Question

if x=9-4√5, find the value of x² + 1/x²​

Answer 1

Question :–

▪︎ If ${ \bold{x = 9 - 4 \sqrt{5} }}$ then find the value of ${ \bold{ {x}^{2} + \dfrac{1}{ {x}^{2} } = ? }}$

ANSWER :–

$\\ \longrightarrow{ \red {\boxed{ \bold{ {x}^{2} + \dfrac{1}{ {x}^{2} } = 322 }}}}$

EXPLANATION :–

GIVEN :–

$\\ { \implies \: \bold{x = 9 - 4 \sqrt{5} }}$

TO FIND :–

$\\ \implies { \bold{ {x}^{2} + \dfrac{1}{ {x}^{2} } = ? }}$

SOLUTION :–

• First , we have to find $\: \: \: { \bold{ \dfrac{1}{x} \: \: - }}$

$\\ \implies{ \bold{ \dfrac{1}{x} \: \: = \frac{1}{9 - 4 \sqrt{5} } }}$

• Now Rationalization –

$\\ \implies{ \bold{ \dfrac{1}{x} \: \: = \frac{1}{9 - 4 \sqrt{5} } \times \frac{9 + 4 \sqrt{5} }{9 + 4 \sqrt{5} } }}$

$\\ \implies{ \bold{ \dfrac{1}{x} \: \: = \frac{9 + 4 \sqrt{5} }{(9)^{2} - (4 \sqrt{5}) ^{2} } }}$

$\\ \implies{ \bold{ \dfrac{1}{x} \: \: = \frac{9 + 4 \sqrt{5} }{81 - 80 } }}$

$\\ \implies{ \bold{ \dfrac{1}{x} \: \: = 9 + 4 \sqrt{5} }}$

• We know a property –

$\\ \implies{ \pink{ \boxed{\bold{ {(a + b)}^{2} = {a}^{2} + {b}^{2} + 2ab }}}}$

• So that –

$\\ \implies{ \bold{ {(x + \frac{1}{x} )}^{2} = {x}^{2} + \dfrac{1}{ {x}^{2} } + 2( \cancel x)( \frac{1}{ \cancel x} ) }}$

$\\ \implies{ \bold{ {(x + \frac{1}{x} )}^{2} = {x}^{2} + \dfrac{1}{ {x}^{2} } + 2 }}$

• We should write this as –

$\\ \implies{ \bold{ {x}^{2} + \dfrac{1}{ {x}^{2} } = {(x + \frac{1}{x} )}^{2} - 2 }}$

• Now put the values –

$\\ \implies{ \bold{ {x}^{2} + \dfrac{1}{ {x}^{2} } = {(9 - \cancel{4 \sqrt{5} } +9 + { \cancel{4 \sqrt{5} }} )}^{2} - 2 }}$

$\\ \implies{ \bold{ {x}^{2} + \dfrac{1}{ {x}^{2} } = {(18)}^{2} - 2 }}$

$\\ \implies{ \bold{ {x}^{2} + \dfrac{1}{ {x}^{2} } = 324 - 2 }}$

$\\ \implies{ \boxed{ \bold{ {x}^{2} + \dfrac{1}{ {x}^{2} } = 322 }}}$