Find the center and radius of the circle represented by the equation below. \[ x^{2}+y^{2}-2 x+2 y-23=0 \] Answer Attempt 1 out of 2 Center: Radius: Submit Answer

To rewrite the equation in standard form, we first group the \(x\) and \(y\) terms: \[ (x^2 - 2x) + (y^2 + 2y) = 23. \] Next, we complete the square for each variable. For \(x^2 - 2x\), we add and subtract 1, transforming it into \((x - 1)^2 - 1\). For \(y^2 + 2y\), we add and subtract 1, turning it into \((y + 1)^2 - 1\). Putting this all together, we get: \[ (x - 1)^2 - 1 + (y + 1)^2 - 1 = 23, \] which simplifies to: \[ (x - 1)^2 + (y + 1)^2 = 25. \] From this, we identify the center of the circle as \((1, -1)\) and the radius as \(5\) (since \(5^2 = 25\)). Center: (1, -1) Radius: 5