Step 1: Group x and y terms together, move the number to the right side.

$$ x^2 - 18 x + y^2 - 2 y = -1 $$

Step 2: Write the equation in the following form.

$$ \left( x^2 - 18 x + \color{blue}{\spadesuit} \right) + \left( y^2 - 2 y + \color{red}{\clubsuit} \right) = -1 + \color{blue}{\spadesuit} + \color{red}{\clubsuit} $$

( 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:

$$ \color{blue}{\spadesuit} = \left( \frac{ -18 }{2} \right)^2 = 81 $$

Step 4: To find $\color{red}{\clubsuit}$ , take the y - term, divide it by 2 an then square the result.

$$ \color{red}{\clubsuit} = \left( \frac{ -2 }{2} \right)^2 = 1 $$

Step 5: Put steps 2, 3 and 4 together.

$$ \begin{aligned} \left( x^2 - 18 x + \color{blue}{ 81 } \right) + \left( y^2 - 2 y + \color{red}{ 1 }\right) &= -1 + \color{blue}{ 81 } + \color{red}{ 1 } \\ \left( x^2 - 18 x + 81 \right) + \left( y^2 - 2 y + 1 \right) &= 81 \\ \left( x -9 \right)^2 + \left( y -1 \right)^2 &= 81 \end{aligned}$$

Step 6: Read off the answer from the last equation. (note that $ r^2 = 81 $ so $ r = \sqrt{ 81 } = 9 $).

Circle equation Calculator