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 side

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

Putting these values in the given equation

`2(y^2 - 2) - y = 11`

``=> 2y^2 - 4 - y - 11 = 0`

`=> 2y^2 - y - 15 = 0`

`=> 2y^2 - 6y + 5y - 15 = 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`