Agolden rectangle has sides of length 1 and x. when a square with side length 1 is removed from the golden rectangle, the remaining rectangle has the same proportions as the original. solve for x.

The original dimensions are is 1 by x
When you take away a square with side length 1, the new dimensions are 0 by x-1.
The only case where 1 by x and 0 by x-1 are proportional are when x=1, since 1 by 1 is proportionate to 0 by 0