SOLUTION: The difference of the reciprocals of two consecutive even integers is 1/24. Find the integers. Thanks -tom

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Question 49954: The difference of the reciprocals of two consecutive even integers is 1/24. Find the integers. Thanks
-tom
Answer by Earlsdon(6103) About Me  (Show Source):
You can put this solution on YOUR website!
Let the two consecutive integers be a and a+2.
You can write:
1%2Fa+-+1%2F%28a%2B2%29+=+1%2F24 Simplify the left side.
%282%29%2F%28a%5E2%2B2a%29+=+1%2F24 Now cross-multiply.
48+=+a%5E2%2B2a Subtract 48 from both sides of the equation.
a%5E2%2B2a-48+=+0 Factor this.
%28a-6%29%28a%2B8%29+=+0 Apply the zero product principle.
a-6+=+0 and/or a%2B8+=+0
If a-6+=+0 then a+=+6
If a%2B8+=+0 then a+=+-8
Since you did not restrict the integers to positive integers only, there are two answers:
6 and 8
-8 and -6
Check the positive integers first:
1%2F6+-+1%2F8+=+1%2F24 Simplify.
%288-6%29%2F48+=+1%2F24
2%2F48+=+1%2F24
1%2F24+=+1%2F24 This checks!
Now the negative integers:
1%2F%28-8%29+-+1%2F%28-6%29+=+1%2F24 Simplify.
%28-6-%28-8%29%29%2F48+=+1%2F24
2%2F48+=+1%2F24
1%2F24+=+1%2F24 This also checks!