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

Video transcript

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 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 B X plus a times B plus a B so if you wanted to go from this form if you want 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 are equal to when I 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 X coefficient the coefficient on X can we think of an A plus a B 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 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 but 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 times negative 10 is equal to 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 so these are going to be the same sign these are going to have the exact same sign and this number right over here was negative then we would have different signs and so if you have two numbers that are going to be the same sign and they add 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 for B is equal to negative 10 and we are done factoring it we can factor this expression as X plus negative 4x 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 and a plus B that's equal to the coefficient on the x-term well the coefficient on the X term here it's this is essentially negative 1 times X so we can say the coefficient is negative 1 and can we think of an a times B where it's going to be equal to a times B is equal to negative 12 is equal to negative 12 well let's think about this a little bit the product of the two numbers is negative so that means that they have different signs so dif different different signs so one will be positive and one will be negative and so when I add them two together I get two negative one well just think about the factors of negative 12 well what about if one is three and maybe one is negative four well that seems to work and you really just have to try these numbers out if a is 3 so 4 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 2 & 6 but that would wouldn't have worked out on this number you wouldn't have got or 2 and negative 6 you wouldn't have gotten the sum to be equal to negative 1 but now that we have we 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