algebra problem, Solve the equation

Question

a nice problem: Solve the equation $$\left|2x-57-2\sqrt{x-55}+\frac{1}{x-54-2\sqrt{x-55}}\right|=|x-1|.$$

It's just for sharing a new ideas, thanks:)

Answer 1

The left hand side of expression, by completing the square, is: \begin{align*} & \left|(\sqrt{x - 55})^2 - 2\sqrt{x - 55} + 1 + x - 3 + \frac{1}{(\sqrt{x - 55})^2 - 2\sqrt{x - 55} + 1}\right| \\ = & \left|(\sqrt{x - 55} - 1)^2 + \frac{1}{(\sqrt{x - 55} - 1)^2} + x - 3\right|. \end{align*} Since $x \geq 55$, the expression inside the absolute symbol is positive, so is the one in the right hand side, so the equation can be written as \begin{align*} (\sqrt{x - 55} - 1)^2 + \frac{1}{(\sqrt{x - 55} - 1)^2} + x - 3 & = x - 1 \\ (\sqrt{x - 55} - 1)^2 + \frac{1}{(\sqrt{x - 55} - 1)^2} & = 2 \tag{1} \end{align*} But by the algebraic-geometric inequality, the left hand side of $(1)$ is at least $2$, and the equality holds if and only if $$(\sqrt{x - 55} - 1)^2 = 1$$ which gives $x = 55$ and $x = 59$.

Answer 2

Now things become easy. Rewriting the LHS in the form

$\begin{align*} &(x-1) + (\sqrt {x-55}-1)^2+\frac{1}{ (\sqrt {x-55}-1)^2}-2\\ &= (x-1)\\ &\;\; +\left((\sqrt {x-55}-1)-\frac{1}{(\sqrt {x-55}-1)}\right)^2 \end{align*}$

So the solution is $x=59$ and $x=55$.