SOLUTION: find two consecutive positive odd numbers given that the difference between their reciprocals is 2 divided by 63

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Question 657932: find two consecutive positive odd numbers given that the difference between their reciprocals is 2 divided by 63
Answer by kevwill(135) About Me  (Show Source):
You can put this solution on YOUR website!
Let the two numbers be x and x+2. Then
1%2Fx+-+1%2F%28x%2B2%29+=+2%2F63
%28x%2B2%29%2F%28x%2B2%29%2A1%2Fx+-+%28x%2Fx%29%2A%281%2F%28x%2B2%29%29+=+2%2F63
%28x%2B2%29%2F%28x%5E2%2B2x%29+-+x%2F%28x%5E2%2B2x%29+=+2%2F63
%28%28x%2B2%29-x%29%2F%28x%5E2%2B2x%29+=+2%2F63
2%2F%28x%5E2%2B2x%29+=+2%2F63
Cross-multiplying:
2%2A%28x%5E2%2B2x%29+=+2%2A63
Dividing by 2:
x%5E2+%2B+2x+=+63
x%5E2+%2B2x+-+63+=+0
We can solve for x using the quadratic equation x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+ with a=2, b=2, and c=-63
x+=+%28-2+%2B-+sqrt%28+2%5E2-4%2A1%2A%28-63%29+%29%29%2F%282%2A1%29+
x+=+%28-2+%2B-+sqrt%28+4-%28-252%29+%29%29%2F2+
x+=+%28-2+%2B-+sqrt%28256%29+%29%2F2+
x+=+%28-2+%2B-+16+%29%2F2+
And since we're looking for positive numbers:
x+=+%28-2+%2B+16+%29%29%2F2+
x+=+14%2F2+=+7+
So the two numbers are 7 and 9.