Question and Answers Forum

All Questions      Topic List

Algebra Questions

Previous in All Question      Next in All Question      

Previous in Algebra      Next in Algebra      

Question Number 204082 by depressiveshrek last updated on 05/Feb/24

$$\sqrt{\frac{\sqrt{x^2+{66}^2}+x}{x}}-\sqrt{x\sqrt{x^2+{66}^2}-x^2}=5$$

Commented by Frix last updated on 05/Feb/24

$$x=\frac{6}{\sqrt{119}}$$

Commented by depressiveshrek last updated on 05/Feb/24

$$Whereisthesolution?$$

Commented by Frix last updated on 06/Feb/24

Inside my �� but soon out here ��

Answered by Frix last updated on 06/Feb/24

$$\mathrm{Obviously}x>0$$$$\mathrm{Let}t=\frac{x+\sqrt{x^2+{66}^2}}{66}\iff x=\frac{33(t^2-1)}{t}\Rightarrow t>1$$$$\frac{\sqrt{2}t}{\sqrt{t^2-1}}-\frac{33\sqrt{2(t^2-1)}}{t}=5$$$$\mathrm{Let}u=t+\sqrt{t^2-1}\iff t=\frac{u^2+1}{2u}\Rightarrow u>1$$$$\frac{66\sqrt{2}}{u^2+1}+\frac{2\sqrt{2}}{u^2-1}-32\sqrt{2}=5$$$$\mathrm{Let}v=u^2\iff u=\sqrt{v}\Rightarrow v>1\mathrm{and}\mathrm{transforming}$$$$v^2-\frac{4(64-5\sqrt{2})}{119}+\frac{2073-320\sqrt{2}}{2023}=0$$$$v=\frac{123+22\sqrt{2}}{119}\Rightarrow u=\frac{11+\sqrt{2}}{\sqrt{119}}\Rightarrow t=\frac{11}{\sqrt{119}}\Rightarrow$$$$x=\frac{6}{\sqrt{119}}$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com