Problem 2

For what real values of

x

\sqrt{x+\sqrt{2x-1}}+\sqrt{x-\sqrt{2x-1}}=A

, given (a)

A=\sqrt2

, (b)

A=1

, (c)

A=2

, where square roots are non-negative?

Step 2 of 4: Square once and simplify

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In plain words

The product of the two outer radicals is $\sqrt{x^2-(2x-1)}=|x-1|$ .

2x+2\sqrt{(x-1)^2}=A^2\Longrightarrow2x+2|x-1|=A^2

Detailed analysis

Squaring the original equation gives $2x+2\sqrt{(x+\sqrt{2x-1})(x-\sqrt{2x-1})}=A^2$ . The product under the remaining radical is $(x-1)^2$ , so the equation becomes $2x+2|x-1|=A^2$ .