tooo solve the equation x^2 + 6x + y^2 + 8y = 52, we need to complete the square for both the x and y terms.
Starting with the x terms, we add (6/2)^2 = 9 to both sides of the equation to get:
x^2 + 6x + 9 + y^2 + 8y = 61
We can simplify the left side of the equation by factoring the perfect square trinomial:
(x + 3)^2 + y^2 + 8y = 61 + 9
Next, we complete the square for the y terms by adding (8/2)^2 = 16 to both sides of the equation:
(x + 3)^2 + y^2 + 8y + 16 = 86
We can simplify the left side of the equation by factoring the perfect square trinomial:
(x + 3)^2 + (y + 4)^2 = 86
This is the equation of a circle with center (-3,-4) and radius sqrt(86)