 This video we'll talk about consecutive integers in applications of quadratics. Consecutive integers would be like starting here at negative four and then we want to put an x above every number or integer up to one just for examples but as you can see as we go from negative four to negative three we went one place and then from negative three over to negative two and so on every time we were just adding one well consecutive odd and even integers let's start with the consecutive odd because those are the funny ones so we start here at negative one and we skip every other so we skip the zero go to one skip the two go to three skip the four go to five skip the six go to seven notice those are all odd numbers and to get from one on to the next we had to add two well is it true to be the same thing if we do starting at an even number so let's start at two if we start here at two and then I skip three and go to four and I skip five and go to six again I'm going two places to get from one to the next so we do add two whether it's even or odd so when we're at consecutive integers then we would say that we had x and then the next integer would be x plus one and then the next integer would be x plus one more or two and the next one would be x plus one more than two or three so if I put zero in here for x then zero plus one would be one zero plus two would be two zero plus three would be three and I'm getting my consecutive integers if I'm doing the consecutive odd or even I start with x the next one would be x plus two and the next one I have to add two more so two plus two would be four and if I were to do the next one I have to add even two more so four plus two would be six okay so let's do some work then the product of two consecutive integers is 90 so that means that my numbers are going to be x and consecutive would be x plus one so if I want the product I'm going to have x times x plus one is or equal 90 or multiplying x squared plus x equal 90 but remember we can't solve a quadratic until it's said equal to zero so we have to subtract the 90 from both sides so we have I'll do it over here x squared plus x minus our 90 then equal to zero so now we're ready to factor and if we do our x we have negative 90 up here and on the bottom our b is one and it's a positive but I have a negative 90 so I have opposite signs that I have to work with let's change our color here opposite signs and I want to get to a one so if I'm thinking about those real quickly it's a positive one so I want the bigger number which would be 10 and nine 10 is positive nine is negative if I add those together I get a positive one and remember again when x is one then we can just say x plus 10 and x minus nine x plus our m and our n now we're ready to see which what the possible integers are so if x plus 10 is equal to zero when we subtract 10 from both sides it's going to be negative 10 and if we come over here and say x minus nine is equal to zero and we add nine to both sides x is going to be equal to nine and now you can see that we have some options here so in those options I would in my sentence I would say the numbers are or integers would be better the integers are negative 10 and if I add one to that x plus one or negative 10 plus one would be negative nine or they are nine and if I take x plus one then that would be equal to nine plus one or 10 so my numbers are either negative 10 and negative nine or positive nine and positive 10 all right so let's look at the second problem the product of two consecutive odd integers so let's look at that real closely we want to talk about odd integers is 143 and we want to find the numbers so in order to do that we need to think about what our variables are going to represent those numbers so x is going to be equal to the first odd and then we'll have to get to the odd one remember that we have to add two to get to the next odd number if I only add one it'll take me to an even number so this is my second odd and it says that I want to have a product of the two consecutive so we would say then the x times x plus two is equal to 143 so we would distribute the x and say that x times x is x squared and x times two is plus 2x is equal to 143 so we can see that we have a quadratic here because we've got this x squared so I want to subtract the 143 because I want to set it equal to zero and it's not like any of those other terms I have here so it's going to be x squared plus 2x minus 143 is going to be equal to zero so if I want to factor it I'm looking for factors of 143 that will add up to two so I could come in here and use my calculator and when I look at my table I'm looking for a difference of two and it looks like we're at 11 and 13 so it's a get rid of this again it's a negative 143 so that means that I need a negative 11 and a positive 13 to add up to positive two so it's x minus 11 and x plus 13 and that's equal to zero so when we set those equal to zero we get x minus 11 equals zero so x is equal to positive 11 x plus 13 is equal to zero so x is equal to negative 13 when I subtract 13 from both sides and now we want to know what we know what x is but what would x plus two be well if we take 11 and add two it'll give us 13 and if we take negative 13 and add two then we should get a negative 11 integers are 11 and 13 or negative 11 and negative 13