x^2 - x - 7 = 0
There's a -1 on the x in the middle. Half of -1 is -1/2. -1/2 squared is 1/4.
x^2 - x + 1/4 - 1/4 - 7 = 0
x^2 - x + 1/4 - 29/4 = 0
x^2 - x + 1/4 = 29/4
(x - 1/2)^2 = 29/4
x - 1/2 = +-sqrt(29/4)
x - 1/2 = +-sqrt(29)/2
x = +-sqrt(29)/2 + 1/2
x = (+-sqrt(29) + 1) / 2
x = (-sqrt(29) + 1) / 2 of x = (sqrt(29) + 1) / 2