Continued Fractions - Exact Solution

[galenSchooled]

I'm dealing with the fraction:
1+1
1+1/(1+1)
1+1/(1+1/(1+1))
...

After viewing another similar forum I found that they came up with the equation

(k+or-sqar(k^2+4))/2

Which here is
(1+or-sqar(5))/2

My question is how did they derive that equation?
I need to show proof of that equation working using the fraction...

 

[CompuChip]

Call the result x, that is
$$x = \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \cdots}}}$$
Then you recognize x again in the right hand side:
$$x = \frac{1}{1 + x}$$
Now solve this for x (it's just a quadratic equation).