To solve the equation \(3x^3 + 9x^2 - 54x = 0\), we can start by factoring out the greatest common factor (GCF).
- First, factor out the GCF which is \(3x\):
\[ 3x(x^2 + 3x - 18) = 0 \]
- Now we have a product of factors equal to zero, so we can set each factor to zero:
\[ 3x = 0 \quad \text{or} \quad x^2 + 3x - 18 = 0 \]
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From the first factor: \[ 3x = 0 \implies x = 0 \]
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Now, we need to solve the quadratic equation \(x^2 + 3x - 18 = 0\). We can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \(a = 1\), \(b = 3\), and \(c = -18\).
- Calculate the discriminant:
\[ b^2 - 4ac = 3^2 - 4 \cdot 1 \cdot (-18) = 9 + 72 = 81 \]
- Now apply the quadratic formula:
\[ x = \frac{-3 \pm \sqrt{81}}{2 \cdot 1} = \frac{-3 \pm 9}{2} \]
- This gives us two solutions:
\[ x = \frac{6}{2} = 3 \quad \text{and} \quad x = \frac{-12}{2} = -6 \]
- Therefore, the complete set of solutions to the equation \(3x^3 + 9x^2 - 54x = 0\) is:
\[ x = 0, \quad x = 3, \quad x = -6 \]
So the complete set of solutions is: \[ { 0, 3, -6 } \]