The equation of the circle is (x + 1)² + (y + 2)² = 25.
The center of the circle is ((-1, -2)) and the radius is 5.
How to find the Equation of the Circle?
To rewrite the equation x² + 2x + y² + 4y = 20 in the form ((x - h)² + (y - k)² = r^2) and identify the center and radius of the circle, we need to complete the square for the (x)-terms and (y)-terms separately.
The standard form of the equation of a circle is ((x - h)² + (y - k)² = r), where:
(h, k) is the center of the circle.
r is the radius
Completing the square for the (x)-terms:
Add ((2/2)² = 1) inside the parentheses: (x² + 2x + 1).
Since we added 1, we need to subtract 1 outside the parentheses to keep the equation balanced.
Completing the square for the (y)-terms:
Add ((4/2)² = 4) inside the parentheses: (y² + 4y + 4).
Since we added 4, we need to subtract 4 outside the parentheses.
After completing the square, the equation becomes:
(x + 1)² + (y + 2)² = 25.
Therefore, the center of the circle is ((-1, -2)) and the radius is 5.