College Algebra, Chapter 2, 2.2, Section 2.2, Problem 70

Show that the equation $x^2 + y^2 + 2x + y + 1 = 0$ represents a circle. Find the center and radius of the circle.


$
\begin{equation}
\begin{aligned}

x^2 + y^2 + 2x + y + 1 =& 0
&& \text{Model}
\\
\\
x^2 + y^2 + 2x + y =& -1
&& \text{Subtract } 1
\\
\\
(x^2 + 2x + \underline{ }) + (y^2 + y + \underline{ }) =& -1
&& \text{Group terms}
\\
\\
(x^2 + 2x + 1) + \left( y^2 + y \frac{1}{4} \right) =& -1 + 1 + \frac{1}{4}
&& \text{Complete the square: add } \left( \frac{2}{2} \right)^2 = 1 \text{ and } \left( \frac{1}{2} \right)^2 = \frac{1}{4}
\\
\\
(x + 1)^2 + \left(y + \frac{1}{2} \right)^2 =& \frac{1}{4}
&& \text{Perfect Square}

\end{aligned}
\end{equation}
$


Recall that the general equation for the circle with
circle $(h,k)$ and radius $r$ is..

$(x - h)^2 + (y - k)^2 = r^2$

By observation,

The center is at $\displaystyle \left( -1, \frac{-1}{2} \right)$ and the radius is $\displaystyle \sqrt{\frac{1}{4}} = \frac{1}{2}$.