The parabolas defined by the equations y = -x^2 - x + 1 and y = x^2 - 3x - 3 intersect at points (a,b) and (c,d), where c >= a. What is c - a? Express your answer as a common fraction.
Set the equations equal
x^2 - 3x - 3 = -x^2 - x + 1 rearrange as
2x^2 - 2x - 4 = 0 divide through by 2
x^2 - x - 2 = 0
(x - 2) ( x +1) = 0
x - 2 = 0 x + 1 = 0
x = 2 = c x = -1 = a
c - a = 2 - (- 1 ) = 3