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 108309 by bemath last updated on 16/Aug/20

$$\frac{△\mathcal{B}e\mathcal{M}ath△}{\therefore }$$$$Given\sqrt{5+\sqrt{9+2\sqrt{15}}}+\sqrt{5-\sqrt{9+2\sqrt{15}}}=x$$$$findthevalueof{(x-\frac{1}{x})}^2$$

Answered by bobhans last updated on 16/Aug/20

$$\frac{\in bobhans\in }{\Bumpeq }$$$$\Rightarrow {(x-\frac{1}{x})}^2=x^2+\frac{1}{x^2}-2$$$$fromtheequation\Rightarrow set\sqrt{9+2\sqrt{15}}=\mathrm{♭}$$$$\sqrt{5+\mathrm{♭}}+\sqrt{5-\mathrm{♭}}=x\Rightarrow 5+\mathrm{♭}+5-\mathrm{♭}+2\sqrt{25-{\mathrm{♭}}^2}=x^2$$$$10+2\sqrt{25-(9+2\sqrt{15})}=x^2$$$$10+2\sqrt{16-2\sqrt{15}}=x^2$$$$10+2(\sqrt{15}-1)=x^2;x^2=8+2\sqrt{15}$$$$and\frac{1}{x^2}=\frac{1}{8+2\sqrt{15}}\times \frac{8-2\sqrt{15}}{8-2\sqrt{15}}=\frac{8-2\sqrt{15}}{4}$$$$\frac{1}{x^2}=2-\frac{\sqrt{15}}{2}.Finallywegot$$$${(x-\frac{1}{x})}^2=8+2\sqrt{15}+2-\frac{\sqrt{15}}{2}-2$$$$=8+\frac{3\sqrt{15}}{2}=\frac{16+3\sqrt{15}}{2}$$

Answered by 1549442205PVT last updated on 16/Aug/20

$$\mathrm{Put}\sqrt{5+\sqrt{9+2\sqrt{15}}}=\mathrm{a},\sqrt{5-\sqrt{9+2\sqrt{15}}}=\mathrm{b}$$$$\mathrm{Then}\mathrm{x}=\mathrm{a}+\mathrm{b}\mathrm{and}$$$$\mathrm{ab}=\sqrt{25-(9+2\sqrt{15})}=\sqrt{16-2\sqrt{15}}=$$$$\sqrt{{(\sqrt{15}-1)}^2}=\sqrt{15}-1\Rightarrow 2\mathrm{ab}=2\sqrt{15}-2$$$${\mathrm{a}}^2+{\mathrm{b}}^2=10\Rightarrow {\mathrm{x}}^2={(\mathrm{a}+\mathrm{b})}^2={\mathrm{a}}^2+{\mathrm{b}}^2+2\mathrm{ab}$$$$=10+2\sqrt{15}-2=8+2\sqrt{15}={(\sqrt{5}+\sqrt{3})}^2$$$$\Rightarrow \mathrm{x}=\mathrm{a}+\mathrm{b}=\sqrt{5}+\sqrt{3}\Rightarrow \frac{1}{\mathrm{x}}=\frac{1}{\sqrt{5}+\sqrt{3}}$$$$=\frac{\sqrt{5}-\sqrt{3}}{2}\Rightarrow {(\mathrm{x}-\frac{1}{\mathrm{x}})}^2={(\sqrt{5}+\sqrt{3}-\frac{\sqrt{5}-\sqrt{3}}{2})}^2$$$$={\left(\frac{\sqrt{5}+3\sqrt{3}}{2}\right)}^2=\frac{32+6\sqrt{15}}{4}=\frac{16+3\sqrt{15}}{2}$$$$\mathrm{Thus},{(\mathbf{x}-\frac{1}{\mathbf{x}})}^2=\frac{16+3\sqrt{15}}{2}$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com