Question

If x=9-4√5 then what is the value of x²-1/x²

Answer 1

Xsquare=81+80-72root5=161-72root5
So,
161-72root5 - 1/161-72root5

Answer 2

Hi ,

x = 9 - 4√5 ---( 1 )

1/x = 1/ ( 9 - 4√5 )

= ( 9 + 4√5 ) /[ (9-4√5 )(9+4√5)]

= ( 9 + 4√5 ) / [ 9² - ( 4√5 )² ]

= ( 9 + 4√5 ) / ( 81 - 80 )

= 9 + 4√5 ---( 2 )

Now ,

x + 1/x = 9 - 4√5 + 9 + 4√5

= 18 ---( 3 )

x - 1/x = 9 - 4√5 - ( 9 + 4√5 )

= 9 - 4√5 - 9 - 4√5

= - 8√5 ---( 4 )

FROM ( 3 ) and ( 4 ) ,

x² - 1/x² = ( x + 1/x ) ( x - 1/x )

= 18 × ( - 8√5 )

= - 144√5

I hope this helps you.

: )