The parabolas defined by the equations $y=-x^2-x+1$ and $y=2x^2-1$ intersect at points $(a,b)$ and $(c,d)$, where $c\ge a$. What is $c-a$? Express your answer as a common fraction.
To find the points of intersection of the two parabolas, I need to set their $y$-coordinates equal to each other and solve for $x$. So I have $-x^2-x+1=2x^2-1$, which simplifies to $3x^2+x-2=0$. I can factor this quadratic equation into $(3x-2)(x+1)=0$, so the solutions are $x=\frac{2}{3}$ and $x=-1$. These are the $x$-coordinates of the points of intersection, so $a=-1$ and $c=\frac{2}{3}$. Therefore, $c-a=\frac{2}{3}-(-1)=\frac{5}{3}$. 