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**topsquark** Start with $\displaystyle \sqrt{x^2-2x+1}+\sqrt{x^2+2x+1}=2$.

As you noticed, this is:

$\displaystyle \sqrt{(x-1)^2}+\sqrt{(x+1)^2}=2$.

But when we take the square root, we get $\displaystyle \pm$ the answer, so:

$\displaystyle \pm (x-1) + \pm (x+1) = 2$, where the $\displaystyle \pm$ signs are independant.

Using the "+ +" version we get x=1.

Using the "+ -" version we get -2=2. Nonsense.

Using the "- +" version we get 2=2. Nothing new.

Using the "- -" version we get x=-1.

I admit that I'm baffled by the x=0 solution. I tried squaring twice to get rid of the radicals (the most general method I know of) and I get $\displaystyle (x+1)^2=(x+1)^2$ (unless I made a mistake) which doesn't help at all.

-Dan