6(x2+1/x2)-35(x-1/x)+62 solve for x

Dear Student,
Here is the solution of your asked query:
$$\begin{gathered} 6\left({\mathrm{x}}^2+\frac{1}{{\mathrm{x}}^2}\right)-35\left(\mathrm{x}-\frac{1}{\mathrm{x}}\right)+62=0 \\ \mathrm{Taking} \mathrm{x}-\frac{1}{\mathrm{x}}=\mathrm{y} ...\left(\mathrm{i}\right) , \mathrm{we} \mathrm{have}; \\ {\left(\mathrm{x}-\frac{1}{\mathrm{x}}\right)}^2={\mathrm{y}}^2 \\ \Rightarrow {\mathrm{x}}^2+\frac{1}{{\mathrm{x}}^2}-2\times \mathrm{x}\times \frac{1}{\mathrm{x}}={\mathrm{y}}^2 \\ \Rightarrow {\mathrm{x}}^2+\frac{1}{{\mathrm{x}}^2}-2={\mathrm{y}}^2 \\ \Rightarrow {\mathrm{x}}^2+\frac{1}{{\mathrm{x}}^2}={\mathrm{y}}^2+2 ...\left(\mathrm{ii}\right) \\ \mathrm{Substituting} \left(\mathrm{i}\right) \mathrm{and} \left(\mathrm{ii}\right) \mathrm{in} \mathrm{the} \mathrm{given} \mathrm{equation} \mathrm{we} \mathrm{get}; \\ 6\left({\mathrm{y}}^2+2 \right)-35\mathrm{y}+62=0 \\ \Rightarrow 6{\mathrm{y}}^2+12-35\mathrm{y}+62=0 \\ \Rightarrow 6{\mathrm{y}}^2-35\mathrm{y}+74=0 \\ \Rightarrow \mathrm{y}=\frac{-\left(-35\right)\pm \sqrt{{\left(-35\right)}^2-4\times 6\times 74}}{2\times 6} \\ =\frac{35\pm \sqrt{551}}{12} \\ =\frac{35\pm 23}{12} \{\mathrm{Since} \sqrt{551}=23.4733, \mathrm{taking} \mathrm{approximate} \mathrm{value} \mathrm{around} 23\} \\ =\frac{35+23}{12} \mathrm{and} \frac{35-23}{12} \\ =\frac{58}{12} \mathrm{and} 1 \\ \mathrm{So} \mathrm{we} \mathrm{have}; \\ \mathrm{x}-\frac{1}{\mathrm{x}}=1 \mathrm{and} \mathrm{x}-\frac{1}{\mathrm{x}}=\frac{58}{12} \\ \Rightarrow {\mathrm{x}}^2-1=\mathrm{x} \mathrm{and} {\mathrm{x}}^2-1=\frac{58}{12}\mathrm{x} \\ \Rightarrow {\mathrm{x}}^2-\mathrm{x}-1=0 \mathrm{and} 12{\mathrm{x}}^2-12=58\mathrm{x} \\ \Rightarrow {\mathrm{x}}^2-\mathrm{x}-1=0 \mathrm{and} 12{\mathrm{x}}^2-58\mathrm{x}-12=0 \\ \Rightarrow \mathrm{x}=\frac{-\left(-1\right)\pm \sqrt{{\left(-1\right)}^2-4\times 1\times \left(-1\right)}}{2\times 1} \mathrm{and} \mathrm{x}=\frac{-\left(-58\right)\pm \sqrt{{\left(-58\right)}^2-4\times 12\times \left(-12\right)}}{2\times 12} \\ \Rightarrow \mathrm{x}=\frac{1\pm \sqrt{5}}{2} \mathrm{and} \mathrm{x}=\frac{58\pm \sqrt{3940}}{24} \end{gathered}$$
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