Step by step
The question
Question: name the conic and write in standard form x² − 6x − 8y + 1 = 0.
1 Name the conic
Look at the squares.
Only x² is here.
So it is a parabola.
2 Keep x on the left
Keep the x terms on the left.
Move − 8y and the plain number to the right: their signs change.
3 Complete the square for x
Half of −6 is −3.
(−3) times (−3) is 9.
Add 9 to both sides.
Careful!
What we add on the left, we add on the right too.
Then the two sides stay equal.
4 Write as a square
The bracket is a full square: (x − 3)².
On the right, −1 plus 9 is 8.
5 Take out the number in front
Take 8 out of the right side: 8(y + 1).
That is the standard form of a parabola.
6 Check
Let's check with the vertex, (3, −1).
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: keep x on the left.
Then: complete the square for x.
Then: write as a square.
Then: take out the number in front.
Last: check.