Question

(क) लभ + क्त्वा
(ख) हन्+अथवा
​

Answer 1

Here we are asked to find the value of

$\sf\red{{x}^{2} + \dfrac{1} {x}^{2}}$and according to the question given that :-

• $\sf \green {\: x = \dfrac{3 - \sqrt{13} }{2}}$

For getting the proper solution, we have to use 2 identities!

• $\sf \purple{ (a + b)² - 2ab = a² + b²}$

• $\sf \purple{ (a - b)² = a² + b² - 2ab}$

Here's our required solution!

$\implies \sf \bigg { {x}^{2} + \dfrac{1}{x}^{2} = ( x + \dfrac{1}{x} \bigg) ^{2} - 2 \times x \times \dfrac{1}{x} } \\ \\ \\ \implies \sf \bigg( \frac{3 - \sqrt{13} }{2} + \frac{1}{ \frac{3 - \sqrt{13} }{2} } \bigg) ^{2} - 2 = {x}^{2} + \frac{1}{ {x}^{2} } \\ \\ \\ \implies \sf \bigg( \frac{3 - \sqrt{13} }{2} + \frac{2}{3 - \sqrt{13} } \bigg) ^{2} - 2 = {x}^{2} + \frac{1}{ {x}^{2} } \\ \\ \\ \implies \sf \bigg( \frac{(3 - \sqrt{13} ) ^{2} + 4}{2(3 - \sqrt{13} )} \bigg)^{2} - 2 = {x}^{2} + \frac{1}{ {x}^{2} }$

$\implies \sf \: \bigg( \dfrac{ {3}^{2} + ( \sqrt{13}) ^{2} - 2(3)( \sqrt{13} ) + 4 }{2(3 - \sqrt{13} )} \bigg) ^{2} - 2 = {x}^{2} + \frac{1}{ {x}^{2} } \\ \\ \\ \implies \sf \: \bigg( \frac{9 + 13 - 6 \sqrt{13} + 4}{2(3 - \sqrt{13} )} \bigg) ^{2} - 2 = {x}^{2} + \frac{1}{ {x}^{2} } \\ \\ \\ \implies \sf \: \bigg( \frac{26 - 6 \sqrt{13} }{2(3 - \sqrt{13} )} \bigg) ^{2} - 2 = {x}^{2} + \frac{1}{ {x}^{2} } \\ \\ \\ \implies \sf \: \bigg( \frac{2(13 - 3 \sqrt{13} )}{2(3 - \sqrt{13}) } \bigg) ^{2} - 2 = {x}^{2} + \frac{1}{ {x}^{2} } \\ \\ \\ \implies \sf \: \bigg( \frac{ - \sqrt{13} (3 - \sqrt{13} )}{ 3 - \sqrt{13} } \bigg) ^{2} - 2 = {x}^{2} + \frac{1}{ {x}^{2} } \\ \\ \\ \implies \sf \: {( - \sqrt{13} )}^{2} - 2 = {x}^{2} + \frac{1}{ {x}^{2} } \\ \\ \\ \sf \implies 13 - 2 = {x}^{2} + \frac{1}{ {x}^{2} } \\ \\ \\ \implies \boxed{ \sf 11 = {x}^{2} + \frac{1}{ {x}^{2} }}$

80

$\implies \sf \: \bigg( \dfrac{ {3}^{2} + ( \sqrt{13}) ^{2} - 2(3)( \sqrt{13} ) + 4 }{2(3 - \sqrt{13} )} \bigg) ^{2} - 2 = {x}^{2} + \frac{1}{ {x}^{2} }$

$\\ \\ \\ \implies \sf \: \bigg( \frac{9 + 13 - 6 \sqrt{13} + 4}{2(3 - \sqrt{13} )} \bigg) ^{2} - 2 = {x}^{2} + \frac{1}{ {x}^{2} } </p><p></p><p>[tex] \\ \\ \\ \implies \sf \: \bigg( \frac{26 - 6 \sqrt{13} }{2(3 - \sqrt{13} )} \bigg) ^{2} - 2 = {x}^{2} + \frac{1}{ {x}^{2} }$

$\\ \\ \\ \implies \sf \: \bigg( \frac{2(13 - 3 \sqrt{13} )}{2(3 - \sqrt{13}) } \bigg) ^{2} - 2 = {x}^{2} + \frac{1}{ {x}^{2} }$

$\\ \\ \\ \implies \sf \: \bigg( \frac{ - \sqrt{13} (3 - \sqrt{13} )}{ 3 - \sqrt{13} } \bigg) ^{2} - 2 = {x}^{2} + \frac{1}{ {x}^{2} }$

$\\ \\ \\ \implies \sf \: {( - \sqrt{13} )}^{2} - 2 = {x}^{2} + \frac{1}{ {x}^{2} }$

$\\ \\ \\ \sf \implies 13 - 2 = {x}^{2} + \frac{1}{ {x}^{2} }$

$\\ \\ \\ \implies \boxed{ \sf 11 = {x}^{2} + \frac{1}{ {x}^{2} }}$