Question

$30\left(x^2+\frac{1}{x^2}\right)-77\left(x-\frac{1}{x}\right)-12=0$

Answer 1

$$\begin{gathered} Given:30\left(x^2+\frac{1}{x^2}\right)-77\left(x-\frac{1}{x}\right)-12=0 \\ \mathrm{We}\mathrm{know}\mathrm{the}\mathrm{identity}{x}^2+\frac{1}{x^2}={\left(x-\frac{1}{x}\right)}^2+2 \\ \mathrm{Thus},\text{the given equation can be written as:} \\ 30\left[{\left(x-\frac{1}{x}\right)}^2+2\right]-77\left(x-\frac{1}{x}\right)-12=0 \\ 30{\left(x-\frac{1}{x}\right)}^2+60-77\left(x-\frac{1}{x}\right)-12=0 \\ 30{\left(x-\frac{1}{x}\right)}^2-77\left(x-\frac{1}{x}\right)+48=0 \\ \mathrm{Let}m=x-\frac{1}{x} \\ \mathrm{Then},\text{the equation can be further written as:} \\ 30m^2-77m+48=0 \\ \mathrm{On}\mathrm{using}\mathrm{the}\mathrm{quadratic}\mathrm{formula},\mathrm{we}\mathrm{get}: \\ m=\frac{-(-77)\pm \sqrt{(-77{)}^2-4(30\left)\right(48)}}{2\left(30\right)} \\ =\frac{77\pm \sqrt{5929-5760}}{60}=\frac{77\pm \sqrt{169}}{60}=\frac{77\pm 13}{60}=\frac{90}{60},\frac{64}{60}=\frac{3}{2},\frac{16}{15} \end{gathered}$$

^{$\text{On sub}s\mathrm{tituting}m=x-\frac{1}{x},\text{we get}:$
$$\begin{gathered} \Rightarrow x-\frac{1}{x}=\frac{3}{2}....\left(i\right)orx-\frac{1}{x}=\frac{16}{15}...\left(ii\right) \\ \mathrm{Now}\mathrm{from}\mathrm{equation}\left(\mathrm{i}\right)\mathrm{we}\mathrm{get}: \\ x-\frac{1}{x}=\frac{3}{2} \\ \Rightarrow \frac{x^2-1}{x}=\frac{3}{2} \\ \Rightarrow 2x^2-2=3x \\ \Rightarrow 2x^2-3x-2=0 \\ \mathrm{On}\mathrm{splitting}\mathrm{the}\mathrm{middle}\mathrm{term}-3\mathrm{x}\mathrm{as}\mathrm{x}-4\mathrm{x},\mathrm{we}\mathrm{get}: \\ 2x^2+x-4x-2=0 \\ \Rightarrow x(2x+1)-2(2x+1)=0 \\ \Rightarrow (2x+1)(x-2)=0 \\ \Rightarrow 2x+1=0orx-2=0 \\ \Rightarrow x=-\frac{1}{2}orx=2 \\ \mathrm{Now}\mathrm{from}\mathrm{equation}\left(\mathrm{ii}\right)\mathrm{we}\mathrm{get}: \\ x-\frac{1}{x}=\frac{16}{15} \end{gathered}$$
$$\begin{gathered} \Rightarrow \frac{x^2-1}{x}=\frac{16}{15} \\ \Rightarrow 15x^2-15=16x \\ \Rightarrow 15x^2-16x-15=0 \\ \mathrm{On}\mathrm{using}\mathrm{the}\mathrm{quadratic}\mathrm{formula},\mathrm{we}\mathrm{get}: \\ x=\frac{-(-16)\pm \sqrt{{\left(-16\right)}^2-4\left(15\right)\left(-15\right)}}{2\left(15\right)} \\ =\frac{16\pm \sqrt{256+900}}{30} \\ =\frac{16\pm 34}{30} \\ =\frac{50}{30},-\frac{18}{30} \\ =\frac{5}{3},-\frac{3}{5} \\ \mathrm{Thus}\mathrm{the}\mathrm{solutions}\mathrm{of}\mathrm{the}\mathrm{given}\mathrm{equation}\mathrm{are}x=-\frac{1}{2},2,\frac{5}{3},-\frac{3}{5} \end{gathered}$$}