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