SOLUTION: Find two positive real numbers that differ by 1 and have a product of 1.

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Question 42036: Find two positive real numbers that differ by 1 and have a product of 1.
Answer by longjonsilver(2297) About Me  (Show Source):
You can put this solution on YOUR website!
Let 2 numbers be x and y

We are told:
x-y=1

and
xy=1

Right then:
xy=1
means that x=1/y

so we then get from x-y=1 that
(1/y)-y=1
--> +1-y%5E2+=+y+
+1+-+y%5E2+-+y+=+0+

and re-arranging, gives +y%5E2+%2B+y+-+1+=+0+

Not easy to factorise, so use the quadratic formula:
+y+=+%28-b+%2B-+sqrt%28b%5E2+-+4ac%29%29%2F%282a%29+

+y+=+%28-%281%29+%2B-+sqrt%28%281%29%5E2+-+4%281%29%28-1%29%29%29%2F%282%281%29%29+

+y+=+%28-1+%2B-+sqrt%281%2B4%29%29%2F%282%29+

+y+=+%28-1+%2B-+sqrt%285%29%29%2F%282%29+

Remember, we want positive answers, so of the 2 answers quoted above, only +y+=+%28-1+%2B+sqrt%285%29%29%2F%282%29+ is acceptable.

Now, if x-y=1 then x = 1+y. So,

+x+=+%28-1+%2B+sqrt%285%29%29%2F%282%29+%2B+1+
+x+=+%28-1+%2B+sqrt%285%29%29%2F%282%29+%2B+2%2F2+
+x+=+%28-1+%2B+2+%2B+sqrt%285%29%29%2F%282%29+
+x+=+%281+%2B+sqrt%285%29%29%2F%282%29+

jon.