Let's solve the equation \( 3x^3 + 48 = 0 \).

### Step 1: Isolate the cubic term
Subtract 48 from both sides:

\[
3x^3 = -48
\]

### Step 2: Divide by 3
Divide both sides of the equation by 3 to solve for \( x^3 \):

\[
x^3 = -16
\]

### Step 3: Take the cube root
To solve for \( x \), take the cube root of both sides:

\[
x = \sqrt[3]{-16}
\]

Since \(-16\) is a negative number, the cube root will also be negative:

\[
x = -\sqrt[3]{16}
\]

So, the solution is:

\[
x = -\sqrt[3]{16}
\]

### Final Answer
The solution to the equation is \( x = -\sqrt[3]{16} \).

Would you like more details on the steps or have any questions?

Here are five related questions:
1. What are the properties of cube roots when applied to negative numbers?
2. How would the solution change if the equation were \( 3x^3 - 48 = 0 \)?
3. What is the approximate decimal value of \( -\sqrt[3]{16} \)?
4. How do you solve higher-degree polynomial equations, such as quartic equations?
5. Can this method be applied to solve any cubic equation?

**Tip:** When dealing with cubic equations, remember that they can have one real root and two complex roots, or all three roots can be real.

Learn how to solve the cubic equation 3x^3 + 48 = 0 with a detailed step-by-step solution. This problem involves understanding cubic equations and cube roots. Suitable for advanced high school students (Grades 10-12).

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