Question

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Answer 1

Appropriate Question :-

$\rm :\longmapsto\:If \: \: x = \dfrac{5 - \sqrt{21}}{2}, \: prove \: that$

$\red{\rm \:\bigg( {x}^{3} + \dfrac{1}{ {x}^{3} } \bigg) - 5\bigg( {x}^{2} + \dfrac{1}{ {x}^{2} } \bigg) + \bigg( x + \dfrac{1}{x} \bigg) = 0}$

$\large\underline{\sf{Solution-}}$

Given that

$\rm :\longmapsto\:x = \dfrac{5 - \sqrt{21}}{2}$

So, Let consider,

$\rm :\longmapsto\:\dfrac{1}{x}$

$\rm \: = \: \: \dfrac{2}{5 - \sqrt{21} }$

On rationalizing the denominator, we get

$\rm \: = \: \: \dfrac{2}{5 - \sqrt{21} } \times \dfrac{5 + \sqrt{21} }{5 + \sqrt{21} }$

$\rm \: = \: \: \dfrac{2(5 + \sqrt{21} )}{ {5}^{2} - {( \sqrt{21}) }^{2} }$

$\rm \: = \: \: \dfrac{2(5 + \sqrt{21} )}{ 25 - 21}$

$\rm \: = \: \: \dfrac{2(5 + \sqrt{21} )}{4}$

$\rm \: = \: \: \dfrac{5 + \sqrt{21} }{2}$

Hence,

$\rm :\longmapsto\:x + \dfrac{1}{x}$

$\rm \: = \: \: \dfrac{5 - \sqrt{21} }{2} + \dfrac{5 + \sqrt{21} }{2}$

$\rm \: = \: \: \dfrac{5 - \sqrt{21} + 5 + \sqrt{21} }{2}$

$\rm \: = \: \: \dfrac{10}{2}$

$\rm \: = \: \: 5$

Hence,

$\red{\rm :\longmapsto\:x + \dfrac{1}{x} = 5}$

Now,

$\rm :\longmapsto\: {x}^{2} + \dfrac{1}{ {x}^{2} }$

$\rm \: = \: \: {\bigg(x + \dfrac{1}{x} \bigg) }^{2} - 2$

$\rm \: = \: \: {5}^{2} - 2$

$\rm \: = \: \: 25 - 2$

$\rm \: = \: \: 23$

Now, Consider

$\rm :\longmapsto\: {x}^{3} + \dfrac{1}{ {x}^{3} }$

$\rm \: = \: \: {\bigg(x + \dfrac{1}{x} \bigg) }^{3} - 3\bigg(x - \dfrac{1}{x} \bigg)$

$\rm \: = \: \: {(5)}^{3} - 3 \times 5$

$\rm \: = \: \: 125 - 15$

$\rm \: = \: \: 110$

Now, Consider

$\red{\rm :\longmapsto\:\bigg( {x}^{3} + \dfrac{1}{ {x}^{3} } \bigg) - 5\bigg( {x}^{2} + \dfrac{1}{ {x}^{2} } \bigg) + \bigg( x + \dfrac{1}{x} \bigg)}$

$\rm \: = \: \: 110 - 5(23) + 5$

$\rm \: = \: \: 110 - 115 + 5$

$\rm \: = \: \: 0$

Hence, proved

Identities Used :-

$\rm :\longmapsto\: {x}^{3} + \dfrac{1}{ {x}^{3} } = \: \: {\bigg(x + \dfrac{1}{x} \bigg) }^{3} - 3\bigg(x - \dfrac{1}{x} \bigg)$

$\rm :\longmapsto\: {x}^{2} + \dfrac{1}{ {x}^{2} } = \: \: {\bigg(x + \dfrac{1}{x} \bigg) }^{2} - 2$

$\red{\rm :\longmapsto\:(x + y)(x - y) = {x}^{2} - {y}^{2}}$