# How do you graph 4x^2 + 4y^2 + 24x - 32y + 51?

Nov 9, 2015

You need to complete the square for both the x and y variables

#### Explanation:

$4 {x}^{2} + 4 {y}^{2} + 24 x - 32 y + 51 = 0$

$4 \left({x}^{2} + 6 x\right) + 4 \left({y}^{2} - 8 y\right) = - 51$

Now, complete the squares remembering to keep the equation balanced :

$4 \left({x}^{2} + 6 x + 9\right) + 4 \left({y}^{2} - 8 y + 16\right) = - 51 + \left(4\right) \left(9\right) + \left(4\right) \left(16\right)$

$4 {\left(x + 3\right)}^{2} + 4 {\left(y - 4\right)}^{2} = 49$

${\left(x + 3\right)}^{2} + {\left(y - 4\right)}^{2} = \frac{49}{4}$

So, this is a circle with:

Center $= \left(- 3 , 4\right)$
Radius $= \sqrt{\frac{49}{4}} = 3.5$

Hope that helped