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 your work.

8 and 10

Step-by-step explanation:

Given

+ =

Combine the fractions on the left side

+ =

= ( cross- multiply )

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

80x + 80 = 9x² + 18x ← subtract 80x + 80 from both sides

0 = 9x² -62x - 80 ← in standard form

To factorise the quadratic

Consider the factors of the product 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

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 = -

However, x is an integer ⇒ x = 8

The integers are 8 and 8 + 2 = 10

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