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