The center of the circle is point: $ C = \left(6, -9\right) $.
The radius of the circle is $ r = 11 $.
Step 1: Group x and y terms together, move the number to the right side.
$$ x^2 - 12 x + y^2 + 18 y = 4 $$Step 2: Write the equation in the following form.
$$ \left( x^2 - 12 x + \color{blue}{\spadesuit} \right) + \left( y^2 + 18 y + \color{red}{\clubsuit} \right) = 4 + \color{blue}{\spadesuit} + \color{red}{\clubsuit} $$( the symbols $ \color{blue}{\spadesuit} $ and $ \color{red}{\clubsuit} $ , are the number needed to complete the square )
Step 3: To find $\color{blue}{\spadesuit}$ , take the x - term, divide it by 2 an then square the result. In this example we have:
$$ \color{blue}{\spadesuit} = \left( \frac{ -12 }{2} \right)^2 = 36 $$Step 4: To find $\color{red}{\clubsuit}$ , take the y - term, divide it by 2 an then square the result.
$$ \color{red}{\clubsuit} = \left( \frac{ 18 }{2} \right)^2 = 81 $$Step 5: Put steps 2, 3 and 4 together.
$$ \begin{aligned} \left( x^2 - 12 x + \color{blue}{ 36 } \right) + \left( y^2 + 18 y + \color{red}{ 81 }\right) &= 4 + \color{blue}{ 36 } + \color{red}{ 81 } \\ \left( x^2 - 12 x + 36 \right) + \left( y^2 + 18 y + 81 \right) &= 121 \\ \left( x -6 \right)^2 + \left( y + 9 \right)^2 &= 121 \end{aligned}$$Step 6: Read off the answer from the last equation. (note that $ r^2 = 121 $ so $ r = \sqrt{ 121 } = 11 $).