Proved as follows:
x- 1/x=a- 1/a
Multiply both sides of the equation by ax at the same time and you get
Axe? -a=a? x-x
Axe? +( 1-a? )x-a=0
(ax+ 1)(x-a)=0
therefore
Ax+ 1=0, the solution is: x=- 1/a,
X-a=0, the solution is: x=a,
Therefore, any fractional equation of the form: x- 1/x=a- 1/a has two roots: x=a and x =-1/a.
Similarly, for a fractional equation in the form of x+ 1/x=a+ 1/a, its two roots are: x=a and x =1/a.
It can also be extended to the fractional equation in the form of x+k/x=a+k/a, and its solution is: x=a, x = k/a.