If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

### Course: High school geometry>Unit 7

Lesson 3: 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?
• 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.
• i don't get why you add the 4 in ?
• 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
• Why is it not -y^2? You put positive y^2 In the above video
• After rewatching the video, I noticed that at , 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...
• 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?
• It depends. In theory, it likely would be, since you don't have a number value for x, y, or C.
(1 vote)
• how can you factor (x^2+2x+2)?
• 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.
• 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)?
• Keep watching the videos. If the y^2 term has a negative coefficient, the curve you end up drawing will be a hyperbola.
• 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??
• If you substitute (a,b) for (x-y) you end up with (a-a)^2 + (b-b)^2 which equals 0.
• 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
• 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
• 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?