Question

if x=3+√8 then (x²+ 1\x²) =?

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Answer 1

Answer:

34

Step-by-step explanation:

Given,

x = 3 + √8

To Find :-

Value of :-

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

How To Do :-

As the given value of 'x' we need to find the value of '1/x' And by rationalizing it and we need to substitute the value of 'x' and '1/x' in the formula.

Formula Required :-

(a + b)² = a² + b² + 2ab

→ a² + b² = (a + b)² - 2ab

Solution :-

x = 3 + √8

1/x = 1/3 + √8

Rationalizing the denominator :-

$=\dfrac{1}{3+\sqrt{8}}\times \dfrac{3-\sqrt{8}}{3-\sqrt{8}}$

$=\dfrac{3-\sqrt{8}}{(3)^2-(\sqrt{8})^2}$

 

$=\dfrac{3-\sqrt{8}}{9-8}$

$=\dfrac{3-\sqrt{8}}{1}$

= 3 - √8

∴ 1/x = 3 - √8

Substituting the values in the formula :-

x²+ 1/x² = (x + 1/x)² - 2 × x × 1/x

= (x + 1/x)² - 2

= (3 + √8 + 3 - √8)² - 2

= (6)² - 2

= 36 - 2

= 34

∴ x²+ 1\x² = 34