The sum of the reciprocals of two consecutive even integers is 9/40. This can be represented by the equation shown.

Question

1/x+1/x+2=9/40
Use the rational equation to determine the integers. Show all work.

Answer 1

1/x + 1/(x + 2) = 9/40

To clear the denominators, we multiply everything by 40x(x + 2), giving:

40(x + 2) + 40x = 9x(x + 2)

Expanding both sides, we get:

40x + 80 + 40x = 9x^2 + 18x

80x + 80 = 9x^2 + 18x

Rearranging the terms, we get a quadratic equation:

9x^2 - 62x - 80 = 0

We can factor this quadratic equation to solve for x:

(9x + 10)(x - 8) = 0

Setting each factor to zero:

9x + 10 = 0 or x - 8 = 0

x = -10/9 or x = 8

Since the integers are consecutive even numbers, we need to choose the positive value of x:

x = 8

So the two consecutive even integers are 8 and 10.