Please answer fast!!!!!! The sum of the reciprocals of two consecutive even integers is 9/40. This can be represented by the eq

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.

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Thanh Thu 4 years 2021-07-25T20:14:54+00:00 1 Answers 104 views 0

Answers ( )

  1. mit
    0
    2021-07-25T20:16:23+00:00
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    Answer:

    8 and 10

    Step-by-step explanation:

    Given

    \frac{1}{x} + \frac{1}{x+2} = \frac{9}{40} ← combine the fractions on the left side

    \frac{x+2+x}{x(x+2)} = \frac{9}{40}

    \frac{2x+2}{x^2+2x} = \frac{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 = – \frac{10}{9}

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

    The integers are 8 and 10

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