Step by step
The question
Question: name the conic and write in standard form x² + y² − 8y = 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.
3 Complete the square for y
Half of −8 is −4.
(−4) times (−4) is 16.
We add 16 inside the bracket.
So we add 16 on the right too.
Careful!
What we add on the left, we add on the right too.
Then the two sides stay equal.
4 Write as squares
Each bracket is now a full square.
On the right, 0 + 16 is 16.
It is the circle rule: center (0, 4), radius 4.
5 Check
Let's check with a point on it: (4, 4), 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 y.
Then: write as squares.
Last: check.