Question

x=1/3+2√2 then find the value of x²+1/x²


Please send complete answer​

Answer 1

$\huge\bf{Answer: 34}$

$\huge\mathbb{Solution=>}$

___________________________✴

Given :-

$x = \frac{1}{3 + \sqrt[2]{2} }$

To Find Value Of :-

${x}^{2} + \frac{1}{ {x}^{2} }$

${Solution=>}$

By Rationalising denominator:-

$x = \frac{1}{3 + \sqrt[2]{2} }$

=$\frac{1}{3 + \sqrt[2]{2} } \times \frac{3 + \sqrt[2]{2} }{(3 + \sqrt[2]{2})(3 - \sqrt[2]{2 }) }$

=$\frac{3 + \sqrt[2]{2} }{ {9}^{2} - {8}^{2} }$

Now, we get :-

$x = 3 + \sqrt[2]{2}$

_ _ _ _ _ (1)

Now,find value of 1/x :-

$\frac{1}{x} = \frac{1}{3 + \sqrt[2]{2} }$

Again Rationalising :-

$\frac{1}{3 + \sqrt[2]{2} } \times \frac{3 - \sqrt[2]{2} }{3 - \sqrt[2]{2} }$

$= \frac{3 - \sqrt[2]{2} }{ ({3})^{2 - { (\sqrt[2]{2} )}^{2} } }$

$= \frac{3 - \sqrt[2]{2} }{1}$

_ _ _ _ _ (2)

in last Putting Value in Equation :-

${x}^{2} + \frac{1}{ {x}^{2} }$

as,

$x = 3 + \sqrt[2]{2} \: and \: \frac{1}{x} = 3 - \sqrt[2]{2}$

So,

${(3 + \sqrt[2]{2}) }^{2} + {(3 - \sqrt[2]{2}) }^{2}$

By using Identities :-

$1.) \: {(x + y)}^{2} = {x}^{2} + {y}^{2} + 2xy$

$2.) {(x - y)}^{2} = { x}^{2} + {y}^{2} - 2xy$

We get,

${3}^{2} + {( \sqrt[2]{2)} }^{2} + 2(3)( \sqrt[2]{2}) + {3}^{2} + { \sqrt[(2]{2} )}^{2} - 2(3)( \sqrt[2]{2} )$

$= 9 + 9 + \sqrt[6]{2} +9 + 8 - \sqrt[6]{2}$

$= 18 + 16 = 34$

Hence, Final Answer = 34.

___________________________✴

Answer 2

Answer:

$\huge\boxed{\boxed{\green{34}}}$

Step-by-step explanation:

x = 1/( 3 + 2√2 )

Rationalizing the denominator

We will rationalize the irrational denominator by multiplying the denominator with a conjugate surd .

Hence we can obtain a rational number easily .

⇒ x = 1/( 3 + 2√2 ) × ( 3 - 2√2 )/( 3 - 2√2 )

⇒ x = ( 3 - 2√2 ) / ( 3² - (2√2)² )

⇒ x = ( 3 - 2√2 ) / ( 9 - 4×2 )

⇒ x = ( 3 - 2√2 ) / ( 9 - 8 )

⇒ x = ( 3 - 2√2 ) / 1

⇒ x = 3 - 2√2

Now we need to find the value of 1/x

x = 1/( 3 + √2 )

1/x = 3 + 2√2

This is because we take reciprocal both sides .

Now we will add the squared values of x and 1/x :

x² = ( 3 - 2√2 )²

1/x² = ( 3 + 2√2 )²

x² + 1/x² = ( 3 - 2√2 )² + ( 3 + 2√2 )²

Using the formulas :

( a + b )² = a² + b² + 2 ab

( a - b )² = a² + b² - 2 ab

⇒ x² + 1/x² = 3² + (2√2)² - 2.3.2√2 + 2.3.2√2 + 3² + ( 2√2 )²

⇒ x² + 1/x² = 9 + 8 + 9 + 8

⇒ x² + 1/x² = 18 + 16

⇒ x² + 1/x² = 34

Hence the answer is 34 .