Solve the polynomial equation $8x^3 = 24x^2 - 18x$ by converting it to standard form and applying factoring techniques.

Step 1 — Write in standard form. Subtract $24x^2$ and add $18x$ to both sides to bring all terms to one side: $8x^3 - 24x^2 + 18x = 0$

Step 2 — Factor the greatest common factor. The three terms share a common factor of $2x$. Factor it out: $2x(4x^2 - 12x + 9) = 0$

Step 3 — Factor the trinomial. The remaining trinomial is a perfect square trinomial: $2x(2x - 3)(2x - 3) = 0$

Step 4 — Apply the Zero Product Property. Set each variable factor to zero: $2x = 0 \quad \text{or} \quad 2x - 3 = 0 \quad \text{or} \quad 2x - 3 = 0$

Step 5 — Solve each equation: $x = 0 \quad \text{or} \quad x = \frac{3}{2} \quad \text{or} \quad x = \frac{3}{2}$

The distinct solutions are $x = 0$ and $x = \frac{3}{2}$.