jmc

Question

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 a. What is c-a? Express your answer as a common fraction.

Accepted Answer

The graph of the two parabolas is shown below:

The graphs intersect when

y

equals both

- x^{2} - x + 1

and

2 x^{2} - 1

, so we have

- x^{2} - x + 1 = 2 x^{2} - 1

. Combining like terms, we get

3 x^{2} + x - 2

. Factoring the quadratic we have

(3 x - 2) (x + 1) = 0

. So either

x = 2/3

or

x = - 1

, which are the two

x

coordinates of the points of intersection. Thus,

c = 2/3

and

a = - 1

, giving

c - a = \frac{5}{3}

.