The sum of the reciprocals of two consecutive even integers is 940.

Question

This can be represented by the equation shown.

1x+1x+2=940

Use the rational equation to determine the integers. Show all work.

Answer 1

1/x + 1/(x+2) = 940

To solve for x, we can first find a common denominator:

(x+2)/(x(x+2)) + x/(x(x+2)) = 940

Combine the fractions:

(x+2 + x)/(x(x+2)) = 940

2x + 2 = 940x(x+2)

Expand the equation:

2x + 2 = 940x^2 + 1880x

Rearrange the equation to set it equal to 0:

940x^2 + 1878x - 2 = 0

Now we can use the quadratic formula to solve for x:

x = (-b ± √(b^2 - 4ac))/(2a)

a = 940, b = 1878, c = -2

x = (-1878 ± √(1878^2 - 4*940*(-2)))/(2*940)
x = (-1878 ± √(3521284 + 7520))/1880
x = (-1878 ± √3528804)/1880
x = (-1878 ± 1876.612)/1880

There are two potential solutions for x:

x1 = (-1878 + 1876.612)/1880 = -0.752
x2 = (-1878 - 1876.612)/1880 = -3.752

Since x represents an even integer, we can discard the negative solution. Therefore, x = -0.752 is not valid.

Thus, the only valid solution is x = -3.752.

Now we can find the consecutive even integers:

If x = -4:
The two consecutive even integers are -4 and -2.