Steps for transforming the equation of a circle into center/radius form. 

Example:    change x² + y² – 4x + 2y – 4 = 0 into the form (x – h)² + (y - k)² = r²

1)  Group the x’s together and y’s together.  Move the constant to the right of the equal sign. 

x² + y² – 4x + 2y – 4 = 0

                            +4    +4    

x² – 4x  + y² + 2y = 4                                           

2)  Leave two blanks.  (Spaces for completing the square.) 

x² – 4x + ___  + y² + 2y + ____ = 4                                           

3)  Complete the square twice, once for the x’s, once for the y’s.            

a)  Take ½ of the linear coefficient on the x, and square it.

(4/2)² =  (2)² = 4 

b)  Add the result to both sides of the equation. 

x² – 4x + 4  + y² + 2y + ____ = 4 + 4 

c)  Take ½ of the linear coefficient on the y, and square it.

(2/2)² = (1)² = 1 

d)  Add the result to both sides of the equation. 

x² – 4x + 4  + y² + 2y + 1 = 4 + 4 + 1 

4)  Factor the 2 perfect squares on the left.  Add the constants on the right.

(x – 2)² + (y+1)² = 9

5)  Compare your equation to the formula for a circle to find the center and radius.

(x – 2)² + (y+1)² = 9

(x – h)² + (y - k)² = r² 

Center = (h,k) =  (2, -1)

Radius= square root of 9 = 3  

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