Question

$\frac{\sqrt{x+13}+\sqrt{x-1}}{\sqrt{x+13}-\sqrt{x-1}}=3$

Answer 1

$$\begin{gathered} \frac{\sqrt{x+13}+\sqrt{x-1}}{\sqrt{x+13}-\sqrt{x-1}}=3 \\ \Rightarrow \frac{\sqrt{x+13}+\sqrt{x-1}+\sqrt{x+13}-\sqrt{x-1}}{\sqrt{x+13}+\sqrt{x-1}-\sqrt{x+13}+\sqrt{x-1}}=\frac{3+1}{3-1}(\mathrm{Componendo}\mathrm{and}\mathrm{dividendo}) \end{gathered}$$

$$\begin{gathered} \Rightarrow \frac{2\sqrt{x+13}}{2\sqrt{x-1}}=\frac{4}{2} \\ \Rightarrow \frac{x+13}{x-1}={\left(\frac{2}{1}\right)}^2 \\ \Rightarrow \frac{x+13}{x-1}=\frac{4}{1} \\ \Rightarrow x+13=4x-4 \\ \Rightarrow -3x=-17 \\ \Rightarrow x=\frac{17}{3} \\ \mathrm{Therefore},x=\frac{17}{3}\mathrm{is}\mathrm{the}\mathrm{solution}. \end{gathered}$$