Question

x = 3 +√5 , then the value of x – 1 / x will be​

Answer 1

ANSWER 》3/2

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Answer 2

Given

• x = 3 + √5

To find

• The value of x - 1/x

Solution

$: \implies \sf \gray{\cfrac{1}{x} = \cfrac{1}{3 + \sqrt{5} } } \\ \\ \\ \qquad: \: \tt \pink{here \: we \: will \: rationalize \: the \: denominator } \\ \\ \\ : \implies \sf \gray{ \cfrac{1}{x} = \cfrac{3 - \sqrt{5} }{(3 + \sqrt{5})(3 - \sqrt{5}) }} \\ \\ \\ \qquad: \implies \sf \gray{ \cfrac{1}{x} = \cfrac{3 - \sqrt{5} }{( {3})^{2} - ( { \sqrt5})^{2}}} \\ \\ \\ \qquad: \implies \sf \gray{ \cfrac{1}{x} = \cfrac{3 - \sqrt{5} }{9 - 5}} \\ \\ \\ \qquad: \implies \sf \boxed {\sf{\underline \purple{\cfrac{1}{x} = \cfrac{3 - \sqrt{5} }{4} } }}\quad \red{ \bigstar}$

$\qquad: \implies \sf \gray{x - \cfrac{1}{x} = 3 + \sqrt{5} - \bigg(\cfrac{3 - \sqrt{5}}{4} \bigg) } \\ \\ \\ \qquad: \implies \sf \gray{x - \cfrac{1}{x} = 3 + \sqrt{5} - \cfrac{3 + \sqrt{5}}{4}} \\ \\ \\ \qquad: \implies \sf \gray{ \cfrac{4(3 + \sqrt{5} ) - 3 + \sqrt{5} }{4} } \\ \\ \\ \qquad: \implies \sf \gray{ \cfrac{12 + 4 \sqrt{5} - 3 + \sqrt{5}}{4} } \\ \\ \\ \qquad: \implies \sf \gray{ \cfrac{9 + 5 \sqrt{5} }{4} } \\ \\ \\ \qquad: \implies \sf \red{x - \cfrac{1}{x} = \cfrac{9 + 5 \sqrt{5} }{4}}$