If x=1/(x-5),find:x^2 - 1/x^2.

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justaguide | College Teacher | (Level 2) Distinguished Educator

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We are given that x = 1 / (x - 5). We need to find x^2 - (1/x^2)

x = 1/(x - 5)

=> x^2 - 5x = 1

=> x^2 = 1 + 5x

=> x^2 - 5x - 1 = 0

solving the quadratic equation, we get two roots:

x1 = 5/2 + sqrt(25 + 4)/2

=> (5 + sqrt 29)/2

=> x1^2 = (25 + 29 + 10*sqrt 29)/4

x2 = (5 - sqrt 29)/2

=> x2^2 = (25 + 29 - 10*sqrt 29)/4

  • x = 1/(x- 5)

=> x - 5 = 1/x

=> x = 1/x + 5

square both the sides

=> x^2 = 1/x^2 + 25 + 10/x

=> x^2 - 1/x^2 = 25 + 10/x

25 + 10/x

for x1 = (5 + sqrt 29)/2

=> 25 + 20/(5 + sqrt 29)

=> 25 + 20(5 - sqrt 29)/(25 - 29)

=> 25 - 5(5 - sqrt 29)

=> 25 - 25 + 5*sqrt 29

=> 5*sqrt 29

for x2 = (5 - sqrt 29)/2

=> 25 + 20/(5 - sqrt 29)

=> 25 + 20(5 + sqrt 29)/(25 - 29)

=> 25 - 5(5 + sqrt 29)

=> 25 - 25 - 5*sqrt 29

=> -5*sqrt 29

The value of x^2 - 1/x^2 is either 5*sqrt 29 or -5*sqrt 29

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