Final Answer:
The center of the circle with the equation x^2 - 4x + y^2 + 2y = 0 is (2, -1), and the radius is 5.
Step-by-step explanation:
To find the center and radius of the circle, complete the square for both the x and y terms. Start with the given equation:
x^2 - 4x + y^2 + 2y = 0
For the x-terms:
x^2 - 4x + 4 = (x - 2)^2
For the y-terms:
y^2 + 2y + 1 = (y + 1)^2
Now, rewrite the equation:
(x - 2)^2 + (y + 1)^2 = 5^2
This is the standard form of the equation for a circle (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center and r is the radius. Thus, the center is (2, -1), and the radius is 5.