To solve the given equation 1/x + 1/(x+2) = 9/40, we need to find the common denominator in order to combine the fractions on the left side of the equation. The common denominator here is x(x+2), so we rewrite the equation as:
(x+2 + x) / (x(x+2)) = 9/40
(2x + 2) / (x^2 + 2x) = 9/40
Now, cross multiply to solve for x:
40(2x + 2) = 9(x^2 + 2x)
80x + 80 = 9x^2 + 18x
9x^2 - 62x - 80 = 0
Now, we can use the quadratic formula to solve for x:
x = (-(-62) ± sqrt((-62)^2 - 4*9*(-80))) / 2*9
x = (62 ± sqrt(62^2 + 2880)) / 18
x = (62 ± sqrt(3844 + 2880)) / 18
x = (62 ± sqrt(6724)) / 18
x = (62 ± 82) / 18
There are two possible solutions for x:
1) x = (62 + 82) / 18 = 144 / 18 = 8
2) x = (62 - 82) / 18 = -20 / 18 = -10/9
Since we are looking for even integers, x = 8 is the integer we are interested in. Therefore, the two consecutive even integers are 8 and 10.
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.
explain how u got both integers and show everything
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