Solve-the-equation-x-1-x-1-x-1-x-1-1-

Question Number 165776 by ZiYangLee last updated on 08/Feb/22

Solve the equation

\sqrt{\frac{\sqrt{x} + 1^{}}{\sqrt{x} - 1}} - \sqrt{\frac{\sqrt{x} - 1^{}}{\sqrt{x} + 1}} = 1

Answered by Rasheed.Sindhi last updated on 08/Feb/22

\sqrt{\frac{\sqrt{x} + 1^{}}{\sqrt{x} - 1}} - \sqrt{\frac{\sqrt{x} - 1^{}}{\sqrt{x} + 1}} = 1

\sqrt{\frac{\sqrt{x} + 1^{}}{\sqrt{x} - 1}} = y (l e t)

y - \frac{1}{y} = 1

y^{2} - y - 1 = 0

y = \frac{1 \pm \sqrt{1 + 4}}{2} = \frac{1 \pm \sqrt{5}}{2}

\sqrt{\frac{\sqrt{x} + 1^{}}{\sqrt{x} - 1}} = \frac{1 \pm \sqrt{5}}{2}

\frac{\sqrt{x} + 1^{}}{\sqrt{x} - 1} = \frac{1 + 5 \pm 2 \sqrt{5}}{4} = \frac{3 \pm \sqrt{5}}{2}

\frac{\sqrt{x} + 1^{}}{\sqrt{x} - 1} - 1 = \frac{3 \pm \sqrt{5}}{2} - 1

\frac{2}{\sqrt{x} - 1} = \frac{1 \pm \sqrt{5}}{2}

\frac{\sqrt{x} - 1}{2} = \frac{2}{1 \pm \sqrt{5}}

\sqrt{x} - 1 = \frac{4}{1 \pm \sqrt{5}} \times \frac{1 \mp \sqrt{5}}{1 \mp \sqrt{5}} = \frac{4 \mp 4 \sqrt{5}}{1 - 5}

= - 1 \pm \sqrt{5}

\sqrt{x} = \pm \sqrt{5} \Rightarrow x = 5

Commented by Tawa11 last updated on 10/Feb/22

Great sir

Answered by Rasheed.Sindhi last updated on 08/Feb/22

\sqrt{\frac{\sqrt{x} + 1}{\sqrt{x} - 1}} - \sqrt{\frac{\sqrt{x} - 1}{\sqrt{x} + 1}} = 1

\sqrt{\frac{{(\sqrt{x} + 1)}^{2}}{(\sqrt{x} - 1) (\sqrt{x} + 1)}} - \sqrt{\frac{{(\sqrt{x} - 1)}^{2}}{(\sqrt{x} + 1) (\sqrt{x} - 1)}} = 1

\frac{\sqrt{x} + 1}{\sqrt{x - 1}} - \frac{\sqrt{x} - 1}{\sqrt{x - 1}} = 1

\sqrt{x} + 1 - \sqrt{x} + 1 = \sqrt{x - 1}

\sqrt{x - 1} = 2

x - 1 = 4

x = 5

Commented by ZiYangLee last updated on 08/Feb/22

Nice sir .

Commented by Tawa11 last updated on 10/Feb/22

Great sir

Commented by Rasheed.Sindhi last updated on 10/Feb/22

T han k s miss!