Step by step
The question
Question: name the conic and write in standard form x² + y² − 6x − 7 = 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: − 7 becomes + 7.
3 Complete the square for x
Half of −6 is −3.
(−3) times (−3) is 9.
We add 9 inside the bracket.
So we add 9 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, 7 + 9 is 16.
It is the circle rule: center (3, 0), radius 4.
5 Check
Let's check with a point on it: (7, 0), 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: write as squares.
Last: check.