81x^3 - 192 = 0
3(27x^3 - 64) = 0
this is now a difference of cubes
(a^3 - b^3) = (a - b)(a^2 + ab + b^2)
where a = 3x and b = 4
so (27x^3 - 64) = (3x - 4)(9x^2 + 12x + 16)
the root to the first factor is x = 4/3
the roots of the second factor are complex, and can be obtained from the quadratic formula
a = 9 b = 12 c = 16
x = [-12 +/- sqrt(144 - 4(9)(16))] / (2*9)
x = [-12 +/- sqrt(-432)] / (18)
note that sqrt(-432) = sqrt(-1 * 144 * 3) = 12i sqrt(3)
so x = [-12 +/- 12i sqrt(3)] / 18
x = -2/3 +/- 2i sqrt(3) / 3
all three roots: x = 4/3 , -2/3 +/- 2i sqrt(3) / 3