What are the roots of - 12x2 + 11x - 3 = 0?

This quadratic does not have any real roots.

It is a parabola that opens downward with vertex at (11/24, -.48) which means the vertex is below the x-axis and therefore will not have any real roots.

Imaginary roots would be

x = [11 + - sqrt (121 - 4 (-12) (-3) ] / (-24)

x = [11 + - i sqrt (23) ] / (-24)