Question

if x = 2 + root3, then find the value of:
a) x + 1/x
b) x^2 + 1/x^2

Answer 1

Answer:

4 and 14

Step-by-step explanation:

   ⇒ x = 2 + √3

⇒ 1 / x = 1 / ( 2 + √3 )

    Multiply as well as divide RHS by 2 - √3 :

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

⇒ 1 / x = ( 2 - √3 ) / ( 2^2 - √3^2 )

⇒ 1 / x = ( 2 - √3 ) / ( 4 - 3 )

⇒ 1 / x = ( 2 - √3 ) / 1

⇒ 1 / x = 2 - √3

  Hence,

⇒ x + 1 / x

  = 2 + √3 + 2 - √3    

  = 2 + 2 = 4

Square on both sides:

⇒ ( x + 1 / x )^2 = 4^2

⇒ x^2 + 1 / x^2 + 2(x*1/x) = 16

⇒ x^2 + 1 / x^2 + 2 = 16

⇒ x^2 + 1 / x^2 = 16 - 2 = 14