Solving a radical equation $\sqrt{x+1} - \sqrt{x-1} = \sqrt{4x-1}$ $$
\sqrt{x+1} - \sqrt{x-1} = \sqrt{4x-1}
$$
How many solutions does it have for $x \in \mathbb{R}$?
I squared it once, then rearranges terms to isolate the radical, then squared again.
I got a linear equation, which yielded $x = \frac54$, but when I put that back in the equation, it did not satisfy.
So I think there is no solution, but my book says there is 1.
Can anyone confirm if there is a solution or not?

$$x+1+x-1-2\sqrt{x+1}\sqrt{x-1}=4x-1\implies(2x-1)^2=4(x^2-1)\implies$$
$$4x^2-4x+1=4x^2-4\implies 4x=5\implies x=\frac54$$
But, indeed
$$\sqrt{\frac54+1}-\sqrt{\frac54-1}\stackrel ?=\sqrt{5-1}\iff\frac32-\frac12=2$$
which is false, thus no solution.
