Expert: Steve Holleran - 9/5/2007

Question
This question is in reference to another question you already answered.
Heres the link the the question.

http://en.allexperts.com/q/Advanced-Math-1363/General-equation-
circle.htm?zIr=5

I was wondering why on the second example, when you solved it you added a
+1 and a -1. Its the third line.

x^2 + y^2 + 2x + 4y - 11 = 0
x^2 + 2x + y^2 + 4y - 11 = 0
(x^2 + 2x + 1 - 1) + (y^2 + 4y + 4 - 4) - 11 = 0
((x + 1)^2 - 1) + ((y + 2)^2 - 4) - 11 = 0
(x + 1)^2 - 1 + (y + 2)^2 - 4 - 11 = 0
(x + 1)^2 + (y + 2)^2 - 1 - 4 - 11 = 0
(x + 1)^2 + (y + 2)^2 - 16 = 0
(x + 1)^2 + (y + 2)^2 = 16

I see how to do the rest, but I have other problems that are the same and I
need to know how, where and why the +1 and -1 are brought up.

Answer
Hi Jason,

Its basically because you have to make sure you balance the equation out.  By adding +1, then-1, I added 0 to the left side.  Its also why theres +4 and -4.

Perhaps it would be clearer this way:

x^2 + y^2 + 2x + 4y - 11 = 0

(x^2 + 2x +   ) + (y^2 + 4y +   )  =  11

(x^2 + 2x + 1)  + (y^2 + 4y + 4)   =  11 + 1 + 4

See, by adding +1 in the x-parenthesis, you could add 1 to the right side to balance, and +4 in the y-parenthesis with +4 on the right side to balance.

Looking back, its probably clearer when you look at it like I just did it above.

Then to finish, you have

       (x+1)^2 + (y+2)^2 = 16

Does that make it a little better?

Steve Holleran