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Question Number 35153 by Victor31926 last updated on 16/May/18

Answered by Rasheed.Sindhi last updated on 16/May/18

$$2=\mathrm{x}+\sqrt{\mathrm{x}+\sqrt{\mathrm{x}+\sqrt{\mathrm{x}+....}}}$$$$2-\mathrm{x}=\sqrt{\mathrm{x}+\sqrt{\mathrm{x}+\sqrt{\mathrm{x}+....}}}$$$${(2-\mathrm{x})}^2={\left(\sqrt{\mathrm{x}+\sqrt{\mathrm{x}+\sqrt{\mathrm{x}+....}}}\right)}^2$$$$4-4\mathrm{x}+{\mathrm{x}}^2=\mathrm{x}+\sqrt{\mathrm{x}+\sqrt{\mathrm{x}+\sqrt{\mathrm{x}+....}}}$$$$4-4\mathrm{x}+{\mathrm{x}}^2=2$$$${\mathrm{x}}^2-4\mathrm{x}+2=0$$$$\mathrm{x}=\frac{4\pm \sqrt{16-8}}{2}=\frac{4\pm 2\sqrt{2}}{2}=2\pm \sqrt{2}$$

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