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 3 of 4: Split at the absolute value
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12≤x≤1: 2=A2;x>1: 4x−2=A2\frac12\le x\le1:\ 2=A^2;\qquad x>1:\ 4x-2=A^2
Detailed analysis

For 12≤x≤1\frac12\le x\le1, ∣x−1∣=1−x|x-1|=1-x, so the left side is 22. For x>1x>1, ∣x−1∣=x−1|x-1|=x-1, so the left side is 4x−24x-2.