how to solve this:

a/a-1 - a-1/a = 4

multiply both sides by a(a-1)

a^2(a-1)/(a-1) - (a(a-1)(a-1))/a=4a(a-1)

or
a^2-(a-1)^2=4a(a-1)

Now you expand it, collect terms, and solve as a quadratic.