Question

rationalize where x=5+2root5 y=1/x find x2 + y2

Answer 1

$Heya$‼

Given that, x = 5+2√5 and
y = 1 / x
= 1 / 5+2√5
= 1 / 5 + 2√5 × 5–2√5 / 5–2√5
= 5–2√5 / ( 5² –2√5² )
= 5 –2√5 / ( 25 – 20 )
= 5 – 2√5 / 5

Now, x² + y²

= ( 5+2√5 )² + ( 5 – 2√5 / 5 )²

= ( 25 + 20 + 20√5 ) + [ ( 25 + 20 – 20√5 ) / 25 ]

= ( 45 + 20√5 ) + [ ( 45 – 20√5 ) / 25 ]

= 5 ( 9 + 4√5 ) + 5 ( 9 – 4√5 ) / 25

= 5 ( 9 + 4√5 ) + ( 9 – 4√5 ) / 5

= [ 25 ( 9 + 4√5 ) + ( 9 – 4√5 ) ] / 5

= ( 225 + 100√5 + 9 – 4√5 ) / 5

= ( 216 + 96√5 ) / 5

= 24 ( 9 + 4√5 ) / 5