Worked example: Rewriting expressions by completing the square
About this video
Sal rewrites X^2 + 16X + 9 in the form (X + A)^2 + B. He matches 2A to 16 to get A = 8, adds and subtracts 64 so the value doesn't change, and gets (X + 8)^2 - 55.
You'll be able to
- find A by matching 2AX to the X-term of an expression.
- add and subtract the same number to complete the square without changing an expression's value.
- rewrite X^2 + bX + c in the form (X + A)^2 + B.
Key words
- completing the square
- Rewriting an expression so part of it is a perfect square, like (X + 8)^2.
- expression
- A mathematical phrase with no equals sign, such as X^2 + 16X + 9.
- perfect square
- Something of the form (X + A)^2 = X^2 + 2AX + A^2.
Transcript
- [Voiceover] Let's see if we can take this quadratic expression here, X squared plus 16 X plus nine and write it in this form. You might be saying, hey Sal, why do I even need to worry about this? And 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 is if we expanded this X plus A squared, we know if we square X plus A it would be X squared plus two 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 nine, and I'm just gonna write it with a few spaces in it. X squared plus 16 X and then plus nine, 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 two A X is the same thing as that, then what's A going to be? So this is two A times X. Well, that means two A 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 eight, I would add an eight 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 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 plus two A X, where A is 8, 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 the stuff that I just squared off, this is going to be X plus eight squared. X plus eight, squared. Once again I know that because A is eight, A is eight, so this is X plus eight squared, and then all of this business on the right hand side.
What is nine minus 64? Well 64 minus nine 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.
Video © Khan Academy, used under CC BY-NC-SA. Watch on Khan Academy. Puzzicle is not affiliated with Khan Academy.