Question

If x = 9 - 4 root 5 find the value of x square plus one equal x square

Answer 1

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Answer 2

Question :

If x = 9 - 4√5 , find the value of x² + 1/x² .

Answer :

x² + 1/x² = 322

Solution :

• Given : x = 9 - 4√5
• To find : x² + 1/x² = ?

We have ;

x = 9 - 4√5

Thus ,

1/x = 1/(9 - 4√5)

Now ,

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

=> 1/x = (9 + 4√5) / (9 - 4√5)(9 + 4√5)

=> 1/x = (9 + 4√5) / [ 9² - (4√5)² ]

=> 1/x = (9 + 4√5) / (81 - 80)

=> 1/x = (9 + 4√5) / 1

=> 1/x = 9 + 4√5

Also ,

We know that ,

(a + b)² = a² + b² + 2ab

Thus ,

=> (x + 1/x)² = x² + (1/x)² + 2•x•(1/x)

=> (9 - 4√5 + 9 + 4√5)² = x² + 1/x² + 2

=> 18² = x² + 1/x² + 2

=> 324 = x² + 1/x² + 2

=> x² + 1/x² = 324 - 2

=> x² + 1/x² = 322

Hence ,

x² + 1/x² = 322