jmc

Question

The parabolas defined by the equations y=2x^2-4x+4 and y=-x^2-2x+4 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

2 x^{2} - 4 x + 4

and

- x^{2} - 2 x + 4

, so we have

2 x^{2} - 4 x + 4 = - x^{2} - 2 x + 4

. Combining like terms, we get

3 x^{2} - 2 x = 0

. Factoring out a

x

, we have

x (3 x - 2) = 0

. So either

x = 0

or

3 x - 2 = 0 \Rightarrow x = 2/3

, which are the two

x

coordinates of the points of intersection. Thus,

c = 2/3

and

a = 0

, and

c - a = \frac{2}{3}

.