Using the following equation, find the center and radius: \begin{tabular}{l}\( x^{2}+2 x+y^{2}+4 y=20 \) \\ The center is located at \( (1,2) \), and the radius is 25 . \\ The center is located at \( (-1,2) \), and the radius is 25 . \\ The center is located at \( (-1,-2) \), and the radius is located at 5 . \( (1,2) \), and the radius is 5 . \\ \hline\end{tabular}

To solve the equation \( x^{2}+2 x+y^{2}+4 y=20 \), we can start by reorganizing it in standard form. Completing the square for both \( x \) and \( y \): 1. For \( x^{2} + 2x \): \[ x^{2} + 2x = (x+1)^{2} - 1 \] 2. For \( y^{2} + 4y \): \[ y^{2} + 4y = (y+2)^{2} - 4 \] Substituting these into the original equation: \[ (x+1)^{2} - 1 + (y+2)^{2} - 4 = 20 \] This simplifies to: \[ (x+1)^{2} + (y+2)^{2} = 25 \] From this equation, we can see that the center is at \((-1, -2)\) and the radius is \(5\) (since \( \sqrt{25} = 5\)). So the correct option is: The center is located at \( (-1,-2) \), and the radius is \( 5 \). --- Did you know that the process of completing the square has been used for centuries in mathematics? Ancient mathematicians discovered that it was a powerful way to derive the properties of conic sections! This method not only helps in transforming equations but also paves the way for understanding the geometry of circles, ellipses, and parabolas. Also, once you discover how to convert equations into standard form, you open doors to geometric insights! The equation of a circle might look complex at first glance, but recognizing it reveals how circles function in real life—from designing roundabouts for traffic flow to creating circular parks for family fun days. Understanding these equations also enhances your ability to tackle problems in physics, engineering, and computer graphics!