Question

Solve:

`2(x^2 + 1/x^2) - (x + 1/x) = 11`

Answer 1

`2(x^2 + 1/x^2) - (x + 1/x) = 11`

Let `x + 1/x = y`

Squaring on both sides

`x^2 + 1/x^2 = y^2 - 2`

Putting these values in the given equation

2(y² – 2) – y = 11

`=>` 2y² – 4 – y – 11 = 0

`=>` 2y² – y – 15 = 0

`=>` 2y² – 6y + 5y – 15 = 0

`=>` 2y(y – 3) + 5(y – 3) = 0

`=>` (y – 3)(2y + 5) = 0

If y – 3 = 0 or 2y + 5  = 0

Then y = 3 or y =` (-5)/2`

`=> x + 1/x = 3` or `x + 1/x = (-5)/2`

`=> (x^2 + 1)/x = 3` or `(x^2 + 1)/x = (-5)/2`

`=> x^2 - 3x+ 1 = 0` or `2x^2 + 5x + 2 = 0`

`=> x = (-3 +- sqrt((-3)^2 - 4(1)(1)))/(2(1))` or `2x^2 + 4x + x + 2 = 0`

`=> x = (-3 +- sqrt5)/2` or `2x(x + 2) + 1(x + 2) = 0`

Then x = –2 and x = `(-1)/2`