In the quadratic equation x^2+((k-(1/k))x-1=0 solve for $x$ in terms of $k$.
x^2+((k-(1/k))x-1 = 0
To simplify things, let [ ( k -1/k ) ] = [ k^2 - 1 ] / k = a
So we have
x^2 + ax - 1 complete the square on x
x^2 + ax + a^2/4 = 1 + a^2/4 factor
( x + a/2)^2 = 1 + a^2/4 take both roots
x + a/2 = ±√ ( [ 4 + a^2 ] / 4 ]
x = ±√[4 + a^2] / 2 - a/2
±√ [ 4 + [ k^2 - 1]^2 ] - [ k^2 -1]
x = ______________________________
2k
±√ [ 4 + k^4 - 2k^2 + 1 ] - k^2 + 1
x = _______________________________
2k
±√ [ k^4 - 2k^2 + 3 ] - k^2 + 1
x = ____________________________
2k