Video Transcripts: Factoring Quadratics - Part 1

Factoring quadratics as (x+a)(x+b) 

So we have a quadratic expression here. X squared minus three x minus 10. And what I'd like to  do in this video is I'd like to factor it as the product of two binomials. Or to put it another way, I  want to write it as the product x plus a, that's one binomial, times x plus b, where we need to  figure out what a and b are going to be. So I encourage you to pause the video and see if you  can figure out what a and b need to be. Can we rewrite this expression as the product of two  binomials where we know what a and b are? So let's work through this together now. I'll  highlight a and b in different colors. I'll put a in yellow and I'll put b in magenta. So one way to  think about it is let's just multiply these two binomials using a and b, and we've done this in  previous videos. You might want to review multiplying binomials if any of this looks strange to  you. But if you were to multiply what we have on the right-hand side out it would be equal to,  you're going to have the x times the x which is going to be x squared. Then you're going to have the a times the x, which is ax. And then you're going to have the b times the x, which is bx.  Actually just let me, I'm not gonna skip any steps here just to see it this time. This is all review,  or should be review. So then we have, so we did x times x to get x squared. Then we have a  times x to get ax, to get a x. And then we're gonna have x times b, so we're multiplying each  term times every other term. So then we have x times b to get bx. So plus bx. b x. And then  finally we have plus the a times the b, which is of course going to be ab. And now we can  simplify this, and you might have been able to go straight to this if you are familiar with  multiplying binomials. This would be x squared plus, we can add these two coefficients because they're both on the first degree terms, they're both multiplied by x. If I have ax's and I add bx's  to that I'm going to have a plus b x's. So let me write that down. a plus b x's, and then finally I  have the plus, I'll do that blue color, finally I have it. I have plus ab. Plus ab. And now we can use  this to think about what a and b need to be. If we do a little bit of pattern matching, we see we  have an x squared there, we have an x squared there. We have something times x, in this case  it's a negative three times x. And here we have something times x. So one way to think about it  is that a plus b needs to be equal to negative three. They need to add up to be this coefficient.  So let me write that down. So we have a plus b needs to be equal to negative three. And we're  not done yet. We finally look at this last term, we have a times b. Well a times b needs to be  equal to negative 10. So let's write that down. So we have a times b needs to be equal to  negative 10. And in general, whenever you're factoring something, a quadratic expression that  has a one on second degree term, so it has a one coefficient on the x squared, you don't even  see it but it's implicitly there. You could write this as one x squared. A way to factor it is to come  up with two numbers that add up to the coefficient on the first degree term, so two numbers  that add up to negative three. And if I multiply those same two numbers, I'm going to get  negative 10. So two numbers that add up to negative three, to add up to the coefficient here.  And now when I multiply it, I get the constant term. I get this right over here. Two numbers  when I multiply I get negative 10. Well what could those numbers be? Well since when you  multiply them we get a negative number, we know that they're going to have different signs.  And so let's see how we could think about it. And since when we add them we get a negative  number, we know that the negative number must be the larger one. So if I were to just factor  10, 10 you could do that as one times ten, or two times five. And two and five are interesting  because if one of them are negative, their difference is three. So if one is negative... So let's see 

if we're talking about negative 10, you could say negative two times five. And when you  multiply them you do get negative 10. But if you add these two, you're going to get positive  three. But what if you went positive two times negative five. Now this is interesting because  still when you multiply them you get negative 10. And when you add them, two plus negative  five is going to be negative three. So we have just figured out our two numbers. We could say  that a is two or we could say that b is two, but I'll just say that a is equal to two and b is equal to  negative five. And so our original expression, we can rewrite as, so we can rewrite x squared  minus three x minus 10. We can say that that is going to be equal to x plus two times x, instead  of saying plus negative five which we could say, we could just say, actually let me write that  down. I could write just plus negative five right over there because that's our b. I could just  write x minus five, and we're done. We've just factored it as a product of two binomials. Now, I  did it fairly involved mainly so you see where all this came from. But in the future whenever you see a quadratic expression, and you have a one coefficient on the second degree term right  over here, you could say alright well I need to figure out two numbers that add up to the  coefficient on the first degree term, on the x term, and those same two numbers when I  multiply them need to be equal to this constant term, need to be equal to negative 10. You say  okay well let's see, they're gonna be different signs because when I multiply them I get a  negative number. The negative one is gonna be the larger one, since when I add them I got a  negative number. So let's see, let's say five and two seem interesting. Well negative five and  positive two, when you add them you're gonna get negative three, when you multiply them  you get negative 10. 

Factoring quadratics as (x+a)(x+b) (example 2) 

To better understand how we can factor second degree expressions like this, I'm going to go  through some examples. We'll factor this expression and we'll factor this expression. And  hopefully it'll give you a background on how you could generally factor expressions like this.  And to think about it, let's think about what happens if I were to multiply x plus something  times x plus something else. Well, if I were to multiply this out, what do I get? Well, you're going to get x squared plus ax plus bx, which is the same thing as a plus bx plus a times b. So if you  wanted to go from this form, which is what we have in these two examples, back to this, you  really just have to think about well, what's our coefficient on our x term, and can I figure out  two numbers that when I take their sum, are equal to that coefficient, and what's my constant  term, and can I think of two numbers, those same two numbers, that when I take the product  equal that constant term? So let's do that over here. If we look at our coefficient on x, can we  think of an a plus ab that is equal to that number negative 14? And can we think of the same a  and b that if we were to take its product, it would be equal to 40? So what's an a and a b that  would work over here? Well, let's think about this a little bit. If I have 4 times 10 is 40, but 4 plus  10 is equal to positive 14. So that wouldn't quite work. What happens if we make them both  negative? If we have negative 4 plus negative 10, well that's going to be equal to negative 14.  And negative 4 times negative 10 is equal to 40. The fact that this number right over here is  positive, this number right over here is positive, tells you that these are going to be the same  sign. If this number right over here was negative, then we would have different signs. And so if  you have 2 numbers that are going to be the same sign and they add up to a negative number, 

then that tells you that they're both going to be negative. So just going back to this, we know  that a is going to be negative 4, b is equal to negative 10, and we are done factoring it. We can  factor this expression as x plus negative 4 times x plus negative 10. Or another way to write  that, that's x minus 4 times x minus 10. Now let's do the same thing over here. Can we think of  an a plus b that's equal to the coefficient on the x term? Well, the coefficient on the x term here  is essentially negative 1 times x. So we could say the coefficient is negative 1. And can we think  of an a times b where it's going to be equal to negative 12? Well, let's think about this a little bit. The product of the 2 numbers is negative, so that means that they have different signs. So one  will be positive and one will be negative. And so when I add the two together, I get to negative  1. Well, just think about the factors of negative 12. Well, what about if one is 3 and maybe one is negative 4. Well, that seems to work. And you really just have to try these numbers out. If a is 3  plus negative 4, that indeed turns out to be negative 1. And if we have 3 times negative 4, that  indeed is equal to negative 12. So that seems to work out. And it's really a matter of trial and  error. You could try negative 3 plus 4, but then that wouldn't have worked out over here. You  could have tried two and six, but that wouldn't have worked out on this number. Or 2 and  negative 6, you wouldn't have gotten the sum to be equal to negative 1. But now that we've  figured out what the a and b are, what is this expression factored? Well, it's going to be x plus 3  times x plus negative 4, or we could say x minus 4. 

More examples of factoring quadratics as (x+a)(x+b) 

In this video I want to do a bunch of examples of factoring a second degree polynomial, which  is often called a quadratic. Sometimes a quadratic polynomial, or just a quadratic itself, or  quadratic expression, but all it means is a second degree polynomial. So something that's going to have a variable raised to the second power. In this case, in all of the examples we'll do, it'll be  x. So let's say I have the quadratic expression, x squared plus 10x, plus 9. And I want to factor it  into the product of two binomials. How do we do that? Well, let's just think about what  happens if we were to take x plus a, and multiply that by x plus b. If we were to multiply these  two things, what happens? Well, we have a little bit of experience doing this. This will be x  times x, which is x squared, plus x times b, which is bx, plus a times x, plus a times b-- plus ab.  Or if we want to add these two in the middle right here, because they're both coefficients of x.  We could right this as x squared plus-- I can write it as b plus a, or a plus b, x, plus ab. So in  general, if we assume that this is the product of two binomials, we see that this middle  coefficient on the x term, or you could say the first degree coefficient there, that's going to be  the sum of our a and b. And then the constant term is going to be the product of our a and b.  Notice, this would map to this, and this would map to this. And, of course, this is the same  thing as this. So can we somehow pattern match this to that? Is there some a and b where a  plus b is equal to 10? And a times b is equal to 9? Well, let's just think about it a little bit. What  are the factors of 9? What are the things that a and b could be equal to? And we're assuming  that everything is an integer. And normally when we're factoring, especially when we're  beginning to factor, we're dealing with integer numbers. So what are the factors of 9? They're  1, 3, and 9. So this could be a 3 and a 3, or it could be a 1 and a 9. Now, if it's a 3 and a 3, then  you'll have 3 plus 3-- that doesn't equal 10. But if it's a 1 and a 9, 1 times 9 is 9. 1 plus 9 is 10. So it does work. So a could be equal to 1, and b could be equal to 9. So we could factor this as being x

plus 1, times x plus 9. And if you multiply these two out, using the skills we developed in the last few videos, you'll see that it is indeed x squared plus 10x, plus 9. So when you see something  like this, when the coefficient on the x squared term, or the leading coefficient on this quadratic is a 1, you can just say, all right, what two numbers add up to this coefficient right here? And  those same two numbers, when you take their product, have to be equal to 9. And of course,  this has to be in standard form. Or if it's not in standard form, you should put it in that form, so  that you can always say, OK, whatever's on the first degree coefficient, my a and b have to add  to that. Whatever's my constant term, my a times b, the product has to be that. Let's do several more examples. I think the more examples we do the more sense this'll make. Let's say we had  x squared plus 10x, plus-- well, I already did 10x, let's do a different number-- x squared plus  15x, plus 50. And we want to factor this. Well, same drill. We have an x squared term. We have a  first degree term. This right here should be the sum of two numbers. And then this term, the  constant term right here, should be the product of two numbers. So we need to think of two  numbers that, when I multiply them I get 50, and when I add them, I get 15. And this is going to  be a bit of an art that you're going to develop, but the more practice you do, you're going to see that it'll start to come naturally. So what could a and b be? Let's think about the factors of 50. It  could be 1 times 50. 2 times 25. Let's see, 4 doesn't go into 50. It could be 5 times 10. I think  that's all of them. Let's try out these numbers, and see if any of these add up to 15. So 1 plus 50  does not add up to 15. 2 plus 25 does not add up to 15. But 5 plus 10 does add up to 15. So this  could be 5 plus 10, and this could be 5 times 10. So if we were to factor this, this would be equal  to x plus 5, times x plus 10. And multiply it out. I encourage you to multiply this out, and see  that this is indeed x squared plus 15x, plus 10. In fact, let's do it. x times x, x squared. x times 10,  plus 10x. 5 times x, plus 5x. 5 times 10, plus 50. Notice, the 5 times 10 gave us the 50. The 5x plus the 10x is giving us the 15x in between. So it's x squared plus 15x, plus 50. Let's up the stakes a  little bit, introduce some negative signs in here. Let's say I had x squared minus 11x, plus 24.  Now, it's the exact same principle. I need to think of two numbers, that when I add them, need  to be equal to negative 11. a plus b need to be equal to negative 11. And a times b need to be  equal to 24. Now, there's something for you to think about. When I multiply both of these  numbers, I'm getting a positive number. I'm getting a 24. That means that both of these need  to be positive, or both of these need to be negative. That's the only way I'm going to get a  positive number here. Now, if when I add them, I get a negative number, if these were positive,  there's no way I can add two positive numbers and get a negative number, so the fact that their sum is negative, and the fact that their product is positive, tells me that both a and b are  negative. a and b have to be negative. Remember, one can't be negative and the other one  can't be positive, because the product would be negative. And they both can't be positive,  because when you add them it would get you a positive number. So let's just think about what  a and b can be. So two negative numbers. So let's think about the factors of 24. And we'll kind  of have to think of the negative factors. But let me see, it could be 1 times 24, 2 times 11, 3  times 8, or 4 times 6. Now, which of these when I multiply these-- well, obviously when I  multiply 1 times 24, I get 24. When I get 2 times 11-- sorry, this is 2 times 12. I get 24. So we  know that all these, the products are 24. But which two of these, which two factors, when I add  them, should I get 11? And then we could say, let's take the negative of both of those. So when  you look at these, 3 and 8 jump out. 3 times 8 is equal to 24. 3 plus 8 is equal to 11. But that 

doesn't quite work out, right? Because we have a negative 11 here. But what if we did negative  3 and negative 8? Negative 3 times negative 8 is equal to positive 24. Negative 3 plus negative 8 is equal to negative 11. So negative 3 and negative 8 work. So if we factor this, x squared minus  

11x, plus 24 is going to be equal to x minus 3, times x minus 8. Let's do another one like that.  Actually, let's mix it up a little bit. Let's say I had x squared plus 5x, minus 14. So here we have a  different situation. The product of my two numbers is negative, right? a times b is equal to  negative 14. My product is negative. That tells me that one of them is positive, and one of them is negative. And when I add the two, a plus b, it'd be equal to 5. So let's think about the factors  of 14. And what combinations of them, when I add them, if one is positive and one is negative,  or I'm really kind of taking the difference of the two, do I get 5? So if I take 1 and 14-- I'm just  going to try out things-- 1 and 14, negative 1 plus 14 is negative 13. Negative 1 plus 14 is 13. So  let me write all of the combinations that I could do. And eventually your brain will just zone in  on it. So you've got negative 1 plus 14 is equal to 13. And 1 plus negative 14 is equal to negative  13. So those don't work. That doesn't equal 5. Now what about 2 and 7? If I do negative 2-- let  me do this in a different color-- if I do negative 2 plus 7, that is equal to 5. We're done! That  worked! I mean, we could have tried 2 plus negative 7, but that'd be equal to negative 5, so that  wouldn't have worked. But negative 2 plus 7 works. And negative 2 times 7 is negative 14. So  there we have it. We know it's x minus 2, times x plus 7. That's pretty neat. Negative 2 times 7 is  negative 14. Negative 2 plus 7 is positive 5. Let's do several more of these, just to really get well  honed this skill. So let's say we have x squared minus x, minus 56. So the product of the two  numbers have to be minus 56, have to be negative 56. And their difference, because one is  going to be positive, and one is going to be negative, right? Their difference has to be negative  1. And the numbers that immediately jump out in my brain-- and I don't know if they jump out  in your brain, we just learned this in the times tables-- 56 is 8 times 7. I mean, there's other  numbers. It's also 28 times 2. It's all sorts of things. But 8 times 7 really jumped out into my  brain, because they're very close to each other. And we need numbers that are very close to  each other. And one of these has to be positive, and one of these has to be negative. Now, the  fact that when their sum is negative, tells me that the larger of these two should probably be  negative. So if we take negative 8 times 7, that's equal to negative 56. And then if we take  negative 8 plus 7, that is equal to negative 1, which is exactly the coefficient right there. So  when I factor this, this is going to be x minus 8, times x plus 7. This is often one of the hardest  concepts people learn in algebra, because it is a bit of an art. You have to look at all of the  factors here, play with the positive and negative signs, see which of those factors when one is  positive, one is negative, add up to the coefficient on the x term. But as you do more and more  practice, you'll see that it'll become a bit of second nature. Now let's step up the stakes a little  bit more. Let's say we had negative x squared-- everything we've done so far had a positive  coefficient, a positive 1 coefficient on the x squared term. But let's say we had a negative x  squared minus 5x, plus 24. How do we do this? Well, the easiest way I can think of doing it is  factor out a negative 1, and then it becomes just like the problems we've been doing before. So  this is the same thing as negative 1 times positive x squared, plus 5x, minus 24. Right? I just  factored a negative 1 out. You can multiply negative 1 times all of these, and you'll see it  becomes this. Or you could factor the negative 1 out and divide all of these by negative 1. And  you get that right there. Now, same game as before. I need two numbers, that when I take their

product I get negative 24. So one will be positive, one will be negative. When I take their sum,  it's going to be 5. So let's think about 24 is 1 and 24. Let's see, if this is negative 1 and 24, it'd be  positive 23, if it was the other way around, it'd be negative 23. Doesn't work. What about 2 and  12? Well, if this is negative-- remember, one of these has to be negative. If the 2 is negative,  their sum would be 10. If the 12 is negative, their sum would be negative 10. Still doesn't work.  3 and 8. If the 3 is negative, their sum will be 5. So it works! So if we pick negative 3 and 8,  negative 3 and 8 work. Because negative 3 plus 8 is 5. Negative 3 times 8 is negative 24. So this  is going to be equal to-- can't forget that negative 1 out front, and then we factor the inside.  Negative 1 times x minus 3, times x plus 8. And if you really wanted to, you could multiply the  negative 1 times this, you would get 3 minus x if you did. Or you don't have to. Let's do one  more of these. The more practice, the better, I think. All right, let's say I had negative x squared  plus 18x, minus 72. So once again, I like to factor out the negative 1. So this is equal to negative  1 times x squared, minus 18x, plus 72. Now we just have to think of two numbers, that when I  multiply them I get positive 72. So they have to be the same sign. And that makes it easier in  our head, at least in my head. When I multiply them, I get positive 72. When I add them, I get  negative 18. So they're the same sign, and their sum is a negative number, they both must be  negative. And we could go through all of the factors of 72. But the one that springs up, maybe  you think of 8 times 9, but 8 times 9, or negative 8 minus 9, or negative 8 plus negative 9,  doesn't work. That turns into 17. That was close. Let me show you that. Negative 9 plus  negative 8, that is equal to negative 17. Close, but no cigar. So what other ones are there? We  have 6 and 12. That actually seems pretty good. If we have negative 6 plus negative 12, that is  equal to negative 18. Notice, it's a bit of an art. You have to try the different factors here. So this will become negative 1-- don't want to forget that-- times x minus 6, times x minus 12. 

Factoring quadratics with a common factor 

Averil was trying to factor six x squared minus 18x plus 12. She found that the greatest common factor of these terms was six and made an area model. What is the width of Averil's area  model? So pause this video and see if you can figure that out, and then we'll work through this  together. All right, so there's a couple of ways to think about it. She's trying to factor six x  squared minus 18x plus 12, and she figured out that the greatest common factor was six. So  one way you could think about it is this could be rewritten as six times something else. And to  help her think about it, she thought about an area model, where if you had a rectangle, if you  had a rectangle like this, and if the height is six and the width, let's just call that the width for  now, so this is the width right over here. If you multiply six times the width, maybe I could write width right over here, if you multiply six times the width, you multiply the height times the  width, you're going to get the area. So imagine that the area of this rectangle was our original  expression, six x squared minus 18x plus 12. And that's exactly what's drawn here. Now, what's  interesting is is that they broke up the area into three sections. This pink section is the six x  squared, this blue section is the negative 18x, and this peach section is the 12. And, of course,  these aren't drawn to scale, 'cause we don't even know how wide each of these are 'cause we  don't know what x is. So this is all a little bit abstract, but it's to show that we can break our  bigger area into three smaller areas. And what's useful about this is we could think about the  width of each of these sub-areas, and then we can add them together to figure out the total 

width. So what is the width of this pink section right over here? Well, six times what is six x  squared? Well, six times x squared is six x squared, so the width here is x squared. Now, what  about this blue area? A height of six times what width is equal to negative 18x? So let's see, if I  take six times negative three, I get negative 18, then I have to multiply it times an x as well to  get negative 18x. So six times negative three x is negative 18x. And then, last but not least, six,  our height of six, times what is going to be equal to 12? Well, six times two is equal to 12. So we  figured out the widths of each of these subregions, and now we know what the total width is.  The total width is going to be our x squared plus our negative three x, plus our two. So the  width is going to be x squared, and I can just write that as, minus three x, plus two. So we have  answered the question. And you could substitute that back in for this, and you could see, if you  multiplied six times all of this, if you distributed the six, you would indeed get six x squared  minus 18x plus 12. 

Factoring completely with a common factor 

So let's see if we can try to factor the following expression completely. So factor this  completely, pause the video and have a go at that. All right, now let's work through this  together. So the way that I like to think about it, I first try to see is there any common factor to  all the terms, and I try to find the greatest of the common factor, possible common factors to  all of the terms. So let's see, they're all divisible by two, so two would be a common factor, but  let's see, they're also all divisible by four, four is divisible by four, eight is divisible by four, 12 is  divisible by four, and that looks like the greatest common factor. They're not all divisible by x,  so I can't throw an x in there. So what I wanna do is factor out a four. So I could re-write this as  four times, now what would it be, four times what? Well if I factor a four out of four x squared,  I'm just going to be left with an x squared. If I factor a four out of negative eight x, negative  eight x divided by four is negative two, so I'm going to have negative two x. And if I factor a four out of negative 12, negative 12 divided by four is negative three. Now am I done factoring? Well it looks like I could factor this thing a little bit more. Can I think of two numbers that add up to  negative two, and when I multiply it I get negative three, since when I multiply I get a negative  value, one of the 'em is going to be positive and one of 'em is going to be negative. I can think  about it this way. A plus B is equal to negative two, A times B needs to be equal to negative  three. So let's see, A could be equal to negative three and B could be equal to one because  negative three plus one is negative two, and negative three times one is negative three. So I  could re-write all of this as four times x plus negative three, or I could just write that as x minus  three, times x plus one, x plus one. And now I have actually factored this completely. Let's do  another example. So let's say that we had the expression negative three x squared plus 21 x  minus 30. Pause the video and see if you can factor this completely. All right now let's do this  together. So what would be the greatest common factor? So let's see, they're all divisible by  three, so you could factor out a three. Let's see what happens if you factor out a three. This is  the same thing as three times, well negative three x squared divided by three is negative x  squared, 21 x divided by three is seven x, so plus seven x, and then negative 30 divided by three  is negative 10. You could do it this way, but having this negative out on the x squared term still  makes it a little bit confusing on how you would factor this further. You can do it, but it still  takes a little bit more of a mental load. So instead of just factoring out a three, let's factor out a 

negative three. So we could write it this way. If we factor out a negative three, what does that  become? Well then if you factor out a negative three out of this term, you're just left with an x  squared. If you factor out a negative three from this term, 21 divided by negative three is  negative seven x. And if you factor out a negative three out of negative 30, you're left with a  positive 10, positive 10. And now let's see if we can factor this thing a little bit more. Can I think  of two numbers where if I were to add them I get to negative seven, and if I were to multiply  them, I get to 10? And let's see, they'd have to have the same sign 'cause their product is  positive. So let's see A could be equal to negative five, and then B is equal to negative two. So I  can re-write this whole thing as equal to negative three times x plus negative five, which is the  same thing as x minus five, times x plus negative two, which is the same thing as x minus two.  And now we have factored completely.