Solve the Cubic Equation: 9x³ - 12x² = 0

To solve this problem, we will factor the given polynomial expression:

Step 1: Identify the greatest common factor (GCF) in the equation $9x^3 - 12x^2 = 0$. The GCF of the terms $9x^3$ and $12x^2$ is $3x^2$.

Step 2: Factor out the GCF from the polynomial:

$9x^3 - 12x^2 = 3x^2(3x - 4) = 0$.

Step 3: Apply the zero-product property. Set each factor equal to zero:

• $3x^2 = 0$
• $3x - 4 = 0$

Step 4: Solve each equation for $x$:

For $3x^2 = 0$, divide by 3:

$x^2 = 0$ → $x = 0$.

For $3x - 4 = 0$, add 4 to both sides and then divide by 3:

$3x = 4$
$x = \frac{4}{3}$.

Thus, the solutions to the equation $9x^3 - 12x^2 = 0$ are $x = 0$ and $x = \frac{4}{3}$.

Therefore, the correct answer is:

$x=0, x=\frac{4}{3}$