Step by step
The question
Question: name the conic and write in standard form x² + y² + 2x − 4y − 4 = 0.
1 Name the conic
Look at the squares.
x² and y² both have 1 in front.
So it is a circle.
2 Group x and y
Put the x terms together in a bracket, and the y terms together.
Move the plain number to the right: − 4 becomes + 4.
3 Complete the square for x
Half of 2 is 1.
1 times 1 is 1.
We add 1 inside the bracket.
So we add 1 on the right too.
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
Half of −4 is −2.
(−2) times (−2) is 4.
We add 4 inside the bracket.
So we add 4 on the right too.
5 Write as squares
Each bracket is now a full square.
On the right, 4 + 1 + 4 is 9.
It is the circle rule: center (−1, 2), radius 3.
6 Check
Let's check with a point on it: (2, 2), the right end of the circle.
Put it in the first equation.
We get 0, like the first equation says.
So the standard form is right!
Let's look back.
First: name the conic.
Then: group x and y.
Then: complete the square for x.
Then: complete the square for y.
Then: write as squares.
Last: check.