Question

Please answer these 2 questions

Answer 1

Answer:

Answer of no 2.1is 10/3

Answer 2

Given : x = (√5 - 2) /(√5 + 2)    , x =  √2 + 1  

To find : x + 1/x  ,  x² + 1/x²    , x³ + 1/x³

Solution:

x = (√5 - 2) /(√5 + 2)

Multiplying & Dividing by  √5 - 2

=> x  =  (√5 - 2)²/(√5 + 2)(√5 - 2)

=> x = ( 5 + 4  - 4√5)/(5 - 4)

=> x = 9 - 4√5

x = (√5 - 2) /(√5 + 2)

=> 1/x = (√5 + 2)/ (√5 - 2)

Multiplying & Dividing by  √5 + 2

=> 1/x =  (5 + 4 + 4√5)/(5 - 4)

=> 1/x = 9 + 4√5

x + 1/x  = 9 - 4√5 + 9 + 4√5   =  18

=> x + 1/x  = 18

x² + 1/x²  = (x + 1/x)²  - 2x(1/x)

= 18² - 2

= 324 - 2

= 322

x² + 1/x²  = 322

x = √2 + 1

1/x = 1/(√2 + 1)  = √2 - 1  

x + 1/x = √2 + 1 +  √2 - 1    = 2√2

x² + 1/x²  =  (x + 1/x)² - 2x(1/x)

= (2√2)² - 2

= 8 - 2

= 6

x² + 1/x²  = 6

x³ + 1/x³  =  (x + 1/x)³  - 3x(1/x)(x + 1/x)

=> x³ + 1/x³  =   (2√2)³ - 3 (2√2)

=  16√2 - 6√2

= 10√2  

x³ + 1/x³  = 10√2  

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