The parabolas defined by the equations y = -x^2 - x + 1 and y = 2x^2 - 3x 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 =
-x^2 -x + 1 = 2x^2 -3x rearrange as
3x^2 - 2x - 1 = 0 factor as
(3x + 1) ( x - 1) = 0
Setting both factors to 0 and solving for x produces
x = a = -1/3 x = c = 1
So
c - a =
1 - ( -1/3) =
4 / 3
