Step by step
The question
Question: find the center and the radius of x² + y² + 4x − 4y − 1 = 0.
1 Group x and y
Put the x terms together in a bracket, and the y terms together.
2 Remove the − 1
We want only the x and y groups on the left.
To remove − 1, we add 1.
We do it on both sides.
Why both sides?
The = sign means both sides are the same amount.
If we change only one side, they are not the same any more.
− 1 and + 1 make zero.
Cross them out.
0 plus 1 is 1.
3 Complete the square for x
To make x² + 4x a full square: half of 4 is 2.
2 times 2 is 4.
So we add 4.
We do it on both sides, so they stay equal.
Careful!
What we add on the left, we add on the right too.
Then the two sides stay equal.
4 Complete the square for y
To make y² − 4y a full square: half of −4 is −2.
(−2) times (−2) is 4.
So we add 4.
We do it on both sides, so they stay equal.
5 Write as squares
Each bracket is now a full square.
On the right, 1 + 4 + 4 is 9.
So (x + 2)² + (y − 2)² = 9.
x² + 4x + 4 is (x + 2)².
y² − 4y + 4 is (y − 2)².
6 Read the center and radius
Compare with the circle rule: (x − h)² + (y − k)² = r².
The signs turn: x + 2 gives h = −2.
y − 2 gives k = 2.
So the center is (−2, 2).
r² is 9.
So r is the square root of 9: 3, because 3 times 3 is 9.
So the center is (−2, 2) and the radius is 3.
7 Check
Let's check with a point on the circle: (1, 2), 3 to the right from the center.
Put it in the first equation.
We get 0, the same as the right side.
So the center and the radius are right!
Let's look back.
First: group x and y.
Then: remove the − 1.
Then: complete the square for x.
Then: complete the square for y.
Then: write as squares.
Then: read the center and radius.
Last: check.