algebra precalculus - 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.

Answer

One rather general strategy is to replace each new root $\sqrt[k]{expression}$ in the equation by a new variable, $r_j$, together with a new equation
$r_j^k = expression$ (so now you will have $m+1$ polynomial equations in $m+1$ unknowns, where $m$ is the number of roots). Then eliminate variables from the system, ending with a single polynomial equation in one unknown, such that your original variable can be expressed in terms of the roots of this polynomial. This procedure can introduce spurious solutions if you only want the principal branch of the $k$'th root, so don't forget to check whether the solutions you get are valid.

For example, in your second equation,
we get the system
$$ \eqalign{r_1 + r_2 + r_3 - 2 \sqrt{2} &= 0\cr
r_1^2-(3x-1) &= 0\cr r_2^2-(5x-3) &= 0\cr
r_3^2-(x-1) &=0\cr}$$

Take the resultant of the first two polynomials with respect to $r_1$, then the resultant of this and the third with respect to $r_2$, and the resultant of this and the fourth with respect to $r_3$. We get
$$ 121 x^4-4820 x^3+28646 x^2-45364 x+21417$$
which happens to factor as
$$ \left( x-1 \right) \left( x-33 \right) \left( 121\,{x}^{2}-706\,x+
649 \right)
$$
However, only the solution $x=1$ turns out to satisfy the original equation.