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.

c - a =

Step-by-step explanation:

Since the parabolas intersect we can equate them, that is

2x² - 1 = - x² - x + 1 ← subtract terms on right side from terms on left side

3x² + x - 2 = 0

Consider the factors of the coefficient of the x² term and the constant term which sum to give the coefficient of the x- term.

product = 3 × - 2 = - 6 and sum = + 1

The factors are + 3 and - 2

Use these factors to split the x- term

3x² + 3x - 2x - 2 = 0 ( factor the first/second and third/fourth terms )

3x(x + 1) - 2(x + 1) = 0 ← factor out (x + 1) from each term

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

Equate each factor to zero and solve for x

x + 1 = 0 ⇒ x = - 1

3x - 2 = 0 ⇒ 3x = 2 ⇒ x =

Since c > a, then

a = - 1 and c =

Thus

c - a = - (- 1) = + 1 = + =

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