x-2-66-2-x-x-x-x-2-66-2-x-2-5-

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

W h e r e i s t h e s o l u t i o n ?

Commented by Frix last updated on 06/Feb/24

Inside my �� but soon out here ��

Answered by Frix last updated on 06/Feb/24

Obviously x > 0

Let t = \frac{x + \sqrt{x^{2} + 66^{2}}}{66} \Leftrightarrow 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

Let u = t + \sqrt{t^{2} - 1} \Leftrightarrow t = \frac{u^{2} + 1}{2 u} \Rightarrow u > 1

\frac{66 \sqrt{2}}{u^{2} + 1} + \frac{2 \sqrt{2}}{u^{2} - 1} - 32 \sqrt{2} = 5

Let v = u^{2} \Leftrightarrow u = \sqrt{v} \Rightarrow v > 1 and 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}}

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