Step by step
The question
Question: name the conic and write in standard form 9x² − 4y² − 18x − 16y − 43 = 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: − 43 becomes + 43.
3 Take out the number in front
Inside each bracket, x² and y² must have 1 in front.
So we take out 9 and 4.
4 Complete the square for x
Half of −2 is −1.
(−1) times (−1) is 1.
We add 1 inside the bracket.
But the bracket is multiplied by 9.
So on the left we really added 9 times 1: 9.
We add 9 on the right too.
Careful!
The 1 is inside a bracket with 9 in front.
It counts 9 times: add 9.
5 Complete the square for y
Half of 4 is 2.
2 times 2 is 4.
We add 4 inside the bracket.
But the bracket is multiplied by −4.
So on the left we really added −4 times 4: −16.
We add −16 on the right too.
6 Write as squares
Each bracket is now a full square.
On the right, 43 + 9 − 16 is 36.
7 Make the right side 1
Divide every part by 36, so the right side is 1.
9/36 is 1/4.
4/36 is 1/9.
That is the standard form of a hyperbola with center (1, −2).
8 Check
Let's check with a point on it: (3, −2), 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: take out the number in front.
Then: complete the square for x.
Then: complete the square for y.
Then: write as squares.
Then: make the right side 1.
Last: check.