x=1/(x-5) implies x^2-5x=1 for x≠5.
x^2-5x+25/4=1+25/4, (x-5/2)^2=29/4, x-5/2=±√29/2=±2.6926 approx, x=2.5±2.6926, so x=5.1926 and -0.1926.
Let's look at x^2-1/x^2. We can substitute for 1/x^2: it's the same as (x-5)^2 so we calculate x^2-(x-5)^2=(x-x+5)(x+x-5)=5(2x-5). But x-5/2=±√29/2, so 2x-5=±√29. Therefore, x^2-1/x^2=±5√29=±26.9258 approx.