Question

How do you simplify `(x-1)/(sqrt x - x)`?

Answer 1

To start lets get rid of the radical sign using substitution. This isn't especially important to the solution, but it does make it easier to recognize some of the identities we will be using.

Let `u=sqrt(x)`.

Substituting `u` into the funtion we get:

`(u^2-1)/(u-u^2)`

We can factor the top and bottom to try and find a term that cancels.

`((u-1)(u+1))/(u(1-u))`

If we multiply the denominator by `(-1)^2` we can find a term to cancel with the denominator.

`((u-1)(u+1))/((-1)^2u(1-u))`

Split up the `(-1)`s and move one out in front of the entire function and the other through the parenthesis in the denominator.

`-1((u-1)(u+1))/(u(-1(1-u)))`

`-((u-1)(u+1))/(u(u-1))`

Now we can cancel the two `(u-1)` terms.

`-(u+1)/u`

Undoing the substitution from before we have our simplified function.

`-(sqrtx +1)/sqrtx`