Question

If (x=9-4√5), find (x^2+1/x^2).​

Answer 1

Answer:

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x = 9 - 4√5

1/x = 1/( 9 - 4√5 )

Rationalize

1/( 9 - 4√5 ) × ( 9 + 4√5 )/( 9 + 4√5 )

( 9 + 4√5 )/[ 9² - (4√5)² ] = ( 9 + 4√5 )/( 81 - 80 )

= 9 + 4√5

x² = ( 9 - 4√5 )² = 81 + 80 - 72√5

(1/x)² = ( 9 + 4√5 )² = 81 + 80 + 72√5

161 - 72√5 - 161 - 72√5 = - 144√5

Or

x² - (1/x)² = ( x + 1/x )( x - 1/x )

( 9 - 4√5 + 9 + 4√5 )[ 9 - 4√5 - ( 9 + 4√5 ) ]

( 18 ) ( - 8√5 ) = - 144√5

Hope it helps

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