is a delightful little rational equation, but unfortunately it has nothing whatever to do with the golden ratio. Close, but no cigar I'm afraid.
The golden ratio is generally defined in terms of a golden section, that is a line segment divided into two parts a and b such that the following proportion holds:
And that is the golden ratio. The above has the unique positive solution for whereas your equation has the unique positive solution -- a very different number indeed.
However, I did say you were close. You only missed it by a sign. Your equation should have been:
Where L is the golden ratio. is actually the reciprocal of the golden ratio, often named
Proof:
Exclude the negative root.
But:
Again, exclude the negative root.
But how do I know that is the correct value for the golden ratio? Back to the original definition:
