Factoring quadratics when a=1 (Let's Learn Factoring)

all right it's now time to get into the good stuff factoring with trinomials this is where most students probably have the most difficulty and if I was going to concede I could probably just make a video on as many different of these examples as possible and just say practice do a hundred of these and you'll be better at it and that is true if you do more practice then you're going to get better at things however the reason why I curated this course the way it was is because I want you to be able to understand things I want you to have a reasoning behind what you're doing because the reality is when you go to summer break when you you know go ahead and take a break from Factory you take another course and then you come back to another course that it has factoring you need to remember what you're doing why you're doing it and otherwise you're just going to forget and all that time you spent practicing a 100 practice problems goes away right you have no reason to relate to it so that's why I spent a lot of the time to get up to this point and I'm not even done yet I'm actually even going to do one more kind of visualization understanding of fact ing right when we have the area factoring is the idea of understanding what are the side lengths for that rectangle so for this example we're just going to focus on each rectangle at a time so if I want to say x squ well what two sides would create an X squ well that could be an x * an X what about a 3X well if I already have X is going to be representing the height then x * what is a 3X that'll be a positive3 over here I have a width of X what time x would give me a 2X that's going to be a positive2 and then we notice that 2 * 3 has to equal 6 which in case it does now that is the factored form of this now technically notice I can combine these two terms right so if I was going to combine these two terms I would have an x^2 plus a 5x plus a 6 right because that's what happened with a 5x and so the factored form of X2 + 5 x + 6 is going to be a x + 3 * an x + 2 but lucky for you I broke down 5x into 3x and 2x and that's what worked but what if you don't know how to break it down well that's what we're going to do in the next video so what we need to do is we need to focus on our last term this is the big one this is what we call our C and remember the last term is being multiplied by two other constants so we need to figure out well what numbers are going to work to multiply by six forget about the negative for a second let's just focus on the six so six can be broken down into 6 * 1 and a 3 * 2 that's it we only got two options now notice it's negative and what negative means is that one of these factors has to be negative right can't do a negative * a negative that's going to be a positive positive times positive never makes a negative so that means one of these numbers is going to be a negative now when we're talking about the middle term what we're talking about is doing this operation I think it's important that you can do enough practice so you can do this operation in your head but sometimes it just doesn't work like that I get it especially when you're first starting out it's sometimes hard to do what I want you to be able to see is we're going to try to combine these factors one of them being negative to give me a positive X so since we have a limited amount of factors why don't we just write out all the possibilities so therefore you can select which one's correct but the more practice you do get at this the easier it does become to identify which would be the correct one in your head okay so now what I did is I broke down my factors with one being positive and one being negative now again remember we're trying to combine these so we're going to be now replacing this multiplication please agree with me that all of these multiply to give me a -6 right because that's what we're looking for we're not really looking for the six we're looking for the6 however what I'm going to do is now to search for my middle term I need to now combine these to go ahead and get a positive X now technically since one of them is negative you're kind of finding the difference but all right well we can just look about this as the addition from on there I'll get to my tip later so which of these add up to give me a positive X and if you're seeing this one and hey you are correct so what I'm going to do in this example is I'm going to break this down one more step just so we can kind of see a factoring technique from here so therefore you can kind of see how this works this will be helpful for the next video but I wouldn't recommend doing this for all of your factoring techniques so just remember these are going to be the terms that com combine here right this is technically a 3X and a -2X because that's how we got to that X so rather than writing them inside a box which I could do right these would just be the numbers of my my coefficients for that box right that's what how we get that middle term cuz remember these two numbers give us that middle term right when we combine them in the area that's how we get down to an X now what we can do here is we can factor out our common terms factor out to the GCF in this case you can see the only thing they have in common is an X so therefore it's an X plus 3 and then over here you can see the only thing they have in common is going to be a -2 and that's going to leave me with an X+ 3 now we factor out the GC f one more time no notice they both share an X+ 3 so factor out the X+ 3 and you have an x + 3 * an x- 2 now you might say miss mclogan that was a lot of work because doesn't the three here translate to the three here and the -2 here translate to the -2 here yes you caught me right I'm doing a little bit extra because this is going to be important for the next lesson but in reality guys we want to do things easy we want to do things fast so once you take your factors that multiply to give your C and you identify which of those factors add or combine to give you your middle term you can use those numbers into your final answer for factoring okay but I wanted you to see this a lot of students have questions with this they get confused with this so hopefully doing it on a little bit of an easier process will be helpful for you when you see it again in the next video all right let's do the next one now I am one that kind of likes to do things in my head but a lot of times that can get me in trouble but there's one time that I think it's important for you to recognize and that is going to be when you have what we call special special products and one thing I always want you to look for is when you have a coefficient of one or really any square number and your last term is also a square number what you're going to want to do is look for a perfect square trinomial now the cool thing about perfect square trinomials is rather than getting brok rather than getting factored to a factor times a factor these get factored to one factor multiplying by itself or one factor being squared so let's go and take a look at four and just kind of see if this works so four I could do four * 4 * 1 and 2 * 2 right I don't know want to use that letter now then we're trying to see well what two numbers multiply to give me a Nega four so obviously if it's negative if this term's negative but that one's positive that means all of these have to both be negative right because a negative * a negative is still a positive a positive times a positive to positive but when I need to combine them right when I need to do this operation and that term is negative that means they both need to be negative so obviously you can see ne4 +1 is A5 so this is my only answer notice how these two numbers are exactly the same 2 and -2 now going through my understanding of going here to here I can rewrite this as an x - 2 * an x -2 but in mathematics we prefer to rewrite that as an x - 2^ squar right so that is going to be your simplii version always look for when you have your last term of squared in your first term of squared always think of factors that would be exactly the same that would work I always like to start with that first and I always like to start with that too when we have something that has a factor term but we're not dealing with trinomials in this case we have a binomial and we don't have anything in common right we have an X2 - 49 so I can't factor out the GCF here but one thing I can do is recognize that I have a square number and again my coefficient over here is also a square number so rather than doing the looking for a perfect square trinomial what I want to do is look for the difference of two squares and again what's really happening here is I'm lost my middle terms that means my middle terms add to zero that means one has to be positive one has to be negative so what two numbers would add to give me a zero well that's going to me two numbers that are exactly the same one positive one negative so what two numbers that are exactly the same give me a - 49 hopefully you got this this is A7 and positive 7 so doing this stuff in my head I can now get a x - 7 * an x + 7 and voila we got the difference of two squares last example whoa whoa whoa you're only going to do one example or where where did this two come from how does this impact it doesn't impact it can I just do what two numbers multiply give me eight add give me -1 no can't do that the process I'm talking about working here is only going to work when my a the coefficient of my x s is equal to one so there is a way to factor it when we have a two or three or four anything out in front but we're going to get to that to the next video what I want you to be able to see here is what happens with students all the time they know that when a is not equal to one the factoring process is a little bit more complicated it's a little bit more confusing so when they see that they just give up they don't even get a shot but remember the video I talked about the GCF factoring out the greatest common factor always look to see is there something all of these terms share that I can divide out out first and you recognize that all of these terms can be divided by two so guess what let's factor out the two that's going to be leaving with an x^2 minus a 5x plus a four all right we're back to four again now again this is a positive four so I have a positive four and I have a Nega five remember we talked about this right up here so we have that4 remember ne4 + 4 gives me a44 +1 gives me A5 so those are going to be my two factors just making this a lot easier look at look at how easy I'm making this for you and for me I don't have to make up or come up with new problems but now you guys can see I have a factored form it's going to be two terms or two expressions and then multiplied by a scaler but I think we all know what time it is it's time now to learn how to factor when there is a two and it cannot be factored out that is all going to come up in the next video