Main content

## Get ready for Geometry

### Unit 3: Lesson 4

Completing the square- Completing the square
- Solving quadratics by completing the square
- Worked example: Completing the square (intro)
- Completing the square (intro)
- Worked example: Rewriting expressions by completing the square
- Worked example: Rewriting & solving equations by completing the square
- Completing the square (intermediate)

© 2023 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# Worked example: Rewriting expressions by completing the square

Sal rewrites x²+16x+9 as (x+8)²-55 by completing the square.

## Want to join the conversation?

- At2:00, why did you subtract 64 from the other side, where in other videos you added 64? What you do to one side you do to the other?(12 votes)
- Sal doesn't have an equation, so there is no other side. He is working with an expression. To ensure he maintains the original value of the expression, he can only add a value that equal 0 (this is the identify property of addition). This is done by adding the 64 and then also subtracting the 64. 64 - 64 = 0. Thus, he hasn't the expression. He's just making it look different.

Hope this helps.(45 votes)

- what does the b mean? I know it is a constant but, in the equation, what is it?(8 votes)
- This is what is known as the vertex form of a quadratic equation: it is for finding the vertex (obviously).

Given the expression (x+a)^2+b, the vertex of this quadratic is (-a,b). So, the b-value is the y-value of the vertex of the quadratic.

For example:

(x-7)^2 - 23

a = -7 and b = -23

So, the vertex of this quadratic is (-(-7),-23), or (7,-23).(16 votes)

- At1:27, how did he geat a=8?(4 votes)
- He tried to find what(a) times 2 = 16x.

So you could simple multiplication to get 8*2=16.

you could also do 16/2=8 (easier for bigger numbers).

Hope this helps :)(15 votes)

- In the last video it was (x-a)^2 and in this video it is (x+a)^2. How do you decide which one to use?(3 votes)
- Well, it depends on your quadratic equation. (x+a)² = x²+ 2ax + a² like in this video. The previous video's equation was of the form x² - 2ax + a², which is the same as (x - a)².(5 votes)

- it's simple, but why choose that formula?(3 votes)
- Completing the Square is quite useful for those equations that are not simply factorable. Also, the quadratic formula is mainly only good for expressions where a>1.(4 votes)

- At about1:45, Sal adds 64, and subtracts 64. There is no equal sign in the expression, so what he's basically doing is making a zero pair?? Uhhh, why would he do that, again?? And then he factors out the (x+a)² part (that I understand), but when he subtracts the 64 from 9... umm, what's the point of doing that? And isn't the WHOLE THING like, an expression?? Huh. So...why does he have to worry about making the expression true (when he added the 64, he said that he had to subtract a 64) when there's
*no equal sign*?? Very confusing.(2 votes)- Even though there is no equal sign we are trying to solve this particular expression, or rather find the zeroes of it. Since we want to do it for this particular expression we want to make sure it doesn't change in value, so basically if you graphed x² + 16x + 9, it would be the same graph as x² + 16x + 64 + 9 - 64. Now since these are the same graphs anything we do to the second expression that keeps its value the same will ake it true for the original too.

Now for that second one if we focus on x² + 16x + 64 hopefully you can see why it turns into (x+8)². Then that just leaves +9 - 64. You could leave that as is and still be right, but there's no reason not to combine like terms.

Basically if you expanded (x + 8)² - 55 you would get back to x² + 16x + 9, which is the whole point of this. This completeing the square method is just a method to change from one form of quadratics to another.

Let me kno if something did not make sense though.(4 votes)

- Sal calls this technique "Completing the square".

Is this form called "Perfect Square" form?:

(x+a)^2+b

Also,

What do we call this form?:

x^2 +2ax +a^2 +b

Is it just called "Completing the square" form?

I'm trying to memorize these forms so I can recall them easier but not sure what they're named.

Thank you K.A. and learners(1 vote)- y=a(x-h)^2+k (similar to your "perfect square" form is actually called vertex form where a is a scale factor and (h,k) is the vertex. Your example just has a=1 and different labels for the vertex which would be at (-a,b). The other two forms are standard y=ax^2+bx+c and factored form y=(ax+b)(cx+d). These are the three forms of quadratics. The idea of a^2x^2 + 2abx + b^2 is just a special case of the standard form which happens to be simple to get into factored form (ax+b)^2. Same with difference of perfect squares where a^2 x^2 - b^2 factors to (ax+b)(ax-b). There are only three forms, there is no perfect square form or completing the square form.(5 votes)

- what about the x in 16x? you keep the x^2 and 16, but not the x from 16x. You also dont do anything with the +64 but you carry the -64 and put it with the +9. So why is it that you dont do anything with those digits?(2 votes)
- Where does Sal ever drop the x from the 16? If you have (x=8)^2=(x+8)(x+8) = x^2 + 16x + 64, so the +64 is what is needed to complete the square. We have 16/2=8 which also goes into the perfect square.(2 votes)

- I didn't understand the part where he subtracted 64 from (9-64)1:44, why did he subtract it and not add, what is the concept or idea behind this step?(2 votes)
- Is this supposed to be a way to rewrite quadratic equations into vertex form when I'm trying to graph a parabola?(2 votes)

## Video 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.