Solve the Cubic Equation: 12x³-9x²-3x = 0

To solve the problem, we follow these steps:

• Step 1: Identify the given cubic equation, $12x^3 - 9x^2 - 3x = 0$.
• Step 2: Look for common factors in the terms of the equation.
• Step 3: Apply the zero-product property to factor and solve the equation.

Let's work through the solution:

Step 1: Observe that each term in the equation $12x^3 - 9x^2 - 3x$ has a common factor of $3x$. So, we can factor $3x$ out of the equation, giving us:

$3x (4x^2 - 3x - 1) = 0$

Step 2: Having factored out $3x$, we now have a product of terms equaling zero. According to the zero-product property, at least one of the factors must be zero:

$3x = 0 \quad \text{or} \quad 4x^2 - 3x - 1 = 0$

This gives us one solution directly:

$x = 0$

Step 3: Solve the quadratic equation $4x^2 - 3x - 1 = 0$ using the quadratic formula, where $a = 4$, $b = -3$, and $c = -1$:

The quadratic formula is:

$x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}$

Applying it to our equation:

$x = \frac{{-(-3) \pm \sqrt{{(-3)^2 - 4 \cdot 4 \cdot (-1)}}}}{2 \cdot 4}$

$x = \frac{{3 \pm \sqrt{{9 + 16}}}}{8}$

$x = \frac{{3 \pm \sqrt{25}}}{8}$

$x = \frac{{3 \pm 5}}{8}$

This gives us two solutions:

When $\sqrt{25} = 5$, $x = \frac{{3 + 5}}{8} = 1$.

When $\sqrt{25} = -5$, $x = \frac{{3 - 5}}{8} = -\frac{1}{4}$.

Therefore, the solutions to the equation $12x^3 - 9x^2 - 3x = 0$ are $x = 0$, $x = 1$, and $x = -\frac{1}{4}$.

Verifying against the provided choices, the correct choice is choice 2: $x=0,x=1,x=-\frac{1}{4}$.