Find all real numbers $x$ such that $\sqrt{x+2\sqrt{x}-1}+\sqrt{x-2\sqrt{x}-1}$ is a real number

Question

I want to find all values of $x\in \mathbb R$ such that the value of $\sqrt{x+2\sqrt{x}-1}+\sqrt{x-2\sqrt{x}-1}$ is a real number.

I solved it as follows:

$x+2\sqrt{x}-1\ge 0$

$(\sqrt{x}+1)^2-2\ge 0$

$(\sqrt{x}+1)^2\ge 2$

$\sqrt{x}+1\ge 2$ or $\sqrt{x}+1\le -\sqrt{2}$

The second can't hold, from the first $x\ge 3-2\sqrt{2}$.

Similarly $(\sqrt{x}-1)^2\ge 2$, hence $x\ge 3+2\sqrt{2}$.

I find my solution to be very ugly. Is my solution correct and is there a neater approach?

Answer 1

The requirement $x\ge0$ is obvious. The expression is real iff $(\sqrt{x}\pm1)^2\ge2$ for both choices of $\pm$, i.e. neither $\sqrt{x}\pm1$ is in $(-\sqrt{2},\,\sqrt{2})$, i.e .$\sqrt{x}\notin[0,\,1+\sqrt{2})\cup[0,\,\sqrt{2}-1)=[0,\,1+\sqrt{2})$, since $\sqrt{x}\ge0$. The solution set is $[3+2\sqrt{2},\,\infty)$, as in @alex.jordan's answer.

Answer 2

Let's use Mathematica for an overview plot:

ReImPlot[Sqrt[x + 2 Sqrt[x] - 1] + Sqrt[x - 2 Sqrt[x] - 1],
         {x, -1, 7}, PlotPoints -> 10^3, 
         GridLines -> {{0, 3 - 2 Sqrt[2], 3 + 2 Sqrt[2]},
                       {0, 2 Sqrt[Sqrt[2] - 1], 2, 2 Sqrt[1 + Sqrt[2]]}}, 
         PlotTheme -> "Scientific"]

• For $x<0$ the function is purely real.
• For $0\le x\le 3-2\sqrt{2}$ the function is purely imaginary.
• For $3-2\sqrt{2}<x<3+2\sqrt{2}$ the function has nonvanishing real and imaginary parts.
• For $x\ge3+2\sqrt{2}$ the function is purely real.