfect square trinomial is. 00:03:05.500 --> 00:03:09.600 And so, recognizing that makes for easy factoring. 00:03:09.600 --> 00:03:17.700 General trinomials are a little more complex to factor; again, you can review all of this in Algebra I--techniques for factoring. 00:03:17.700 --> 00:03:23.200 And an example of a general trinomial would be x² + x - 12; 00:03:23.200 --> 00:03:29.800 it doesn't fit into any of these special cases, like the difference of two squares, or perfect square trinomials. 00:03:29.800 --> 00:03:32.900 So, I have to do a little more work in factoring that out. 00:03:32.900 --> 00:03:35.800 Recall from earlier lessons that you need to look at the signs. 00:03:35.800 --> 00:03:38.300 I have a positive sign here; I have a negative here. 00:03:38.300 --> 00:03:46.100 And if I have a negative, that must be the result of multiplying a positive and a negative. 00:03:46.100 --> 00:03:59.300 I then look at what the factors of 12 are; and the factors of 12 are 1 and 12, 2 and 6, and 3 and 4. 00:03:59.300 --> 00:04:09.400 And one will be positive, and one will be negative; and I need to have factors that sum to this middle term, which is actually 1. 00:04:09.400 --> 00:04:19.100 And I can try different combinations; and I know that -1 and 12 is not going to work; -12 and 1 certainly will not work. 00:04:19.100 --> 00:04:28.200 And I go on down, and from that, you can see that, if I want them to sum to 1, I am going to need factors that are close to each other. 00:04:28.200 --> 00:04:31.500 And the closest two I have here are 3 and 4. 00:04:31.500 --> 00:04:46.100 So, if I make 4 positive and 3 negative, then when I add the outer term and the inner terms, I will get my middle term x. 00:04:46.100 --> 00:04:52.500 And you can always check this using the FOIL method: First terms (that is x²), 00:04:52.500 --> 00:05:03.100 Outer terms (-3x), Inner terms (4x), and then the Last is -12. 00:05:03.100 --> 00:05:06.500 Simplifying this, I get my original back. 00:05:06.500 --> 00:05:12.700 General trinomials take a little more work, and you can always check those by multiplying it back out to make sure you did it correctly. 00:05:12.700 --> 00:05:17.000 So, we are making sure you have all these and know how to use them well. 00:05:17.000 --> 00:05:22.000 And then, we are going to be using them today to actually solve some quadratic equations. 00:05:22.000 --> 00:05:30.800 Now, once you have factored, you need to use the zero product rule to actually find the solutions. 00:05:30.800 --> 00:05:39.000 And what the zero product rule says is that, for any number a and b, if ab is zero, then either a is zero or b is zero-- 00:05:39.000 --> 00:05:46.300 because if a equals 0, then you would get 0 times b; that would work as a solution; 00:05:46.300 --> 00:05:53.200 if b is 0, then 0 times a would give you 0, and that is also a solution. 00:05:53.200 --> 00:06:01.500 For example, if I was given x² - 16 = 0 and asked to solve that, I would first recognize that it is 00:06:01.500 --> 00:06:12.500 in the form a² - b², and that it is therefore the difference of two squares. 00:06:12.500 --> 00:06:19.800 That allows me to factor it pretty quickly into (x + 4) (x - 4). 00:06:19.800 --> 00:06:22.800 So, I am factoring this, and it still equals 0. 00:06:22.800 --> 00:06:37.100 Now, use the zero product rule: the zero product rule tells me that, if this is 0 or this is 0, then this entire thing will equal 0, which is what I want. 00:06:37.100 --> 00:06:47.000 So, if x + 4 equals 0, this will be solved; if x - 4 equals 0, this will be solved. 00:06:47.000 --> 00:06:58.200 So, I am going to set this factor equal to 0 and solve for x; I am going to set this factor equal to 0 and solve for x. 00:06:58.200 --> 00:07:08.200 And that is going to give me...let's see...x = 4. 00:07:08.200 --> 00:07:14.200 And if you wanted to check that, you could go back up here and say, "OK, let x equal -4." 00:07:14.200 --> 00:07:24.700 So, that is -4² - 16 = 0; that is 16 - 16 = 0, and that checks out; that is a valid solution. 00:07:24.700 --> 00:07:34.000 I could do the same thing for 4; x = -4, or I could say x = 4; and that is going to give me 4² - 16 = 0. 00:07:34.000 --> 00:07:38.000 16 - 16 does equal 0, so there are two solutions here. 00:07:38.000 --> 00:07:47.100 And I was able to find those using factoring and the zero product rule. 00:07:47.100 --> 00:07:58.400 So, trying this out: first I am just asked to factor; and recall that the first thing you want to do is factor out any greatest common factor, 00:07:58.400 --> 00:08:02.400 because that is going to make whatever is left much easier to work with. 00:08:02.400 --> 00:08:10.100 And I see that I have a greatest common factor of 4. 00:08:10.100 --> 00:08:15.800 This factors into 4 times x² - x - 6. 00:08:15.800 --> 00:08:22.000 Now, all I have here is a general trinomial, so I want to think about what I have. 00:08:22.000 --> 00:08:30.200 And I want to make sure that I bring my 4 along with me, because that is part of the solution. 00:08:30.200 --> 00:08:40.900 I have a negative sign here; and the only way you are going to end up with that is if one of these is positive and one is negative. 00:08:40.900 --> 00:08:49.400 Now, I am going to think about what my factors of 6 are; I have 1 and 6, and 2 and 3. 00:08:49.400 --> 00:08:56.600 I need factors that sum to a middle term of -1; and that is not a very large number, so I am going to look for factors that are close together. 00:08:56.600 --> 00:09:01.900 I am going to try these first: now one is negative, and the other is positive. 00:09:01.900 --> 00:09:09.400 Since this is negative, I am going to look for making the larger number negative; so let me try if I have -3 plus 2. 00:09:09.400 --> 00:09:25.900 That equals -1, and that gives me the coefficient of that middle term; so this is what I have. 00:09:25.900 --> 00:09:30.400 And I can always check that by using FOIL to go back and multiply these out; 00:09:30.400 --> 00:09:33.800 and then I would have to multiply the 4 back into it. 00:09:33.800 --> 00:09:41.400 But this is factored; so I first factored out the GCF, and then I saw that I can take it farther, because this is not factored out all the way. 00:09:41.400 --> 00:09:48.800 It is a general trinomial that factors into (x + 2) (x - 3), times that greatest common factor, 4. 00:09:48.800 --> 00:09:56.900 OK, now we are asked to actually solve; and this is 3x² = 27. 00:09:56.900 --> 00:10:05.400 Before you can solve a quadratic equation, you need to put it in standard form. 00:10:05.400 --> 00:10:12.900 Recall that standard form is ax² + bx + c = 0. 00:10:12.900 --> 00:10:24.800 So, I have 3x² = 27; to put this in standard form, I am going to subtract 27 from both sides. 00:10:24.800 --> 00:10:34.000 And I have an ax² term; b must be 0, because there is no x term; and then I have a c of -27. 00:10:34.000 --> 00:10:42.900 So, I am going to solve by factoring; I have this in standard form--now I am going to factor out the GCF. 00:10:42.900 --> 00:10:53.200 And the GCF is 3; I am going to pull that out. 00:10:53.200 --> 00:11:05.000 Now, I am looking at this, and I see that what I have is something in the form a² - b², which is the difference of two squares. 00:11:05.000 --> 00:11:09.200 Always make sure you bring this GCF down--don't leave it behind. 00:11:09.200 --> 00:11:27.400 (x - 3) (x + 3) = 0; so now, I am going to use the zero product property (or zero product rule) to find my solutions. 00:11:27.400 --> 00:11:38.700 And the zero product rule says that if a times b equals 0, then a equals 0 or b equals 0, or they both have to be 0. 00:11:38.700 --> 00:11:56.800 So, first I am going to have 3 times (x - 3) equals 0; and if I divide 0/3, I am just going to get x - 3 = 0, or x = 3. 00:11:56.800 --> 00:12:01.900 So, that is one solution; the other solution is going to be x + 3 = 0. 00:12:01.900 --> 00:12:13.100 Using the zero product rule, that tells me that x equals -3; so my two solutions are that x equals 3 and x equals -3. 00:12:13.100 --> 00:12:23.100 These are my two solutions for this quadratic equation. 00:12:23.100 --> 00:12:35.900 And I found that just by factoring: x = 3 and x = -3; and I made my factoring a lot easier by first pulling out the GCF. 00:12:35.900 --> 00:12:40.600 Again, solve by factoring: and we have another situation where it is not in standard form. 00:12:40.600 --> 00:12:51.700 So, I am going to put it in standard form, which is ax² + bx + c = 0. 00:12:51.700 --> 00:13:04.200 All right, that is going to give me 4x² - 24x + 36 = 0, just subtracting 24x from both sides. 00:13:04.200 --> 00:13:12.100 This is another situation where I have a greatest common factor; so I have a GCF equal to 4--factor that out. 00:13:12.100 --> 00:13:19.900 Pull that out: that is 4, times x² - 6x + 9, equals 0. 00:13:19.900 --> 00:13:25.000 Again, I have something much easier to work with since I have pulled that out. 00:13:25.000 --> 00:13:36.800 So, figuring out how to work with this, I am going to go ahead and factor this out, because it is not all the way factored. 00:13:36.800 --> 00:13:58.600 And here, I have (x - 3) times (x -3); and it is actually a perfect square trinomial; this is really just (x - 3)². 00:13:58.600 --> 00:14:09.800 OK, so this is a perfect square trinomial, because you have x², and then (just check this using FOIL) 00:14:09.800 --> 00:14:21.700 I would have x² - 3x - 3x + 9; so I know I did that correctly. 00:14:21.700 --> 00:14:38.900 Now, I am going to use the zero product rule, which tells me that, if one of these is equal to 0, then the entire thing is equal to 0. 00:14:38.900 --> 00:14:41.000 So, I can use that to find the solution. 00:14:41.000 --> 00:14:49.600 Now, you can just go ahead and divide both sides by 4, in which case this will move over to here, and that just stays 0. 00:14:49.600 --> 00:14:59.300 Really, I just need to work with this and this: x - 3 = 0, and this is the same thing: x - 3 = 0. 00:14:59.300 --> 00:15:06.200 So, actually, when I figure this out, I just get the same thing for both; and it is x = 3. 00:15:06.200 --> 00:15:12.400 So, I only have one solution, or one real root, in this case; so, x equals 3. 00:15:12.400 --> 00:15:19.400 And again, put it in standard form; factor out the greatest common factor; complete your factoring; 00:15:19.400 --> 00:15:28.900 and then use the zero product rule to determine that there is one real solution, and that is that x equals 3. 00:15:28.900 --> 00:15:38.600 OK, again, solve by factoring; this is already in standard form; however, I have to get rid of this greatest common factor, which is 2. 00:15:38.600 --> 00:15:45.000 So, I am pulling that out to get 2x² + 5x - 12. 00:15:45.000 --> 00:15:50.400 When the leading coefficient is not 1, the factoring is a bit more difficult. 00:15:50.400 --> 00:15:53.200 So, let's move over here and work on this. 00:15:53.200 --> 00:15:59.200 When the leading coefficient is 2, I am going to have something in this form. 00:15:59.200 --> 00:16:05.000 And I have a negative here, so I also know that one of these is going to be a positive, and one is going to be a negative. 00:16:05.000 --> 00:16:14.100 But I don't actually know which one yet; but I know I am going to have +/-, or I am going to have this. 00:16:14.100 --> 00:16:25.600 OK, let's look at some factors of 12: factors of 12 would be 1 and 12, 2 and 6, and 3 and 4. 00:16:25.600 --> 00:16:32.500 Now, 5 is not that large of a number, especially when we think about the fact that we are going to have to be also multiplying by 2. 00:16:32.500 --> 00:16:39.200 So, if I go and use something like 6, and it ends up getting multiplied by 2, it is going to be very large; the difference is going to be great. 00:16:39.200 --> 00:16:48.000 I want factors that are smaller, since the difference between those, even with this 2x thrown in, is only going to be 5x. 00:16:48.000 --> 00:16:52.200 So, I am going to start with these, because they don't have a large difference between them. 00:16:52.200 --> 00:17:02.300 So, I am going to start out just trying (2x + 3) (x - 4) and seeing what I get. 00:17:02.300 --> 00:17:08.900 I am not worried about the first term; I am worried about the outer terms added to the inner terms. 00:17:08.900 --> 00:17:10.900 And see if I get the correct middle term. 00:17:10.900 --> 00:17:16.400 I am looking for the middle term equal to 5x. 00:17:16.400 --> 00:17:28.800 This is going to give me 2x times -4; that is -8x, plus 3x; that is 5x. 00:17:28.800 --> 00:17:33.400 I have the right idea, but I actually have the wrong signs here. 00:17:33.400 --> 00:17:43.300 So, I am going to try reversing the signs, because (this is -5x) I want this to be 5x, because here I have -8x + 3x is -5x. 00:17:43.300 --> 00:17:50.600 So, the same idea: let's try different signs, though. 00:17:50.600 --> 00:18:03.100 This time, I am making this negative and this positive; this is going to give me 2x(4); that is going to give me +8x; -3x is 5x. 00:18:03.100 --> 00:18:08.100 So, this is correct; I got the correct middle term, so this is the correct factoring. 00:18:08.100 --> 00:18:12.800 And this can be a lot of work to factor these; so it is important to go logically-- 00:18:12.800 --> 00:18:18.700 for example, seeing that I don't have a very large term here, especially when I am dealing with the 2x also 00:18:18.700 --> 00:18:23.300 (it is going to amplify things), to look for factors that are not very far apart. 00:18:23.300 --> 00:18:28.500 OK, so now, I am back here, and I am solving by factoring. 00:18:28.500 --> 00:18:37.400 This is going to give me (2x - 3) (x + 4) = 0. 00:18:37.400 --> 00:18:43.400 Dividing both sides by 2, this 2 is just going to drop out. 00:18:43.400 --> 00:18:53.500 So, when I use the zero product property, I am going to get 2x - 3 = 0, and I am also going to get x + 4 = 0. 00:18:53.500 --> 00:19:11.500 So, I just need to go ahead and solve those to get 2x = 3, or x = 3/2. 00:19:11.500 --> 00:19:19.300 Here, I just have x + 4 = 0, and that is simple: it is x = -4. 00:19:19.300 --> 00:19:30.600 I have two solutions: x = 3/2 and x = -4. 00:19:30.600 --> 00:19:37.600 I solved this by pulling out the greatest common factor, factoring that out, then factoring this trinomial into this, 00:19:37.600 --> 00:19:49.000 and using the zero product rule to give me 3/2 for a solution from here, and x = -4 for a solution from here. 00:19:49.000 --> 00:19:53.000 Thanks for visiting Educator.com, and I will see you next lesson!