The sum of the reciprocals of two consecutive even integers is 9/40. This can be represented by the equation shown. 1/x + 1/(x+2) = 9/40. Use the rational equation to determine the integers.

Answer 1

To solve the equation 1/x + 1/(x+2) = 9/40, we can first find a common denominator and combine the fractions on the left side:

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

Multiplying both sides by x(x+2) to clear the fractions:

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

Expanding both sides:

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

Combining like terms:

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

Rearranging to set the equation equal to zero:

9x^2 + 18x - 80x - 80 = 0

9x^2 - 62x - 80 = 0

This is a quadratic equation that can be factored:

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

Setting each factor to zero:

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

Since we're dealing with even integers, the possible values for x are x = 10. Therefore, the two consecutive even integers are 10 and 12.