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