4x^2 - x - 3 = 0
x^2 - (1/4)x - 3/4 = 0
There's a -1/4 on the x in the middle. Half of -1/4 is -1/8. -1/8 squared is 1/64.
x^2 - (1/4)x + 1/64 - 1/64 - 3/4 = 0
(x - 1/8)^2 - 1/64 - 3/4 = 0
(x - 1/8)^2 - 1/64 - 48/64 = 0
(x - 1/8)^2 - 49/64 = 0
(x - 1/8)^2 = 49/64
x - 1/8 = +- sqrt(49/64)
x - 1/8 = +- 7/8
x = +- 7/8 + 1/8
x = -6/8 or 8/8
x = -3/4 or x = 1