Is there a better way to solve this equation? I came across this equation:
$x + \dfrac{3x}{\sqrt{x^2 - 9}} = \dfrac{35}{4}$
Wolfram Alpha found 2 roots: $x=5$ and $x=\dfrac{15}{4}$, which "coincidentally" add up to $\dfrac{35}{4}$. So I'm thinking there should be a better way to solve it than the naïve way of bringing the fractions together and then squaring. Is there any?

It' not a coincidence.   Let $f(x)=x + g(x)$ where $g(g(x))=x$, that is, $g$ is an involution.  We'll show that $f(x)=f(a-x)$, provided that $f(x)=a$.
Assume $f(x)=a$, that is $a-x=g(x)$. Now
$$f(a-x)=f(g(x))=g(x)+g(g(x))=g(x)+x=f(x)$$.
In our case define 
$$g(x)=\frac{3x}{\sqrt{x^2 - 9}}.$$
Indeed, $g$ is an involution as
$$g(g(x))=\dfrac{3\frac{3x}{\sqrt{x^2 - 9}}}{\sqrt{\frac{9x^2}{x^2 - 9} - 9}}=
\dfrac{x\frac{9}{\sqrt{x^2 - 9}}}{3\sqrt{\frac{x^2}{x^2 - 9} - 1}}=
\dfrac{x\frac{9}{\sqrt{x^2 - 9}}}{3\sqrt{\frac{9}{x^2 - 9}}}=x.
$$
