Question

if x = 1/9+4√5 find: 1. x +1/x 2.x²+1/x²

Answer 1

Answer :

• x + 1/x = 18

• x² + 1/x² = 322

Solution :

• Given : x = 1/(9 + 4√5)
• To find : x + 1/x , x² + 1/x²

We have ;

x = 1/(9 + 4√5)

Now ,

Rationalising the denominator of the term in RHS , we have ;

=> x = (9 - 4√5) / (9 + 4√5)(9 - 4√5)

=> x = (9 - 4√5) / [ 9² - (4√5)² }

=> x = (9 - 4√5) / (81 - 80)

=> x = (9 - 4√5) / 1

=> x = 9 - 4√5

Also ,

=> x = 1/(9 + 4√5)

=> 1/x = 9 + 4√5

Now ,

=> x + 1/x = 9 - 4√5 + 9 + 4√5

=> x + 1/x = 18

Now ,

Squaring both sides , we have ;

=> (x + 1/x)² = 18²

=> x² + (1/x)² + 2•x•(1/x) = 324

=> x² + 1/x² + 2 = 324

=> x² + 1/x² = 324 - 2

=> x² + 1/x² = 322

Hence ,

• x + 1/x = 18

• x² + 1/x² = 322