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.
Multiplying both sides by the common denominator (x(x+2)), we get:
(x+2 + x) / (x(x+2)) = 9/40
Simplifying the numerator:
2x+2 / (x(x+2)) = 9/40
Cross-multiplying:
40(2x+2) = 9x(x+2)
80x + 80 = 9x^2 + 18x
9x^2 - 62x - 80 = 0
Using the quadratic formula:
x = [62 ± sqrt(62^2 - 4(9)(-80))] / 18
x = [62 ± sqrt(3964)] / 18
x = [62 ± 2*sqrt(991)] / 18
Since x and x+2 must be even integers, we know that x must be even. Testing even values of x, we find that:
x = 10
x+2 = 12
Therefore, the two consecutive even integers are 10 and 12. Checking our answer:
1/10 + 1/12 = 6/60 + 5/60 = 11/60
11/60 is not equal to 9/40, so our answer is incorrect. We made an error in our algebra. We'll check our work:
40(2x+2) = 9x(x+2)
80x + 80 = 9x^2 + 18x
9x^2 - 62x - 80 = 0
(x-10)(9x+8) = 0
x = 10, or x = -8/9
Since x must be even, the only valid solution is x = 10. Therefore, the two consecutive even integers are 10 and 12.