Step by step
The question
Question: name the conic and write in standard form x² − y² − 4x − 5 = 0.
1 Name the conic
Look at the squares.
x² and y² have different signs.
So it is a hyperbola.
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: − 5 becomes + 5.
3 Complete the square for x
Half of −4 is −2.
(−2) times (−2) is 4.
We add 4 inside the bracket.
So we add 4 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, 5 + 4 is 9.
5 Make the right side 1
Divide every part by 9, so the right side is 1.
That is the standard form of a hyperbola with center (2, 0).
6 Check
Let's check with a point on it: (5, 0), a vertex.
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.
Then: make the right side 1.
Last: check.