Find all real values of x for which
. . . . .\(\displaystyle \dfrac{1}{\sqrt{\strut x\,}\,+\,\sqrt{\strut x\,-\,2\,}}\,+\,\dfrac{1}{\sqrt{\strut x\,}\,+\,\sqrt{\strut x\,+\, 2\,}}\,=\,\dfrac{1}{4}\)
I tried rationalizing and expanding everything, but it becomes too complicated and takes a lot of time. The question was meant to be done in 10-15 minutes.
How
much time does it take for you? (This time estimate may be based on the assumption that you've learned the conjugation trick to rationalization, so that the process does
not take terribly long. It may be that you're doing everything correctly; you just need to practice more.)
On the left-hand side, you started with this:
. . . . .\(\displaystyle \dfrac{1}{\sqrt{\strut x\,}\,+\,\sqrt{\strut x\,-\,2\,}}\,+\,\dfrac{1}{\sqrt{\strut x\,}\,+\,\sqrt{\strut x\,+\, 2\,}}\)
After adequate practice, you wrote down the next step as:
. . . . .\(\displaystyle \dfrac{\sqrt{\strut x\,}\, -\, \sqrt{\strut x\, -\, 2\, }}{(x)\, -\, (x\, -\,2)}\, +\, \dfrac{\sqrt{\strut x\,}\, -\, \sqrt{\strut x\, +\, 2\, }}{(x)\, -\, (x\, +\, 2)}\)
You simplified as:
. . . . .\(\displaystyle \dfrac{\sqrt{\strut x\,}\, -\, \sqrt{\strut x\, -\, 2\, }}{2}\, +\, \dfrac{\sqrt{\strut x\,}\, -\, \sqrt{\strut x\, +\, 2\, }}{-2}\)
You multiplied through by "2", and... then what?
(By the way, the above took me about five minutes to type-set. Obviously, it would take
much less time, writing by hand.)
