What are the possible solution sets of the equation x2 + y2 + cx + dy + e = 0?

Hello:

x² + y² + cx + dy + e = 0

(x²+cx) + (y²+dy) + e = 0

(x² + c/2 x + (c/2) ² - (c/2) ²) + y² + d/2y + (d/2) ² - (d/2) ²) + e = 0

(x+c/2) ² + (y + d2) ² - (c/2) ² - (d/2) ² + e = 0

(x+c/2) ² + (y + d/2) ² = (c/2) ² + (d/2) ² - e

case 1:

(c/2) ² + (d/2) ² - e < 0 : no solutions because : (x+c/2) ² + (y + d/2) ² ≥ 0

case 2:

(c/2) ² + (d/2) ² - e = 0 : (x+c/2) ² + (y + d/2) ² = 0 so : one solution (-c/2, - d/2)

case 3:

(c/2) ² + (d/2) ² - e > 0 : infinity solutions