Question

if x = 3+2√2, find the value of (√x-1/√x)​

Answer 1

$\blue{\bold{\underline{\underline{Answer:}}}}$

$\green{\tt{\therefore{(\sqrt{x}-\frac{1}{\sqrt{x}})=2}}}$

$\orange{\bold{\underline{\underline{Step-by-step\:explanation:}}}}$

$\green{\underline \bold{Given :}} \\ \tt: \implies x= 3 +2\sqrt{2} \\ \\ \red{\underline \bold{To \: Find :}} \\ \tt: \implies( \sqrt{x} - \frac{1}{\sqrt{x}} ) =?$

• According to given question :

$\bold{As \: we \: know \: that} \\ \tt: \implies (\sqrt{x} - \frac{1}{ \sqrt{x} } )^{2} = { (\sqrt{x} )}^{2} + \frac{1}{ (\sqrt{x})^{2} } - 2 \times \sqrt{x} \times \frac{1}{ \sqrt{x} } \\ \\ \tt: \implies {(\sqrt{x}- \frac{1}{\sqrt{x}}) }^{2} = x + \frac{1}{x} - 2 \\ \\ \tt: \implies {(\sqrt{x}- \frac{1}{\sqrt{x}}) }^{2} =3 + 2 \sqrt{2} + \frac{1}{3 + 2 \sqrt{2} } \times \frac{3 - 2 \sqrt{2} }{3 - 2 \sqrt{2} } - 2 \\ \\ \tt: \implies {(\sqrt{x}- \frac{1}{\sqrt{x}}) }^{2} =3 + 2 \sqrt{2} + \frac{3 - 2 \sqrt{2} }{ {3}^{2} - {(2 \sqrt{2}) }^{2} } - 2 \\ \\ \tt: \implies {(\sqrt{x}- \frac{1}{\sqrt{x}}) }^{2} =3 + 2 \sqrt{2} + \frac{3 - 2 \sqrt{2} }{9 - 8} - 2 \\ \\ \tt: \implies {(\sqrt{x}- \frac{1}{\sqrt{x}}) }^{2} =3 + 2 \sqrt{2} + 3 - 2\sqrt{2} - 2 \\ \\ \tt: \implies {(\sqrt{x}- \frac{1}{\sqrt{x}}) }^{2} =6 - 2 \\ \\ \tt: \implies {(\sqrt{x}- \frac{1}{\sqrt{x}}) }^{2} =4 \\ \\ \tt: \implies {(\sqrt{x}- \frac{1}{\sqrt{x}}) }= \sqrt{4} \\ \\ \green{\tt: \implies {(\sqrt{x}- \frac{1}{\sqrt{x}}) } =2}$

Answer 2

Aɴꜱᴡᴇʀ

Your answer is 2

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Gɪᴠᴇɴ

$\large \sf \longmapsto{}x =3 + 2 \sqrt{2}$

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Tᴏ ꜰɪɴᴅ

$\large \sf \longmapsto{} value \: of \: (\sqrt{x } - \dfrac{1}{ \sqrt{x} } )$

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Sᴛᴇᴘꜱ

$\sf \leadsto{ (\sqrt{x} - \frac{1}{ \sqrt{x} }{)}^{2} = { (\sqrt{x}) }^{2} - 2 \times \cancel{ \sqrt{x} } \times \frac{1}{ \cancel{ \sqrt{x}} } + {( \frac{1}{ \sqrt{x} } )}^{2} } \\ \\ \leadsto \sf( \sqrt{x } - \frac{1}{ \sqrt{x} }{)}^{2} = x + \frac{1}{x} - 2 \\ \\ \leadsto \sf ( \sqrt{x} - \frac{1}{ \sqrt{x} } {)}^{2} = 3 + 2 \sqrt{2} + \frac{1}{3 + 2 \sqrt{2} } \times \frac{3 - 2 \sqrt{2} }{3 - 2 \sqrt{2} } - 2 \\ \\ \leadsto \sf ( \sqrt{x} - \frac{1}{ \sqrt{x} } {)}^{2} = 3 + \cancel{ 2 \sqrt{2}} + {3 - \cancel{ 2 \sqrt{2}}} - 2 \\ \\ \leadsto \sf( \sqrt{x} - \frac{1}{ \sqrt{x} }{)}^{2} = 6 - 2 \\ \\ \leadsto \sf\sqrt{ x} - \frac{1}{ \sqrt{x} }= \sqrt{4} \\ \\ \pink{ \leadsto \sf \sqrt{x} - \frac{1}{ \sqrt{x} } = \large 2 }$

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