Simplify $A=\frac{y^\frac12}{y^\frac12-2}+\frac{y^\frac12}{y^\frac12+2}-2$
Simplify $$A=\dfrac{y^\frac12}{y^\frac12-2}+\dfrac{y^\frac12}{y^\frac12+2}-2$$ So $$A=\dfrac{y^\frac12\left(y^\frac12+2\right)+y^\frac12\left(y^\frac12-2\right)}{y-4}-2=\dfrac{2y^\frac14}{y-4}-2=\dfrac{2y^\frac14-2y+8}{y-4}$$ Can we simplify further?
Solution 1:
Substitute $t=y^\frac12$ and you get: $$\frac{t}{t-2}+\frac{t}{t+2}-2 =$$ $$=\frac{t(t+2)+t(t-2)}{(t-2)(t+2)}-2=$$ $$=\frac{t^2+2t+t^2-2t}{t^2-4}-2=$$ $$=\frac{2t^2-2t^2+8}{t^2-4}=$$ $$=\frac{8}{t^2-4}$$ Substitute back and you get: $$\frac{8}{y-4}$$