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Question Number 114676 by ZiYangLee last updated on 20/Sep/20

$$\mathrm{If}{a}^{\frac{1}{2}}-a^{-\frac{1}{2}}=1,\mathrm{prove}\mathrm{that}a+a^{-1}=3$$

Answered by Olaf last updated on 20/Sep/20

$$\sqrt{a}+\frac{1}{\sqrt{a}}=1$$$$\Rightarrow {(\sqrt{a}+\frac{1}{\sqrt{a}})}^2=1$$$$a+\frac{1}{a}+2=1\Rightarrow a+a^{-1}=-1!$$$$\mathrm{But}\mathrm{if}\sqrt{a}-\frac{1}{\sqrt{a}}=1$$$$\Rightarrow {(\sqrt{a}-\frac{1}{\sqrt{a}})}^2=1$$$$a+\frac{1}{a}-2=1$$$$a+a^{-1}=3$$

Commented by ZiYangLee last updated on 20/Sep/20

$$\mathrm{oops}\mathrm{typo}\mathrm{haha}\mathrm{sry}\mathrm{sir}$$

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