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

Feb 9, 2016

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.

$4 {x}^{2} + 24 x + 4 {y}^{2} - 24 y + 56 = 0$
$4 \left({x}^{2} + 6 x + {y}^{2} - 6 y + 14\right) = 0$

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

${\left(x + 3\right)}^{2} - 9 + {\left(y - 3\right)}^{2} - 9 + 14 = 0$
${\left(x + 3\right)}^{2} + {\left(y - 3\right)}^{2} - 4 = 0$
${\left(x + 3\right)}^{2} + {\left(y - 3\right)}^{2} = 4$

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