"Complete the square" is a method that can be used to turn this in to the standard equation of a circle, which is set up in a way that makes it very easy to find the center and the radius.
Standard Equation is: (X - A)² + (Y - B)²=r²
Where (A, B) is the center of the circle, and r is the radius.
x² + 6x + y² - 2y + 6 = 0
First step, get the 6 on the other side.
x² + 6x + y² - 2y = -6
Second, take your first 'group' of X terms: x² + 6x
We cannot factor x² + 6x into (x + A)² but, we CAN turn x² + 6x + 9 into (x + 3)², which works.
So we add the 9 to both sides, and factor the x² + 6x + 9 into (x + 3)², which gives us this:
(x + 3)² + y² - 2y = 3
Now we do the same with our y's. It may not always seem obvious what our number needs to be that we add to both sides, so I'll tell you the process to use: halve the coefficient of the non-squared term, then square it.
So with y² - 2y we take the -2, halve it, making it -1, then square it, which gives us 1.
This makes y² - 2y + 1 which makes (y - 1)²
Since '1' was our number that we added to make it a perfect square (the name of the (x + 3)² and (y - 1)² and such), we will add 1 to other side as well.
(x + 3)² + (y - 1)² = 4
Remember that the center is comprised of those two numbers in the parentheses, but multiplied by -1 (look back at our original equation (Distribute the negatives!)). Therefore our center is C(-3,1). Our radius is the square root of the number on the other side. The square root of 4 is 2 (or -2, but you can't have a negative length).
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"Complete the square" is a method that can be used to turn this in to the standard equation of a circle, which is set up in a way that makes it very easy to find the center and the radius.
Standard Equation is: (X - A)² + (Y - B)²=r²
Where (A, B) is the center of the circle, and r is the radius.
x² + 6x + y² - 2y + 6 = 0
First step, get the 6 on the other side.
x² + 6x + y² - 2y = -6
Second, take your first 'group' of X terms: x² + 6x
We cannot factor x² + 6x into (x + A)² but, we CAN turn x² + 6x + 9 into (x + 3)², which works.
So we add the 9 to both sides, and factor the x² + 6x + 9 into (x + 3)², which gives us this:
(x + 3)² + y² - 2y = 3
Now we do the same with our y's. It may not always seem obvious what our number needs to be that we add to both sides, so I'll tell you the process to use: halve the coefficient of the non-squared term, then square it.
So with y² - 2y we take the -2, halve it, making it -1, then square it, which gives us 1.
This makes y² - 2y + 1 which makes (y - 1)²
Since '1' was our number that we added to make it a perfect square (the name of the (x + 3)² and (y - 1)² and such), we will add 1 to other side as well.
(x + 3)² + (y - 1)² = 4
Remember that the center is comprised of those two numbers in the parentheses, but multiplied by -1 (look back at our original equation (Distribute the negatives!)). Therefore our center is C(-3,1). Our radius is the square root of the number on the other side. The square root of 4 is 2 (or -2, but you can't have a negative length).
So there you go! Hope it helped. :)
C = (-3, 1)
R = 2
thus
(x+3)^2 + (y-1)^2 = 4
( x + 3 )^2 + ( y - 1 )^2 + 6 - 9 - 1 = 0
( x + 3 )^2 + ( y - 1 )^2 - 4 = 0
( x + 3 )^2 + ( y - 1 )^2 = 4
centre ( - 3, 1 )
radius = sqrt ( 4 ) = 2
Change that to the standard form for a circle. See my source.