Question

if x = 2+√5, find the value of (x+1/x)²

Answer 1

Answer:

Given that ,

x = 2 + √5

x^2 + 1 / x ^ 2 = ?

Solution

1/x = 1 / 2 + √ 5 .

BY rationalization ,

= 1 × ( 2 - √ 5 ( ) + 2 + √ 5 ) × ( 2 - √ 5)

= 2 - √ 5 / (2 ) ^ 2 - ( √ 5 ) ^ 2

= 2 - √ 5 / 4 - 5

= 2 - √ 5 ) 4 - 1

= - 2 + √ 5

x^ 2 + 1 / x ^ 2

= ( 2 + √ 5 ) ^ 2 + ( -2 + √ 5 ) ^ 2

= (2)^ 2 + ( √ 5 ) ^ 2 + 2 × 2 + √ 5 + ( - 2 ) ^ 2 + ( √ 5 ) ^ 2 + 2 × ( - 2 )

= 4 + 5 + 4 √ 5 + 4 + 5 - 4 √ 5

= 9 + 4 √ 5 79 - 4 √ 5

= 18 .