Question

x²+1/x²=51 find x-1/x​

Answer 1

Given :

$\sf \: {x}^{2} + \dfrac{1}{ {x}^{2} } \: = \: 51$

To find :

$\sf \: x - \dfrac{1}{x}$

Identity used :

• ( a+b )² = a² + b² - 2ab

Solution :

$\sf \: \: x - \dfrac{1}{x}$

Taking square

$\sf \: \implies \: \bigg( x - \dfrac{1}{x} { \bigg)}^{2}$

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

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

As we are given that

$\bigg[ \: \sf \: {x}^{2} \: + \: \bigg( \dfrac{1}{x} { \bigg)}^{2} \: = \:51 \: \: \bigg]$

Therefore

$\sf \implies \: \bigg( x - \dfrac{1}{x} { \bigg)}^{2} \: = \: {x}^{2} \: + \: \bigg( \dfrac{1}{x} { \bigg)}^{2} \: - \:2 \: = \: 51 - 2$

$\sf \implies \: \: \bigg( x - \dfrac{1}{x} { \bigg)}^{2} = \: 49$

$\sf \implies \: \bigg( x - \dfrac{1}{x} { \bigg)} \: = \sqrt{49}$

$\sf \implies \: \bigg( x - \dfrac{1}{x} { \bigg)} \: = \: \pm \: 7$

ANSWER : ± 7