Question

if x^3/2=1/x^3/2=2 root 5, x>0, then find the value of x^3-1/x^3. (from rd sharma)

Answer 1

Sorry I don't know this answer

Answer 2

Given :   x^3/2=1/x^3/2=2 root 5, x>0

To Find :  value of x^3-1/x^3

Solution:

x^3/2=1/x^3/2=2 root 5  can be

x^3/2 -  1/x^3/2 = 2√5   or  x^3/2 + 1/x^3/2 = 2√5

case 1 :

if  x^3/2 -  1/x^3/2 = 2√5

Squaring both sides

=> x³  + 1/x³  - 2 = 20

=> x³  + 1/x³  = 22

(x³  + 1/x³ )² = (x³  -  1/x³)²  + 4

=> 22² = (x³  -  1/x³)²  + 4

=> 484 =  (x³  -  1/x³)²  + 4

=>  (x³  -  1/x³)² = 480

=> x³  -  1/x³  = 4√30

case 2 :

If  x^3/2 + 1/x^3/2 = 2√5

Squaring both sides

=> x³  + 1/x³  + 2 = 20

=> x³  + 1/x³  = 18

(x³  + 1/x³ )² = (x³  -  1/x³)²  + 4

=> 18² = (x³  -  1/x³)²  + 4

=> 324 =  (x³  -  1/x³)²  + 4

=>  (x³  -  1/x³)² = 320

=> x³  -  1/x³  = 8√5

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