Question

which of the following is a complete set of solutions to 3x^3+9x^2−54x=0 ?

Answer 1

To solve the equation \(3x^3 + 9x^2 - 54x = 0\), we can start by factoring out the greatest common factor (GCF).

1. First, factor out the GCF which is \(3x\):

\[
3x(x^2 + 3x - 18) = 0
\]

2. Now we have a product of factors equal to zero, so we can set each factor to zero:

\[
3x = 0 \quad \text{or} \quad x^2 + 3x - 18 = 0
\]

3. From the first factor:
\[
3x = 0 \implies x = 0
\]

4. 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 \(c = -18\).

5. Calculate the discriminant:

\[
b^2 - 4ac = 3^2 - 4 \cdot 1 \cdot (-18) = 9 + 72 = 81
\]

6. Now apply the quadratic formula:

\[
x = \frac{-3 \pm \sqrt{81}}{2 \cdot 1} = \frac{-3 \pm 9}{2}
\]

7. This gives us two solutions:

\[
x = \frac{6}{2} = 3 \quad \text{and} \quad x = \frac{-12}{2} = -6
\]

8. 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:
\[
\{ 0, 3, -6 \}
\]