Squaring binomials of the form (x+a)²

let's see if we could figure out what X plus seven let me write that a little bit neater x plus 7 squared is and I encourage you to pause the video and work through it on your own all right now let's work through this together so we just remember the whole trick we're squaring the entire binomial so this thing is going to be the same thing as X plus 7 times X plus 7 I'm going to write the second X plus 7 in a different color which is going to be helpful when we actually multiply things out when we see it like this that we can multiply these out the way we would multiply any binomials and I'll first do it the I guess you could say the slower way but the more intuitive way applying the distribute distributive property twice and then we'll think about maybe some shortcuts or some patterns we might be able to recognize especially when we are squaring binomials so let's start we're just applying the distributive property twice so let's look let's distribute let's distribute this yellow X plus 7 over this magenta X plus 7 so we can multiply it by the X this magenta X so it's going to be X let me do it in that same color so it's going to be magenta X times X plus 7 plus magenta 7 times yellow X plus 7 X plus 7 and now we can apply the distributive property again we can take this magenta X and multi and distribute it over the X plus 7 so x times X is x squared X times 7 is 7x and then we can do it again over here this 7 we did a different color so this 7 times that X is going to be plus another 7 X and then the 7 times the 7 is going to be 49 and we're in the homestretch we can then simplify it this is going to be x squared and then these two middle terms we can add together 7 X 7 X 7 deuce in orange 7x + 7 X that's going to be 14x plus 14x plus 49 plus 49 so plus 49 and we're done now the key question is do we see some patterns here do we see some patterns that we can generalize and that might help us square binomials a little bit faster in the future well when we first looked at just multiplying binomials we saw a pattern like X plus a times X plus B is going to be equal to x squared let me write it this way is going to be equal to x squared plus a plus B X plus B squared and so if both a and B are the same thing we could say that X plus a times X plus a is going to be equal to x squared and this is the case when we have a we have a coefficient of 1 on both of these X's x squared so now in this case a and B are both a so it's going to be a plus a times X or we could just say plus 2 a X let me be clear what I just did instead of writing a plus B I could just view this is a plus a times X and then plus a squared or that's the same thing as x squared plus 2ax plus plus a squared this is a general way of expressing of expressing a squared binomial like this a squared binomial where the coefficients on both X's are one we can see that's exactly what we saw over here and this in the example we did seven is our a so we got x squared x squared right over there let me circle it so we have this blue x squared that corresponds to that over there and then seven is our a so 2 ax 2 times 7 is 14x notice we have the 14x right over there so this 14x corresponds to 2 ax and then finally if a is 7 a squared is 49 a squared is 49 so in general if you're squaring a binomial you could a fast way to do is to do this this pattern here and we could do another example real fast just to make sure that we've we've understood things if I were to tell you what is X minus off to upload a negative in here X minus 3 squared encourage you to pause the video and think about it think about expressing this using this pattern well this is going to be in this case our a we have to be careful our a is going to be negative 3 so that is our a right over there so this is going to be equal to this is going to be equal to x squared now to a X let me do it in the same colors actually just for fun so it's going to be x squared now what is 2 times a times X a is negative 3 so 2 times a is negative 6 so it's going to be negative 6x so minus 6x that's 2 times a is the coefficient and then we have our X there and then plus a squared well if a is negative 3 what is negative 3 times negative 3 it's going to be positive 9 and just like that when we looked at this pattern we were able to very quickly figure out what this binomial squared actually is and I encourage you you can do it again with you applying the distributive property twice to verify that this is indeed the same thing as X -3 squared