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CCSS.Math: , , , , , ,

let's see if we can take this quadratic expression here x squared plus 16x plus 9 and write it in this form and you might be saying hey Sal why why do I even need to worry about this and 1 it is just good algebraic practice to be able to manipulate things but as we'll see in the future what we're about to do is called completing the square it's a really valuable technique for solving quadratics and it's actually the basis for the proof of the quadratic formula which you will use which you'll learn in the future so it's actually a pretty interesting technique so how do we write this in this form well one way to think about is if we expanded this X plus a squared we know if we Square X plus a would be x squared plus 2 a X plus a squared and then you still have that plus B right over there so one way to think about it is let's take this expression this x squared plus 16 X plus 9 I'm just going to write it with a little few spaces in it x squared plus 16 X and then plus 9 just like that and so if we say alright we have an x squared here we have an x squared here if we say that to a X is the same thing as that then what's a going to be so if this is to a times X well that means 2a is 16 or that a is equal to 8 and so if I want to have an a squared over here well if a is 8 I would add a I would add an 8 squared which would be a 64 well I can't just add numbers willy-nilly to an expression without changing the value of an expression so if I don't want to change the value of the expression I still need to subtract 64 so notice all that I have done now is I just took our original expression and I added 64 and I subtracted 64 so I have not changed the value of that expression but what was valuable about me doing that is now this first part of the expression this part right over here it fits the pattern of a perfect square quadratic right over here we have x squared + 2 ax where a is a plus a squared 64 once again how did I get 64 I took half of the 16 and I squared it to get to the 64 and so this stuff in this that I just squared off this is going to be X plus 8 squared X plus 8 squared once again I know that because a is 8 a is 8 so this is X plus 8 squared and then all of this business on the right hand side what is 9 minus 64 well 64 minus 9 is 55 so this is going to be negative 55 so minus 55 and we're done we've written this expression in this form and what's also called completing the square