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CCSS.Math: , , , , ,

we're asked to solve for s and we have s squared minus 2 s minus 35 is equal to 0 now this is the first time that you've seen this type of what's essentially a quadratic equation you might be tempted to try to solve for s using traditional algebraic means but the best way to solve this especially when it's explicitly equal to 0 is to factor the left-hand side and then think about whether think about the fact that those binomials that you factor into that they have to be equal to 0 so let's just do that so how can we factor this we've seen it several ways I'll show you kind of the standard way we've been doing it by grouping and then there's a little bit of a shortcut when you have a 1 as a coefficient over here so when you look at something when you do something by grouping when you factor by grouping you think about two numbers whose sum is going to be equal to negative 2 so you think about two numbers whose sum a plus B is equal to negative 2 and whose product is going to be equal to negative 35 a times B is equal to negative 35 since the product is a negative number one has to be positive one has to be negative and so if you think about it ones that are about two apart you have five and negative seven I think that'll work five plus negative seven is equal to negative two so the factor by grouping you split this middle term into a we can split this into a let me write it this way we have s squared and then this middle term right here I'll do it in pink this middle term right there I can write it as plus five s minus seven s and then we have the minus 35 and of course all of that is equal to zero now we call it factoring by grouping because we group it so we can group these first two terms and these first two terms they have a common factor of s so let's factor that out you have s times s plus five that's the same thing as s squared plus 5 s now in this second in the second two terms right here you have a common factor of negative seven so let's factor that out so you have negative seven times s plus five and of course all of that is equal to zero now we have two terms here where both of them have s plus five is a factor both of them have this s plus 5 as a factor so we can factor that out so let's do that so you have s plus 5 times times this s times this S right here right F plus 5 times s would give you this term and then you have minus that 7 right there - that 7i undistributed the S plus 5 and then this is going to be equal this is going to be equal to 0 now that we've factored it we just have to think a little bit about what happens when you take the product of two numbers I mean s plus 5 is the number S minus 7 is another number and we're saying that the product of those two numbers is equal to 0 if I told you that I had two numbers if I told you that I had the numbers a times B and that they equal to zero what do we know about either A or B or both of them well at least one of them has to be equal to 0 or both of them have to be equal to 0 so the fact that this number times that number is equal to 0 tells us that either either s plus 5 is equal to 0 or or and maybe both of them or s minus 7 is equal to 0 or divided in just green or s minus 7 is equal to 0 and so you have these two equations actually you could say and or it could be or either way and both of them could be equal to 0 so let's see let's see how we can solve for this well we can just subtract 5 from both sides of this equation right there and so you get on the left hand side you have s is equal to negative 5 that is one solution to the equation or you have so you can add 7 to both sides of that equation and you get s is equal to 7 so if s is equal to negative 5 or s is equal to 7 then we have satisfied this equation we can never even verify it if you make s equal to negative 5 you have positive 25 plus 10 we 35 - 35 that does equal zero if you have 749 - 14 - 35 does equal zero so we've solved for s now I mentioned there's an easier way to do it and when you have when you have something like this where you have one as the leading coefficient you just you don't have to do this two-step factoring let me just show you an example if I just have X plus X plus a times X plus B what is that equal to x times X is x squared x times B is BX a times X is plus ax a times B is a B plus a B so you get x squared plus these two can be added plus a plus B X plus a B and that's the pattern that we have right here we have one as a leading coefficient here we have one as a leading coefficient here so once we have our two numbers that add up to negative two so once we have our two numbers that add up to negative two that's our a plus B and we have our product that gets to negative 35 then we can straight just factor it into the product of those two things so it'll be or the product of the binomials where those will be the A's and the B so it'll be so we figured out it's five and negative seven five plus negative seven is negative 2 5 times negative 7 is negative 35 so we could have just straight factored at this point 2 - well actually this was the case of s so we could have factored it straight to the case of s plus 5 times s minus 7 we could have done that straight away and we've gotten to that right there and of course that whole thing was equal to 0 so that have been a little bit of a shortcut but but factoring by grouping will it'll-it'll it's a completely appropriate way to do it as well