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we're now going to explore factoring a type of expression called a difference of squares and the reason why it's called a difference of squares is because it's expressions like x squared minus 9 this is a difference we're subtracting between two quantities that are each squares this is literally x squared let me do a different color this is x squared minus 3 squared it's the difference between two quantities that have been squared it turns out that this is pretty straightforward to factor and to see how it can be factored let me pause there for a second and get a little bit of review of multiplying binomials so put this on the backburner a little bit before I give you the answer how do you factor this let's do a little bit of an exercise let's multiply X plus a times X minus a where a is some number and we can use that do that using either the foil method but I like just thinking of this as a distributive property twice we could take X plus a and distribute it onto the X and onto the a so when we multiply it by X we would get x times X is x squared a times X is plus a X and then when we multiply it by the negative a well it'll become negative a times X minus a squared so these middle two terms cancel out and you are left with x squared minus a squared you're left with a difference of squares x squared minus a squared so we have an interesting result right over here that x squared minus a squared is equal to is equal to X plus a X plus a times X minus a and so we can use and this is for any a so we could use this pattern now to factor this here what is our a our a is 3 this is x squared minus 3 squared or we could say minus our a squared if we say three is a and so to factor it this is just going to be equal to X plus R a which is three times X minus R a which is three so X plus three times X minus three now let's do some examples to really reinforce this idea of factoring differences of squares so let's say we want to factor let me say Y squared minus 25 and it has to be a difference of squares Kant doesn't work with a sum of squares well in this case this is going to be Y and you have to confirm okay yeah 25 is 5 squared + y squared is well Y squared this is going to be y plus something times y minus something and what is it something well this right here is 5 squared so it's y plus 5 times y minus 5 and the variable doesn't have to come first we could write 121 - I'll introduce a new variable - B squared well this is a difference of squares because 121 is 11 squared so this is going to be 11 plus something times 11 minus something in this case that something is going to be the thing that was squared so 11 plus B times 11 minus B so in general if you see a if you see a difference of squares one square being subtracted from another Square and it could be a numeric perfect square or it could be a variable that has been square that can be that you could take the square root of well then you could say all right well that's just going to be the first thing that squared plus the second thing that has that has been squared times the first thing that was squared minus the second thing that was squared now some common mistakes that I've seen people do including my son when they first learned this is they say okay it's easy to recognize the difference of squares but then they say oh oh is this Y squared plus 25 times y squared minus 25 no the important thing to realize is is that what is getting squared over here why is the thing getting squared and over here it is five that is getting squared those are the things that are getting squared in this difference of squares and so it's going to be y plus five times y minus five I encourage you to just try this out we have a whole practice section on Khan Academy where you can do many many more of these to become familiar