Step by step
The question
Question: name the conic and write in standard form 9x² + 4y² + 36x − 8y + 4 = 0.
1 Name the conic
Look at the squares.
x² and y² have the same sign, but 9 and 4 are different numbers.
So it is an ellipse.
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: + 4 becomes − 4.
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 4 is 2.
2 times 2 is 4.
We add 4 inside the bracket.
But the bracket is multiplied by 9.
So on the left we really added 9 times 4: 36.
We add 36 on the right too.
Careful!
The 4 is inside a bracket with 9 in front.
It counts 9 times: add 36.
5 Complete the square for y
Half of −2 is −1.
(−1) times (−1) is 1.
We add 1 inside the bracket.
But the bracket is multiplied by 4.
So on the left we really added 4 times 1: 4.
We add 4 on the right too.
6 Write as squares
Each bracket is now a full square.
On the right, −4 + 36 + 4 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 an ellipse with center (−2, 1).
8 Check
Let's check with a point on it: (0, 1), 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.