Question

if x=2+√3, find the value of x²+1/x²

Answer 1

Heya ☺

Given that

x = 2 + √3

1/x = 1/2 + √3

= 1 × (2 - √3)/(2 + √3) (2 - √3)

= (2 - √3)/(2^2 - √3^2)

= (2 - √3)/4 - 3

= (2 - √3)

Therefore ,

x^2 = (2 + √3)

= (2)^2 + (√3)^2 + 2 × 2 × √3

= 4 + 3 + 4√3

= 7 + 4√3

1/x^2 = (2 - √3)^2

= (2)^2 + (√3)^2 - 2 × 2 × √3

= 4 + 3 - 4√3

= 7 - 4√3

x^2 + 1/x^2

= (7 + 4√3) + (7 - 4√3)

= 7 + 4√3 + 7 - 4√3

= 7 + 7 + 4√3 - 4√3

= 14



Thanks

Answer 2

Hey mate ^_^

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Given :

x = 2 + √3

To find :

x² + 1 / x²

Solution ;

x = 2 + √3

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

⇒ 1 / x = 2 - √3 / 2² - √3²

⇒ 1 / x = 2 - √3 / 4 - 3

⇒ 1 / x = 2 - √3

Now,

x + 1 / x = 2 + √3 + 2 - √3

⇒ x + 1 / x = 2 + 2

⇒ x + 1 / x = 4

And, on squaring both sides.

( x + 1 / x ) ² = (4)²

⇒ x² + 1 / x² + 2 = 16

⇒ x² + 1 / x² = 16 - 2

⇒ x² + 1 / x² = 14.

Hence,

x² + 1 / x² = 14.

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Thanks for the question!

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