Question

How do you graph `4x^2 + 4y^2 + 24x + 56 = 24y`?

Answer 1

First rearrange the terms to get all the `x` terms together and all the `y` terms together, then complete the squares to get the expression into the standard form of an ellipse or circle.

`4x^2 +24x +4y^2 -24y +56 = 0`
`4(x^2 +6x +y^2 -6y +14)=0`

We can divide the whole expression by `4` without affecting the final result.

`(x+3)^2 -9 +(y-3)^2 - 9 +14=0`
`(x+3)^2 +(y-3)^2 -4=0`
`(x+3)^2 +(y-3)^2 = 4`

This is now a circle with centre `(-3,3)` and radius `2` and can be graphed as such.