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

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