Conservation problem

Squaring both sides gives

$\begin{aligned} 4(x-1) + 25(x+1) + 20\sqrt{x^2-1} &= &(x+6)^2 + (x^2-1) + 2(x+6)\sqrt{x^2-1} \\ \Leftrightarrow -2x^2+17x-14 &=& (2x-8)\sqrt{x^2-1} \; . \end{aligned}$

Squaring both sides again gives

$\begin{aligned} 4x^4-68x^3 + 345x^2 - 476x + 196 &=& 4x^4-32x^3 + 60x^2 + 32x-64 \\ \Leftrightarrow 36x^3 - 285x^2 + 508x - 260 &=& 0 \; . \end{aligned}$

By rational root theorem , the rational root of $x$ is $\dfrac 54$ . Factoring this cubic polynomial and apply quadratic formula , to get the other two roots: $\dfrac{10}3 \pm \dfrac4{\sqrt3}$ .

By trial and error , we see that only one of these values satisfy the given equation (because we introduced extraneous roots by squaring the equation (at least once). Hence, our answer is $\boxed1$ .

@Pi Han Goh Check my solution =)))) The equation can be rewrite as: $(\sqrt{x-1}-2\sqrt{x+1}+3)(-\sqrt{x-1}-\sqrt{x+1}+1)=0$ The right polynomial give us no solution and left polynomial give us only 1 solution.