I want to find positive solution to an equation $ \text{Solve}\left[t==\frac{\sqrt{\left(\frac{L}{\gamma }+t v\right)^2+x^2}}{c}\land \text{assume},t\right]$ .

Although using many assumptions, Mathematica provides a negative solution as explained in what follows.

A similar but simpler equation works as expected.

How can I obtain the positive solution to the equation

### First a little context:

I’m using Solve to solve for t:

`t11s1 = Solve[t == Sqrt[(L + v*t)^2 + (x)^2]/(c), t] `

The solution are two roots:

$ $ \left\{\left\{t\to \frac{L v-\sqrt{c^2 L^2+c^2 x^2-v^2 x^2}}{c^2-v^2}\right\},\left\{t\to \frac{\sqrt{c^2 L^2+c^2 x^2-v^2 x^2}+L v}{c^2-v^2}\right\}\right\} $ $

This I Refine with some assumptions, and get a positive solution:

$ $ \left\{\frac{\sqrt{c^2 L^2+c^2 x^2-v^2 x^2}}{c^2-v^2}+\frac{L v}{c^2-v^2}\right\} $ $

However, if instead of $ L$ I use $ L\rightarrow\frac{L}{\sqrt{1-\frac{v^2}{c^2}}}$ which is a positive, real multiplicative factor on $ L$ , the solution changes to

$ $ \left\{\left\{t\to \text{ConditionalExpression}\left[\text{Root}\left[\text{$ \#$ 1}^4 \left(c^8-2 c^6 v^2+c^4 v^4\right)+\text{$ \#$ 1}^2 \left(-2 c^6 L^2-2 c^6 x^2+2 c^4 v^2 x^2+2 c^2 L^2 v^4\right)+c^4 L^4+2 c^4 L^2 x^2+c^4 x^4-2 c^2 L^4 v^2-2 c^2 L^2 v^2 x^2+L^4 v^4\&,4\right],v>0\land c>v\land x>0\land L>x\land 0<a<x\land 0<n<\frac{c}{v}\right]\right\}\right\} $ $

which if I force to be expressed as radicals (to eliminate the Root[,4]) and again I refine using the same assumptions I get

$ $ \left\{-\sqrt{\frac{c^2 \left(L^2+x^2\right)+2 c L v \sqrt{L^2+x^2}+L^2 v^2}{c^4-c^2 v^2}}\right\} $ $

which is a negative value because all variables are Reals !!

My assumptions are:

$ $ \text{assume}=v>0\land v\in \mathbb{R}\land L>0\land L\in \mathbb{R}\land \text{L2}>0\land \text{L2}\in \mathbb{R}\land a>0\land a\in \mathbb{R}\land t>0\land t\in \mathbb{R}\land c>0\land c\in \mathbb{R}\land n>0\land n\in \mathbb{R}\land \gamma >0\land \gamma \in \mathbb{R}\land \beta \geq 0\land \beta <1\land \beta \in \mathbb{R}\land c>v\land x\in \mathbb{R}\land x>0\land n<\frac{c}{v}\land L>x\land L>a\land x>a $ $

and I use them as:

$ $ \text{t11s2}=\text{Solve}\left[t==\frac{\sqrt{\left(\frac{L}{\gamma }+t v\right)^2+x^2}}{c}\land \text{assume},t\right] $ $

$ $ \text{t2}=\text{Simplify}[\text{ToRadicals}[\text{Refine}[t\text{/.}\, \text{t11s2},\text{assume}]],\text{assume}] $ $

to get

$ $ t2=\left\{-\sqrt{\frac{c^2 \left(L^2+x^2\right)+2 c L v \sqrt{L^2+x^2}+L^2 v^2}{c^4-c^2 v^2}}\right\} $ $

What can I do to recover a positive real values for t? Why is Mathematica behaving like this?