solve for x by completing the square?

Question

x^2 - x - c^2 - c = 0

Answer 1

add c^2 and c for both sides

x^2 - x = c^2 + c

use (b/2)^2 and what you add on the left, you must add on the right.

x^2 - x + (1/2)^2 = c^2 + c + (1/2)^2

x^2 - x + 1/4 = c^2 + c + 1/4

factor
(x - 1/2)^2 = (c + 1/2)^2

take a square root
x - 1/2 = c + 1/2

x = c + 1

there are two posible answer.
x - 1/2 = -(c + 1/2)
x - 1/2 = -c - 1/2
x = -c

so the two asnwers are:
x = c + 1 and x = -c

Answer 2

x^2 - x - c^2 - c = 0
x^2 - x + (1/2)^2 - (c^2 + c + (1/2)^2) = 0
(x - 1/2)^2 - (c + 1/2)^2
(x - 1/2 + c + 1/2)(x - 1/2 - c - 1/2) = 0
x = - c, c + 1/2

Answer 3

x^2 - x - c^2 - c = 0

factor c
x^2 - x - c(c+1) = 0

factor out
(x + c)(x - (c+1)) = 0

first value of x is
x = -c

second value of x is
x = c +1

Answer 4

x = (1+- sqrt(1+4c^2 +4c)) /2