General Principles of Solving Radical Equations What are the general ways to solve radical equations similar to questions like

$\sqrt{x+1}+\sqrt{x-1}-\sqrt{x^2 -1}=x$  $\sqrt{3x-1}+\sqrt{5x-3}+\sqrt{x-1}=2\sqrt2$$\sqrt{\frac{4x+1}{x+3}}-\sqrt{\frac{x-2}{x+3}}=1$

Are there just a few known ways to solve them? How do you know the best way to solve such questions? I have trouble with a lot of square root equations, and when I ask them on this site, I get good answers, but for one question. I was wondering if there were any general principles of solving such questions.

I would write the equation in the form $$\sqrt{x+1}+\sqrt{x-1}=x+\sqrt{x^2-1}$$
with $$x\geq 1$$ after squaring one times and isolating the square root we get
$$2\sqrt{1-x^2}(1-x)=2x^2-2x-1$$ squaring again we obtain
$$(x^2-1)(2-2x)^2=(2x^2-2x-1)^2$$ this gives the equation $$4x-5=0$$ and we get $$x=\frac{5}{4}$$ fulfills our equation.
in the second equation we get $$x=1$$
write your third equation in the form
$$\sqrt{\frac{4x+1}{x+3}}=1+\sqrt{\frac{x-2}{x+3}}$$ we get after squaring two times
$$\left(\frac{2x}{x+3}\right)^2-4\left(\frac{x-2}{x+3}\right)=0$$
after simplifying we obtain$$-4\,{\frac {x-6}{ \left( x+3 \right) ^{2}}}=0$$
