Question

if x=3+^8,find the volue of x2+ 1÷x2​

Answer 1

Question :

If x = 3 + √8 , then find the value of x² + 1/x² .

Answer :

x² + 1/x² = 34

Solution :

• Given : x = 3 + √8
• To find : x² + 1/x² = ?

We have ,

x = 3 + √8

Thus ,

1/x = 1/(3 + √8)

Now ,

Rationalising the denominator in RHS ,

We have ;

=> 1/x = (3 - √8) / (3 + √8)(3 - √8)

=> 1/x = (3 - √8) / [ 3² - (√8)² ]

=> 1/x = (3 - √8) / (9 - 8)

=> 1/x = 3 - √8

Now ,

We know that ,

(a + b)² = a² + b² + 2ab

Thus ,

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

=> (x + 1/x)² = x² + 1/x² + 2

=> (3 + √8 + 3 - √8)² = x² + 1/x² + 2

=> 6² = x² + 1/x² + 2

=> 36 = x² + 1/x² + 2

=> x² + 1/x² = 36 - 2

=> x² + 1/x² = 34

Hence,

x² + 1/x² = 34

Answer 2

Question :-

• If x = 3 + √8 , find the value of x² + 1/x².

Answer :-

• x² + 1/x² = 34

Solution :-

• Given = x = 3 + √8
• To find = x² + 1/x² = ?

We have ,

x = 3 + √8

Thus ,

1/x = 1/(3 + √8)

Now ,

Retionalisling the denominator in RHS ,

We have ,

→ 1/x = (3 - √8) / (3 + √8)(3 - √8)

→ 1/x = (3 - √8) / [ 3² - (√8)² ]

→ 1/x = (3 - √8) / (9 - 8)

→ 1/x = 3 - √8

Now ,

★ We know that ★

(a6 + b)² = a² + b² + 2ab

Thus ,

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

→ (x + 1/x)² = x² + 1/x² + 2

→ (3 + √8 + 3 - √8)² = x² + 1/x² + 2

→ 6² = x² + 1/x² + 2

→ 36 = x² + 1/x² + 2

→ x² + 1/x² = 36 - 2

→ x² + 1/x² = 34

Hence,

• x² + 1/x² = 34