Question

if X = 3+2√2 , find the value of X ^ 2 + 1/x^2 ​

Answer 1

Solution

Given :-

• x = 3 + 2√2

Find :-

• Value of x² + 1/x²

Explanation

First Calculate 1/x

➡ 1/x = 1/(3+2√2)

Rationalize denominator

➡1/x = (3-2√2)/(3-2√2)(3+2√2)

➡1/x = (3-2√2)/(9-8)

➡1/x = (3-2√2)

Squaring both side,

➡ 1/x² = (3-2√2)²

➡1 )/x² = [3² + (2√2)² - 2×3×(2√2)]

➡1/x² = 9 + 8 - 12√2

➡1/x² = 17 - 12√2

Now, calculate,

➡ x² + 1/x² = (3+2√2) + (17 - 12√2)

➡x² + 1 /x² = 20 - 10√2

Hence

• Value of x² + 1 /x² be = 10(2 - √2)

_______________

Answer 2

Step-by-step explanation:

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

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

$= \frac{3 - 2 \sqrt{2} }{9 - 8} = 3 - 2 \sqrt{2}$

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

$( {x}^{2} + \frac{1}{ {x}^{2} } ) = 3 + 2 \sqrt{2} + 3 - 2 \sqrt{3} {)}^{2} - 2$

$( {x}^{2} + \frac{1}{ {x}^{2} }) = 36 - 2 \\ = 34$