Question

if x= 2+under root 5 then x-1/x =?
Q. If $x=2+\sqrt{5}$ then $x-\frac{1}{x}=?$
Solve

Answer 1

Dear Student,

$$\begin{gathered} x=2+\sqrt{5} \\ \frac{1}{x}=\frac{1}{2+\sqrt{5}}\times \frac{2-\sqrt{5}}{2-\sqrt{5}}\left[\mathbf{U}\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{g}\mathbf{}{\mathbf{a}}^{\mathbf{2}}\mathbf{-}{\mathbf{b}}^{\mathbf{2}}\mathbf{=}\left(\mathbf{a}\mathbf{-}\mathbf{b}\right)\left(\mathbf{a}\mathbf{+}\mathbf{b}\right)\right] \\ \frac{1}{x}=\frac{2-\sqrt{5}}{4-5} \\ \frac{1}{x}=-\left(2-\sqrt{5}\right) \\ \frac{1}{x}=-2+\sqrt{5} \\ Now,x-\frac{1}{x}=2+\sqrt{5}-\left[-2+\sqrt{5}\right] \\ x-\frac{1}{x}=2\cancel{+\sqrt{5}}+2\cancel{-\sqrt{5}} \\ \mathbf{}\mathit{x}\mathbf{-}\frac{\mathbf{1}}{\mathbf{x}}\mathbf{}\mathbf{=}\mathbf{4} \end{gathered}$$

Regards