Question

Please answer fast!!!!!!

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. Show all work.

Answer 1

Answers: Integers are x=8, x=10.

Answer 2

8 and 10

Explanation:

Given

`(1)/(x)` +
`(1)/(x+2)` =
`(9)/(40)` ← combine the fractions on the left side

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

`(2x+2)/(x^2+2x)` =
`(9)/(40)` ( cross- multiply )

9(x² + 2x) = 40(2x + 2) ← distribute both sides

9x² + 18x = 80x + 80 ( subtract 80x + 80 from both sides )

9x² - 62x - 80 = 0

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

product = 9 × - 80 = - 720 and sum = - 62

The factors are - 72 and + 10

Use these factors to split the x- term

9x² - 72x + 10x - 80 = 0 ( factor the first/second and third/fourth terms )

9x(x - 8) + 10(x - 8) = 0 ← factor out (x - 8) from each term

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

Equate each factor to zero and solve for x

x - 8 = 0 ⇒ x = 8

9x + 10 = 0 ⇒ 9x = - 10 ⇒ x = -
`(10)/(9)`

x is an integer , then x = 8 and x + 2 = 8 + 2 = 10

The integers are 8 and 10