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: \[ x^2 + y^2 + 4x - 2y - 20 = 0 \] Rearranging, we find: \[ x^2 + 4x + y^2 - 2y = 20 \] Now, we complete the square for both \(x\) and \(y\). 1. For \(x^2 + 4x\): - Take half of the coefficient of \(x\): \( \frac{4}{2} = 2\), and square it: \(2^2 = 4\). - This gives us: \(x^2 + 4x + 4 = (x + 2)^2\). 2. For \(y^2 - 2y\): - Take half of the coefficient of \(y\): \( \frac{-2}{2} = -1\), and square it: \(-1^2 = 1\). - This gives us: \(y^2 - 2y + 1 = (y - 1)^2\). Now we rewrite the equation including these squares: \[ (x + 2)^2 + (y - 1)^2 = 20 + 4 + 1 \] \[ (x + 2)^2 + (y - 1)^2 = 25 \] Thus, the equation of the circle in standard form is: \[ (x + 2)^2 + (y - 1)^2 = 25 \] Now we can identify the center and radius: The center of the circle is \((-2, 1)\) and the radius \(r = \sqrt{25} = 5\). To summarize: The center of the circle is \((-2, 1)\). The radius of the circle is \(r = 5\). You can graph this circle centered at \((-2, 1)\) with a radius of \(5\), creating a beautiful round shape on your graph!