Worked example: Rewriting expressions by completing the square | High School Math | Khan Academy

Let's see if we can take this quadratic expression here, ( x^2 + 16x + 9 ), and write it in this form.

You might be saying, "Hey Sal, why do I even need to worry about this?" One, 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'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 it is if we expanded this ( (x + a)^2 ). We know if we square ( (x + a) ) we would get ( x^2 + 2ax + a^2 ), 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^2 + 16x + 9 ). I'm just going to write it with a little few spaces in it: ( x^2 + 16x ) and then ( + 9 ), just like that.

If we say, "Alright, we have an ( x^2 ) here, we have an ( x^2 ) here," if we say that ( 2ax ) is the same thing as that, then what's ( a ) going to be? So if this is ( 2a \times x ), well that means ( 2a = 16 ) or that ( a = 8 ).

And so if I want to have an ( a^2 ) over here, well if ( a ) is 8, I would add ( 8^2 ), which would be 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 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, fits the pattern of what? A perfect square quadratic.

Right over here we have ( x^2 + 2ax ) where ( a ) is 8, plus ( a^2 ) which is 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 that I've just squared off, this is going to be ( (x + 8)^2 ). Once again, I know that because ( a ) is 8. So this is ( (x + 8)^2 ).

And then all of this business on the right-hand side, what is ( 9 - 64 )? Well, ( 64 - 9 ) is ( -55 ). So this is going to be ( -55 ).

So minus 55, and we're done! We've written this expression in this form, and what's also called completing the square.