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### Course: Geometry (all content) > Unit 14

Lesson 12: Expanded equation of a circle# Features of a circle from its expanded equation

Sal finds the center and the radius of a circle whose equation is x^2+y^2+4x-4y-17=0, and then he graphs the circle. Created by Sal Khan.

## Want to join the conversation?

- In the first equation, why does Sal make the y^2 positive?(36 votes)
- This annoyed me for a solid 5 minutes. His line goes over the vertical part of the +. If you look back at the original equation you can see it's actually a positive y^2. Hope this helps.(66 votes)

- i don't get why you add the 4 in1:34?(17 votes)
- because, we have to add 4 to make it a square or to make it in the form of (a+b)^2.If you add 4 to (x^2 + 4x):

=(x^2 + 4x) +4

=(x^2 + 4x + 4) = (x^2 + 2x + 2x +4)

=[x(x+2) + 2(x+2)]

=(x+2) (x+2) = (x+2)^2

this is how a quadratic equation is solved. we have to bring it into (x-h)^2 form which is a part of (x-h)^2 + (y-k)^2 = y^2(26 votes)

- Why is it not -y^2? You put positive y^2 In the above video(7 votes)
- After rewatching the video, I noticed that at1:05, his circling of +y^2 covers up the | part of the plus sign, thus making it look like -y^2.

The equation should be x^2 + y^2...(15 votes)

- So how would you solve an equation if it goes x^2+y^2= C? Would the center just be (0,0) and radius the square root of C?(5 votes)
- It depends. In theory, it likely would be, since you don't have a number value for x, y, or C.(2 votes)

- how can you factor (x^2+2x+2)?(2 votes)
- Evan,

If you want to find the factors of

x²+2x+2

You can set the expression equal to 0 making it an equation and then find its roots. Then take (x - first root)(x-second root) and that will be the factors of the original expression.

While the method works for all equations. you sometimes get irrational factors and/or complex factors using the imaginary number.

As Sid said, for your expression, you will get a complex factor using the imaginary number.

Here is how to do it.

x²+2x+2 = 0 To use "complete the square" method you start my eliminating the constant on the left by subtracting the constant form both sides.

x²+2x=-2

Now to "complete the square" on the left, you need to

take the coefficient next to the x divided it by 2 and then square it.

You have a 2x in the equation, so divide the 2 by 2 which is 1 and square it. 1² is still 1. So add 1 to both sides.

x²+2x+1 = -2+1

x²+2x+1 = -1 Now factor the expression on the left

(x+1)(x+1) = -1 Which is

(x+1)² = -1 Now take the square root of each side.

x+1 =±√(-1) The √(-1) is the imaginary number i or more correctly i² = -1 so

x+1 = ±i Now subtract 1 from both sides

x=-1±i The ± means you have two answers so you could write it as

x=-1+i and x=-1-i

These are your two roots. To find the factors you need to subtract the roots from x

(x-first root)(x-second root)

And putting in the roots you have found

(x-(-1+i))(x-(-1+i)) Distribute the negative signs

(x+1-i)(x+1+i)

So that is how you find that

x²+2x+2 factors into

(x+1-i)(x+1-i)

just as Sid said,

I hope that is of some help.(10 votes)

- What did Sal mean when he said the center is the point (a,b) that essentially sets both (x-a)^2+(y-b)^2 equal to
**zero**??(3 votes)- If you substitute (a,b) for (x-y) you end up with (a-a)^2 + (b-b)^2 which equals 0.(7 votes)

- Is it possible to have a negative y^2 term? If yes, then how do you solve such kind of equations? I mean won't the negative coefficient make it more confusing? (eg. x^2 + 2x -y^2 =0) How would you solve this problem ( that is if the y^2 term can have a negative coefficient)?(4 votes)
- Keep watching the videos. If the y^2 term has a negative coefficient, the curve you end up drawing will be a hyperbola.(4 votes)

- I'm doing Geometry but didnt do Algebra yet (mostly because that seems to be the order it comes in KA)

The subject completing the square suddenly appears here in Geometry but the videos are in the algebra section I didnt do yet

Am I missing a simple intro to completing the square enough to do this problem without working through algebra first?

Thanks for any guidance(5 votes)- I would advice learning how to solve the square first, because that's used over and over again in Conic Sections. Conic Sections are technically speaking Pre-Calc, not geometry. That's why we use solving the square. Last year, my math class was an accommodation of both geometry and algebra to make math easier (we did circles back then). And believe me, it was a lot easier that way. I would just go ahead and watch those videos, the concept really isn't that hard to grasp and if you do understand it, then you'll be ahead of your classmates!(1 vote)

- how would I be able to solve

x^2+y^2-10x+16=0(2 votes)- x^2 - 10x + (y+0)^2 = -16

x^2 - 10x + 25 + (y+0)^2 = -16 + 25

(x-5)^2 + (y+0)^2 = 9

r = 3, h,k = 5,0(1 vote)

- I have question for you why is it you use the long equation of this next lesson about the equation of circle?(3 votes)
- to have you make it into a (x-h)^2 + (y-k)^2 = r^2 equation yourself

to teach you how to do it(3 votes)

## Video transcript

We're asked to graph the circle. And they give us this somewhat
crazy looking equation. And then we could graph
it right over here. And to graph a circle, you have
to know where its center is, and you have to know
what its radius is. So let me see if
I can change that. And you have to know
what its radius is. So what we need to do
is put this in some form where we can pick out its
center and its radius. Let me get my little scratch pad
out and see if we can do that. So this is that same equation. And what I
essentially want to do is I want to complete
the square in terms of x, and complete the square in terms
of y, to put it into a form that we can recognize. So first let's take
all of the x terms. So you have x squared and
4x on the left-hand side. So I could rewrite this
as x squared plus 4x. And I'm going to put some
parentheses around here, because I'm going to
complete the square. And then I have my y terms. I'll circle those
in-- well, the red looks too much like the purple. I'll circle those in blue. y squared and negative 4y. So we have plus y
squared minus 4y. And then we have a minus 17. And I'll just do that
in a neutral color. So minus 17 is equal to 0. Now, what I want
to do is make each of these purple expressions
perfect squares. So how could I do that here? Well, this would be a perfect
square if I took half of this 4 and I squared it. So if I made this plus 4,
then this entire expression would be x plus 2 squared. And you can verify
that if you like. If you need to review on
completing the square, there's plenty of videos
on Khan Academy on that. All we did is we took
half of this coefficient and then squared it to get 4. Half of 4 is 2,
square it to get 4. And that comes straight out of
the idea if you take x plus 2 and square it, it's going
to be x squared plus twice the product of 2 and
x, plus 2 squared. Now, we can't just
willy-nilly add a 4 here. We had an equality before,
and just adding a 4, it wouldn't be equal anymore. So if we want to
maintain the equality, we have to add 4 on the
right-hand side as well. Now, let's do the same
thing for the y's. Half of this coefficient right
over here is a negative 2. If we square negative 2,
it becomes a positive 4. We can't just do that
on the left-hand side. We have to do that on the
right-hand side as well. Now, what we have in blue
becomes y minus 2 squared. And of course, we
have the minus 17. But why don't we
add 17 to both sides as well to get rid of
this minus 17 here? So let's add 17 on the left
and add 17 on the right. So on the left, we're just left
with these two expressions. And on the right, we
have 4 plus 4 plus 17. Well, that's 8 plus 17,
which is equal to 25. Now, this is a form
that we recognize. If you have the form x minus a
squared plus y minus b squared is equal to r squared, we know
that the center is at the point a, b, essentially,
the point that makes both of these equal to 0. And that the radius
is going to be r. So if we look over
here, what is our a? We have to be careful here. Our a isn't 2. Our a is negative 2. x minus
negative 2 is equal to 2. So the x-coordinate
of our center is going to be negative 2, and
the y-coordinate of our center is going to be 2. Remember, we care about the
x value that makes this 0, and the y value
that makes this 0. So the center is negative 2, 2. And this is the radius squared. So the radius is equal to 5. So let's go back to the
exercise and actually plot this. So it's negative 2, 2. So our center is negative 2, 2. So that's right over there. X is negative 2,
y is positive 2. And the radius is 5. So let's see, this
would be 1, 2, 3, 4, 5. So you have to go a little
bit wider than this. My pen is having trouble. There you go. 1, 2, 3, 4, 5. Let's check our answer. We got it right.