<s>[INST] 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?
 [/INST] Like Solve the equation $x^2+\frac{9x^2}{(x+3)^2}=27$
Clearly $x>0$
Let $\dfrac{3x}{\sqrt{x^2-9}}=y$ so that $x+y=\dfrac{35}4$
$$\implies9x^2=x^2y^2-9y^2\iff\dfrac19=\dfrac1{x^2}+\dfrac1{y^2}=\dfrac{(x+y)^2-2xy}{(xy)^2}$$
Use $x+y=\dfrac{35}4$ and $xy>0$ for find $xy=\dfrac{75}4$
So, $x,y$ are the roots of $$t^2-\dfrac{35}4t+\dfrac{75}4=0$$
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