MathLabs

Problem 2

For what real values of xx is x+2x−1+x−2x−1=A\sqrt{x+\sqrt{2x-1}}+\sqrt{x-\sqrt{2x-1}}=A, given (a) A=2A=\sqrt2, (b) A=1A=1, (c) A=2A=2, where square roots are non-negative?
Step 4 of 4: Evaluate the three cases
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A=2: x∈[12,1];A=1: ∅;A=2: x=32A=\sqrt2:\ x\in[\frac12,1];\quad A=1:\ \varnothing;\quad A=2:\ x=\frac32
Detailed analysis

If A=2A=\sqrt2, the constant branch gives every x∈[12,1]x\in[\frac12,1]. If A=1A=1, neither branch is possible. If A=2A=2, the second branch gives 4x−2=44x-2=4, hence x=32x=\frac32.