Question

If x = 5 - 2√6, find `x^2 + 1/x^2`

Answer 1

Given `x = 5 - 2sqrt6`

We need to find `x^2 + 1/x^2`

Since x = `5 - 2sqrt6` , we have

`1/x = 1/[ 5 - 2sqrt6]`

⇒ `1/x = 1/[ 5 - 2sqrt6 ] xx [ 5 + 2sqrt6 ]/[ 5 + 2sqrt6 ]`

⇒ `1/x = ( 5 - 2sqrt6 )/[( 5 - 2sqrt6 )( 5 + 2sqrt6 )]`

⇒ `1/x = ( 5 + 2sqrt6)/ [ 5^2 - (2sqrt6)^2 ]`

⇒ `1/x = ( 5 + 2sqrt6)/ [ 25 - 24 ]`

⇒ `1/x = ( 5 + 2sqrt6)/1`

⇒ `1/x = ( 5 + 2sqrt6)`                          .....(1)

Thus,`( x - 1/x ) = ( 5 - cancel(2sqrt6) ) - ( 5 + cancel(2sqrt6))` = 10

`x^2 + 1/x^2 = (x + 1/x)^2 - 2`

`x^2 + 1/x^2= (10)^2 - 2`

 `x^2 + 1/x^2= 100 - 2`

`x^2 + 1/x^2 = 98`