Determine root of equation square root(x+2square root(x-1))+root(x-2square root(x-1))=square root 2?

Expert Answers
embizze eNotes educator| Certified Educator


Square both sides -- note that the left side is a binomial and will have 4 terms; also squaring might introduce extraneous solutions so we must check any answer to see that it is a solution in the original problem.

`x+2sqrt(x-1)+2sqrt((x+2sqrt(x-1))(x-2sqrt(x-1)))+x-2sqrt(x-1)=2` Add like terms and simplify the radicand:


Divide both sides by 2 and rewrite the radicand as a perfect square:




If x>2 then x-2=1-x ==> 2x=3 ==> `x=3/2`

If x<2 then -x+2=1-x ==> 2=1 which is impossible.

The only possible solution is `x=3/2` .

Check in the original equation:




=2 not `sqrt(2)`

So there is no solution.

The graphs -- note that they do not intersect: