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The equation provided is:

\[
9(x - 8)(2x + 9) = 0
\]

We can solve this using the **zero-product principle**, which states that if the product of two or more factors equals zero, at least one of the factors must be zero. So, we will set each factor equal to zero and solve for \(x\).

### Step 1: Solve each factor

1. From \( 9(x - 8) = 0 \):
   \[
   x - 8 = 0 \implies x = 8
   \]

2. From \( (2x + 9) = 0 \):
   \[
   2x + 9 = 0 \implies 2x = -9 \implies x = -\frac{9}{2}
   \]

### Step 2: Write the solution set

Thus, the solution set is:
\[
x = 8, -\frac{9}{2}
\]

Would you like any further details or have any questions?

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Here are some related questions for further exploration:

1. How does the zero-product principle work for more than two factors?
2. Can you explain why the zero-product principle applies in this case?
3. What happens if one of the terms had been a quadratic equation?
4. How would the solution change if the equation was not set equal to zero?
5. How would you factor a more complex expression, like a cubic equation?

**Tip:** Always ensure the equation is set to zero before using the zero-product principle to solve it!

Solve the equation using the zero-product principle: 9(x - 8)(2x + 9) = 0. What is the solution set?

Learn how to solve quadratic equations using the Zero-Product Principle with a detailed step-by-step solution. This guide helps you understand how setting each factor to zero helps find the solution set. Suitable for high school students studying algebra.

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