Question

if a=3+√5/2 then find the value of a^2+1/a^2​

Answer 1

Answer:

7

Step-by-step explanation:

Given ----> a = ( 3 + √5 ) / 2

To find -----> Value of a² + ( 1 / a² )

Solution ----> ATQ,

a = ( 3 + √5 ) / 2

1 / a = 1 / ( 3 + √5 ) / 2

= 2 / ( 3 + √5 )

= 2 ( 3 - √5 ) / ( 3 + √5 ) ( 3 - √5 )

= 2 ( 3 - √5 ) / { ( 3 )² - ( √5 )² }

= 2 ( 3 - √5 ) / ( 9 - 5 )

= 2 ( 3 - √5 ) / 4

1 / a = ( 3 - √5 ) / 2

Now,

a + ( 1 / a ) = ( 3 + √5 ) / 2 + ( 3 - √5 ) / 2

= ( 3 + √5 + 3 - √5 ) / 2

= 6 / 2

=>a + ( 1 / a ) = 3

Squaring both sides we get,

=> { a + ( 1 / a ) }² = ( 3 )²

We have an identity ,

( a + b )² = a² + b² + 2ab , applying it here we get,

=> a² + ( 1 / a )² + 2 a ( 1 / a ) = 9

=> a² + ( 1 /a² ) + 2 = 9

=> a² + ( 1 / a² ) = 9 - 2

=> a² + ( 1 / a² ) = 7