 Hey guys, welcome to tutor terrific. This is part two of a solving quadratic equations series in this Video we are going to look at more complex methods of factoring quadratic equations Namely factoring by grouping and the complex diamond. So we're going to look at those two methods right now All right, so let's start with factor by grouping We are used factor by grouping when we have four terms in our polynomial So it's not a trinomial, but it has four terms and they're of this type for quadratic equation a x squared plus b x plus c x plus d so we see two terms in the middle that have an x to the first power and The first term has x to the second power and the a can be one or not. It could be anything So for example number one here We have x squared plus 9x plus 6x plus 54 The premise of this type of factoring is to do GCF greatest common factor style factoring twice We will separate the first and second binomial to begin with and factor those by GCF Then we will factor the two generated terms and I'll show you what I mean by GCF again Okay, so you're gonna rope off so to speak Both the first binomial and the second binomial now I know it's thought of as a single polynomial But polynomial of four terms has two binomials in it now if there's a plus between the first and second binomial You're fine if there is a minus you have to be a little more careful And I'll go over that in the next example now we will factor this first one by GCF What can I divide out of both x squared and 9x I could divide out an x They both share at most an x factor So now I have that x divided out all that's left is x times x plus nine Now in the second one What is the largest factor that those two terms share six Six does go into fifty four nine times so I could pull out of six and I would get x plus nine Now the way these are set up in your math books is that if you do it correctly The two binomials inside parentheses that remain will be the same now I want you to notice something I have created from four sums to sums with products so now I really have Well a single sum with of two of a product now I have two terms Okay, and I've bracketed them off for you. What factor do they share? They share each an x plus nine so I'm gonna take this is weird I'm gonna factor out that entire x plus nine factor I know it's a binomial that I'm factoring out totally legal and so you get That x plus nine is being factored out what are divided out what is left in the first term and x What is left in the second term a six Look at that Now we've got a wonderful simple Binomial product that we use the zero product property on and we'll get the following results X equals negative nine and negative six Okay, now for a more complex one this one has an a that's not equal to one Now what you sometimes have to do when you do any type of factoring the first step will be To factor by GCF the entire thing it makes it simpler And that's totally fine. So what I'm gonna do is I'm gonna see is there anything that all four terms share What's the largest factor? They all share. It's not one It's seven because each of these numbers is a multiple of seven so I can factor by GCF right from the start Before I even break up the first the two binomials, then I get seven x squared minus I have twenty eight divided by seven. That's four Negative fourteen divided by seven negative two with an x 56 divided by seven eight And here's the beauty of this By the zero product property this factor or this factor can equal zero well I know seven doesn't equal zero. So when I divide seven It disappears and that's always true With numerical factors So it's just divided out and then it's erased because it would never help the expression equals zero So it can be removed entirely through division, which is great. Now look what I have a very simple one That can only be done if you are solving the Quadratic equation not just factoring it. We are solving. That's what this video series is about now. We're gonna break up the two See a minus sign right here. We have to handle that properly the way we handle this properly is We include it in the second binomial and put a plus an extra plus in between the two Okay, this will assure us that we do not forget to factor out a negative from both terms in case it's necessary All right, let's look at the first binomial. What is The greatest common factor of this first binomial. It's x. So now we have x times x minus four Now remember from up here The goal is to get a binomial in the second from the second binomial that is identical to this one You see how the minus signs are reversed What that means is that I want to pull out a negative one. Can I pull out anything more? 2x and 8 I can they both have a 2 as well. So I can divide out negative 2 so that's a minus sign now and I get negative 2x divided by negative 2. That's just x negative 8 divided by positive 8 divided by negative 2 is negative 4 Aha, now I have the same binomial. Here are my now two terms I need to effectively think about and they both have that binomial x minus four as a factor So I will pull that out X minus four now that what's left when I divide that out of both terms is x and negative 2 So this is equal to zero by the zero product property. I'm gonna get the following two solutions x equals positive four and Positive two So here we have factoring by grouping and this is a part of the solution To my final factoring method for solving quadratic equations, which we'll go over next All right guys and for the last type of factoring in this video the complex diamond. That's my name for it It's gonna be used to solve the following type of quadratic equations a x squared plus bx plus c this time a is greater than one or Something other than one it could be negative as well The basic premise for factoring this way is to determine how you're gonna break up the Trinomial into four term polynomial that could be group factored So the middle term bx we are gonna figure out how to split that up so that we can group factor two binomials into two other binomials and That's the idea. So let's look at this first one now. You may be saying to yourself, but hey mister I can take out negative three from all the terms and then turn it into a Binomial or trinomial, excuse me where a is one that is true. Let's not do that for the sake of this example So what we're gonna do is we're going to split it into two binomials Well, we don't know how the 6x term was split up So it's gonna look like this we're gonna create just like it shows here in the pictorial two Binomials that are added together. We are going to place the negative 3x squared In the first spot and we're gonna place the nine in The last spot there and we're gonna figure out how 6x is gonna be broken up We are going to use the diamond for this step. That's quite nice We're going to put the six down here from the middle term, but in the top term, we're not gonna just put nine We're gonna put negative three times nine negative 27 ah Okay We've seen this before in another video now in the negative 27 Those are the numbers need to multiply the two numbers we're looking to need to multiply to negative 27 They need to add to six that means one of them must have been negative So that they can add to something positive one of them had to be positive Let's think of how we could get 27. Well, the first thing to pop to mind is three times nine Well, if we do nine and negative three those add to six and multiply To 27 does the order matter? No, it does not it works both ways, which is quite nice So here we have a Nine so we're gonna place that nine positive nine in the second spot in the first parentheses, but we're gonna put an X by it Then we're gonna put the negative three with an X by it at the first spot in the second parentheses and connect it to nine With a plus sign because that nine was positive Aha, now we saw how six X broke up into nine X and negative three X The next step is to do what we did in the previous factory method, which is group factor from here on out now we can see That in this first binomial that we have an X in both terms at least and we have three in both terms at least But I hate a leading negative and so I'm going to factor out negative 3x to get rid of that leading negative So I'll move over here Negative 3x will be factored out of my first binomial My binomial, excuse me and what I have left is X minus three X minus three In the second one, I'm gonna divide out negative 3x Also, no, excuse me just negative three both terms don't have an X But they both do have three and again, I hate leading negative So I'm gonna factor out that negative which will change this to a negative and make this term positive It's gonna be minus three and then what I have left is X and then minus three Okay, so what we see here Again, we will look at these as two separate terms. We will factor out the binomial X minus three What we'll have left is negative 3x minus three Okay, we set each of these equal to zero separately From the first one we'll get X equals three from the second one We would subtract we'd add three over and then divide by negative three. We're gonna get negative one Okay, so we see how this works it involves factoring by grouping, which is why I showed you that first next one now this one 5 18 and 9 it is completely Unclear as to if I can factor out any numbers in fact, I can't their greatest common factor is one Whereas before I know I can factor out a negative three and I postponed it to show you the method here I cannot do greatest common factoring. I have to do this complex diamond method So I will make two parentheses with a plus sign in between them and Then I am going to do the following. I'm gonna put the 5x squared the first spot in the first parentheses I'm gonna put the nine in the last spot in the second parentheses Figuring out how to break up negative 18. It's where the diamond comes in we put the negative 18 Down in the bottom of the Xbox or diamond and then the top we put five times nine so 45 What two numbers multiply to 45 but add to negative 18 Aha, this time that the product term is positive and the additive term is negative That means the two numbers must both be negative It's the only way for them to multiply to a positive and add to a negative. Let's think about 45 45 Five times nine. That's probably the first thing that comes to mind negative five and negative nine do not add to negative 18 so we need to try another set of things that multiply to 45 and What it looks like to me is 15 and three if I do negative 15 They do negative three Those multiply Deposit 45 and add to negative 18. So those are the two numbers with an x attached to them that go in these two spots That's how negative 18 broke up so it's minus 15 x in the first one and Minus 3x in the second one with a plus after it. Okay So now we will do the group factoring of both these binomials and we are going to see here Sorry, that's a five. It's hard to see That we could take out 5x from this first binomial. So divide that out you get x minus three Now remember, that's what we want in the second binomial as well. So to get this From here. I must divide out negative three So it's to switch the sign. So this is positive and this one's negative. So minus three. What's left inside is x minus Three Okay, so now looking at the two terms I could see they both share that binomial x minus three So I'm gonna factor that out now x minus three times 5x minus Three and that's equal to zero Okay Now by doing the zero product property. I'm gonna get the following two solutions x equals positive three from this one From this one when I add three over and divide by five I get three-fifths Yes, this time it's a fraction for one of the solutions All right guys that about sums up the ways to solve quadratic equations By factoring if you look at part one and part two together of this series in the third part We are gonna do methods for solving quadratic equations where factoring is not possible All right guys. Thanks for watching this video. This is Falconator signing out