Question

solve this question please​

Answer 1

Answer:

18 and 322

Step-by-step explanation:

for explanation look at the solution I posted.

Answer 2

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

Given that,

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

On rationalizing the denominator, we get

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

$\rm :\longmapsto\:x = \dfrac{ {( \sqrt{5} - 2)}^{2} }{ {( \sqrt{5})}^{2} - {(2)}^{2} }$

$\red{\bigg \{ \because \:(x + y)(x - y) = {x}^{2} - {y}^{2} \bigg \}}$

$\rm :\longmapsto\:x = \dfrac{5 + 4 - 4 \sqrt{5} }{5 - 4}$

$\red{\bigg \{ \because \: {(x - y)}^{2} = {x}^{2} + {y}^{2} - 2xy \bigg \}}$

$\rm :\longmapsto\:x = \dfrac{9 - 4 \sqrt{5} }{1}$

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

Now,

Consider,

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

$\rm \: \: = \: \dfrac{1}{9 - 4 \sqrt{5} }$

$\rm \: \: = \: \dfrac{1}{9 - 4 \sqrt{5} } \times \dfrac{9 + 4 \sqrt{5} }{9 + 4 \sqrt{5} }$

$\rm \: \: = \: \dfrac{9 + 4 \sqrt{5} }{ {(9)}^{2} - {(4 \sqrt{5}) }^{2} }$

$\rm \: \: = \: \dfrac{9 + 4 \sqrt{5} }{81 - 80}$

$\rm \: \: = \: \dfrac{9 + 4 \sqrt{5} }{1}$

$\rm \: \: = \: 9 + 4 \sqrt{5}$

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

Hence,

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

$\rm \: \: = \:9 - 4 \sqrt{5} + 9 + 4 \sqrt{5}$

$\rm \: \: = \: 18$

Therefore,

$\rm :\implies\:x + \dfrac{1}{x} = 18$

On squaring both sides, we get

$\rm :\longmapsto\: {\bigg(x + \dfrac{1}{x} \bigg) }^{2} = {(18)}^{2}$

$\rm :\longmapsto\: {x}^{2} + \dfrac{1}{ {x}^{2} } + 2 \times x \times \dfrac{1}{x} = 324$

$\rm :\longmapsto\: {x}^{2} + \dfrac{1}{ {x}^{2} } + 2 = 324$

$\rm :\longmapsto\: {x}^{2} + \dfrac{1}{ {x}^{2} } = 324 - 2$

$\rm :\longmapsto\: {x}^{2} + \dfrac{1}{ {x}^{2} } = 322$

Additional Information :-

More Identities to know:

(a + b)² = a² + 2ab + b²

(a - b)² = a² - 2ab + b²

a² - b² = (a + b)(a - b)

(a + b)² = (a - b)² + 4ab

(a - b)² = (a + b)² - 4ab

(a + b)² + (a - b)² = 2(a² + b²)

(a + b)³ = a³ + b³ + 3ab(a + b)

(a - b)³ = a³ - b³ - 3ab(a - b)