To solve the equation \(3x^3 + 9x^2 - 54x = 0\), we can start by factoring out the common term \(3x\):
\[ 3x(x^2 + 3x - 18) = 0 \]
This gives us one solution:
\[ 3x = 0 \implies x = 0 \]
Next, we need to solve the quadratic \(x^2 + 3x - 18 = 0\). We can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \(a = 1\), \(b = 3\), and \(c = -18\).
Calculating the discriminant:
\[ b^2 - 4ac = 3^2 - 4(1)(-18) = 9 + 72 = 81 \]
Now we can find the roots:
\[ x = \frac{-3 \pm \sqrt{81}}{2 \cdot 1} = \frac{-3 \pm 9}{2} \]
Calculating the two possible values:
- \(x = \frac{-3 + 9}{2} = \frac{6}{2} = 3\)
- \(x = \frac{-3 - 9}{2} = \frac{-12}{2} = -6\)
Thus, the complete set of solutions to the equation \(3x^3 + 9x^2 - 54x = 0\) is:
\[ x = 0, \quad x = 3, \quad x = -6 \]
The correct answer is:
0, 3, −6 (0, 3, negative 6).