The problem asks us to find all roots of the equation $\frac{x^3}{6} - 8x = \frac{x^2}{3}$.

1. Problem Description

The problem asks us to find all roots of the equation $\frac{x^3}{6} - 8x = \frac{x^2}{3}$.

2. Solution Steps

First, let's multiply both sides of the equation by 6 to eliminate the fractions:

$6(\frac{x^3}{6} - 8x) = 6(\frac{x^2}{3})$

$x^3 - 48x = 2x^2$

Next, let's move all terms to one side of the equation:

$x^3 - 2x^2 - 48x = 0$

Now, we can factor out an $x$ from the equation:

$x(x^2 - 2x - 48) = 0$

We can further factor the quadratic expression:

$x(x-8)(x+6) = 0$

Finally, we can find the roots by setting each factor equal to zero:

$x = 0$

$x - 8 = 0 \implies x = 8$

$x + 6 = 0 \implies x = -6$

Therefore, the roots are $x = 0, x = 8, x = -6$.

3. Final Answer

$x = -6, 0, 8$