Question

If x=2+3 (under root)2, find the value of (x+4/x)

Answer 1

QUESTION :–

▪︎ If $\: \: { \bold{x = 2 + 3\sqrt{2} }} \: \:$ , then find the value of $\: \: { \bold{x + \dfrac{4}{x} = ? }} \: \:$

ANSWER :–

GIVEN :–

$\\ \: \:{ \huge{. \: \: \: }} { \bold{x = 2 + 3\sqrt{2} }} \: \:$

TO FIND :–

$\\ \: \:{ \huge{. \: \: \: }} { \bold{x + \dfrac{4}{x} = ? }} \: \:$

SOLUTION :–

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

• So that –

$\\ \implies { \bold{ \dfrac{1}{x} = \dfrac{1}{2 + 3 \sqrt{2} } }} \: \:$

• Now rationalization –

$\\ \implies { \bold{ \dfrac{1}{x} = \dfrac{1}{2 + 3 \sqrt{2} } \times \dfrac{2 - 3 \sqrt{2} }{2 - 3 \sqrt{2} } }} \: \:$

$\\ \implies { \bold{ \dfrac{1}{x} = \dfrac{2 - 3 \sqrt{2} }{(2 + 3 \sqrt{2} )(2 - 3 \sqrt{2} )} }} \: \:$

$\\ \implies { \bold{ \dfrac{1}{x} = \dfrac{2 - 3 \sqrt{2} }{ {(2)}^{2} - {(3 \sqrt{2} )}^{2} } \: \: \: \: \: [ \because \: \: (a + b)(a - b) = {a}^{2} - {b}^{2} ]}} \: \:$

$\\ \implies { \bold{ \dfrac{1}{x} = \dfrac{2 - 3 \sqrt{2} }{ 4 - 18 } }} \: \:$

$\\ \implies { \bold{ \dfrac{1}{x} = \dfrac{2 - 3 \sqrt{2} }{ - 14} }} \: \:$

• Now –

$\\ \implies { \bold{x + \dfrac{4}{x} = 2 + 3 \sqrt{2} + 4(\dfrac{2 - 3 \sqrt{2} }{ - 14}) }} \: \:$

$\\ \implies { \bold{x + \dfrac{4}{x} = 2 + 3 \sqrt{2} + \dfrac{6}{7} \sqrt{2} - \dfrac{4}{7} }} \: \:$

$\\ \implies \large { \red{ \boxed{ \bold{x + \dfrac{4}{x} = \frac{10}{7} + \frac{27 \sqrt{2} }{7} }}}} \: \:$