Question

if x = 3 + root 8 ; then what is x^2 + 1 /x^2
$3 + \sqrt{8}$

Answer 1

x = 3 + √8

x² = ( 3 + √8 )²

[ ( a + b )² = a² + b² + 2ab ]

= ( 3 )² + ( √8 )² + 2 ( 3 ) ( √8 )

= 9 + 8 + 6√8

= 17 + 6√( 2³ )

x² = 17 + 12√2

1 / x² = 1 / ( 17 + 12√2 )

= [ 1 / ( 17 + 12√2 ) ] × [ ( 17 - 12√2 ) / ( 17 - 12√2 ) ]

[ ( a + b ) ( a - b ) = a² - b² ]

= ( 17 - 12√2 ) / ( 17 )² - ( 12√2 )²

= ( 17 - 12√2 ) / ( 289 - 288 )

1 / x² = 17 - 12√2

Now,

x² + ( 1 / x² ) :

= 17 + 12√2 + 17 - 12√2

= 34

Hence, the answer is 34.