 This video will talk about consecutive integers in applications of quadratics. Consecutive integers would be like starting here at negative 4 and then we want to put an x above every number or integer up to 1, just for examples. But as you can see, as we go from negative 4 to negative 3, we went one place and then from negative 3 over to negative 2 and so on every time we were just adding 1. 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 1 and we skip every other. So we skip the 0, go to 1, skip the 2, go to 3, skip the 4, go to 5, skip the 6, go to 7. Notice those are all odd numbers and to get from 1 on to the next we had to add 2. Well is it true to be the same thing if we do starting at an even number? So let's start at 2. If we start here at 2 and then I skip 3 and go to 4 and I skip 5 and go to 6, again I'm going two places to get from 1 to the next. So we do add 2 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 1 and then the next integer would be x plus 1 more or 2 and the next one would be x plus 1 more than 2 or 3. So if I put 0 in here for x then 0 plus 1 would be 1, 0 plus 2 would be 2, 0 plus 3 would be 3 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 2 and the next one I have to add 2 more so 2 plus 2 would be 4 and if I were to do the next one I have to add even 2 more so 4 plus 2 would be 6. Okay so let's do some work then. The product of 2 consecutive integers is 90 so that means that my numbers are going to be x and consecutive would be x plus 1. So if I want the product I'm going to have x times x plus 1 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 0 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 0. 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 1 and it's a positive but I have a negative 90 so I have opposite signs that I have to work with if I just change our color here opposite signs and I want to get to a 1 so if I'm thinking about those real quickly it's a positive 1 so I want the bigger number which would be 10 and 9. 10 is positive 9 is negative if I add those together I get a positive 1 and remember again when x is 1 then we can just say x plus 10 and x minus 9 x plus our m and our n. So now we're ready to see which what the possible integers are so if x plus 10 is equal to 0 when we subtract 10 from both sides it's going to be negative 10 and if we come over here and say x minus 9 is equal to 0 and we add 9 to both sides x is going to be equal to 9 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 1 to that x plus 1 or negative 10 plus 1 would be negative 9 or they are 9 and if I take x plus 1 then that would be equal to 9 plus 1 or 10 so my numbers are either negative 10 and negative 9 or positive 9 and positive 10 okay so the next one says consecutive two consecutive odd integers so that's important for us to see that we have odd integers and it's again a product of those two odd integers and then it is 143 well I don't really know what those are right off the bat so I'm going to go back to my calculator and bring that in here and I'm going to go to y equal and I put my 143 divided by x the trick is I have to look for consecutive odd integers but when I go to my table and I looked in here and there's a whole bunch of decimals anyway but when I get down to 11 I see that it's 13 11 and 13 would be 143 and those are consecutive odd integers so those must be my values so I'm going to say then that I have when I factor over here it's 143 and let's write the factors first my numbers are x always x is always the first one times a consecutive odd so that would be x plus 2 and that's going to be equal to 143 so when I multiply I get x squared plus 2x equal 143 and if I subtract that 143 from both sides I have x squared plus 2x minus 143 now we have negative 143 b is 2 so that makes our m and n negative 11 and positive 13 again we have x squared having a coefficient of one so we can just write x minus 11 x plus 13 and those two factors are equal to zero and if we work those two out so x minus 11 equals zero add 11 and x is equal to 11 if I take x plus 13 equal to zero and I subtract 13 from both sides I'm going to get a negative 13 on this side integers are negative 11 and negative 13 if you take 11 plus 2 you're going to get 13 so you have 11 and 13