Step by step
The question
Question: name the conic and write in standard form 2x² + 2y² − 8x + 4y − 8 = 0.
1 Name the conic
Look at the squares.
x² and y² both have 2 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: − 8 becomes + 8.
3 Take out the number in front
Inside each bracket, x² and y² must have 1 in front.
So we take out 2 and 2.
4 Complete the square for x
Half of −4 is −2.
(−2) times (−2) is 4.
We add 4 inside the bracket.
But the bracket is multiplied by 2.
So on the left we really added 2 times 4: 8.
We add 8 on the right too.
Careful!
The 4 is inside a bracket with 2 in front.
It counts 2 times: add 8.
5 Complete the square for y
Half of 2 is 1.
1 times 1 is 1.
We add 1 inside the bracket.
But the bracket is multiplied by 2.
So on the left we really added 2 times 1: 2.
We add 2 on the right too.
6 Write as squares
Each bracket is now a full square.
On the right, 8 + 8 + 2 is 18.
7 Make x² and y² alone
Divide every part by 2: 18/2 is 9.
It is the circle rule: center (2, −1), radius 3.
8 Check
Let's check with a point on it: (5, −1), 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: take out the number in front.
Then: complete the square for x.
Then: complete the square for y.
Then: write as squares.
Then: make x² and y² alone.
Last: check.