Complete the square and write the given equation in standard form. Then give the center and radius of the circle and graph the equation. \( \mathrm{x}^{2}+\mathrm{y}^{2}+4 \mathrm{x}-2 \mathrm{y}-20=0 \) The equation of the circle in standard form is (Simplify your answer.) \begin{tabular}{l} The center of the circle is \\ (Type an ordered pair.) \\ The radius of the circle is \( \mathrm{r}=\square \). \\ Use the graphing tool to graph the circle. \\ \hline Click to \\ enlarge \\ graph \end{tabular}

To complete the square and write the given equation in standard form, we start with the equation: \[ x^2 + y^2 + 4x - 2y - 20 = 0 \] First, we rearrange the equation to group the \(x\) and \(y\) terms: \[ x^2 + 4x + y^2 - 2y = 20 \] Next, we complete the square for the \(x\) terms and the \(y\) terms. ### Completing the square for \(x\): 1. Take the coefficient of \(x\) (which is 4), divide it by 2 to get 2, and then square it to get \(2^2 = 4\). 2. Add and subtract 4 inside the equation. ### Completing the square for \(y\): 1. Take the coefficient of \(y\) (which is -2), divide it by 2 to get -1, and then square it to get \((-1)^2 = 1\). 2. Add and subtract 1 inside the equation. Now we rewrite the equation: \[ (x^2 + 4x + 4) + (y^2 - 2y + 1) = 20 + 4 + 1 \] This simplifies to: \[ (x + 2)^2 + (y - 1)^2 = 25 \] ### Standard Form The equation of the circle in standard form is: \[ (x + 2)^2 + (y - 1)^2 = 5^2 \] ### Center and Radius From the standard form \((x - h)^2 + (y - k)^2 = r^2\), we can identify: - The center \((h, k)\) is \((-2, 1)\). - The radius \(r\) is \(5\). ### Summary - The center of the circle is \((-2, 1)\). - The radius of the circle is \(r = 5\). ### Graphing the Circle To graph the circle, plot the center at \((-2, 1)\) and draw a circle with a radius of 5 units around this point. Here’s a summary of the results: \[ \text{The center of the circle is } (-2, 1). \] \[ \text{The radius of the circle is } r = 5. \]