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To solve the inequality \( x^2 + 15x + 54 < 0 \), we first need to find the roots of the quadratic equation. These roots will help us determine the intervals where the expression is negative. Here's how we solve it step by step:

### Step 1: Solve the equation \( x^2 + 15x + 54 = 0 \).
To find the roots, we can use the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \( a = 1 \), \( b = 15 \), and \( c = 54 \).

\[
x = \frac{-15 \pm \sqrt{15^2 - 4 \cdot 1 \cdot 54}}{2 \cdot 1}
\]
\[
x = \frac{-15 \pm \sqrt{225 - 216}}{2}
\]
\[
x = \frac{-15 \pm \sqrt{9}}{2}
\]
\[
x = \frac{-15 \pm 3}{2}
\]

This gives us two roots:
\[
x_1 = \frac{-15 + 3}{2} = -6 \quad \text{and} \quad x_2 = \frac{-15 - 3}{2} = -9
\]

### Step 2: Determine the intervals
The roots are \( x = -9 \) and \( x = -6 \). These roots divide the number line into three intervals:

1. \( (-\infty, -9) \)
2. \( (-9, -6) \)
3. \( (-6, \infty) \)

### Step 3: Test the sign of the quadratic in each interval
We need to test each interval to see where \( x^2 + 15x + 54 < 0 \):

- **Interval 1: \( (-\infty, -9) \)**  
  Pick \( x = -10 \):  
  \( x^2 + 15x + 54 = 100 - 150 + 54 = 4 \) (Positive)

- **Interval 2: \( (-9, -6) \)**  
  Pick \( x = -8 \):  
  \( x^2 + 15x + 54 = 64 - 120 + 54 = -2 \) (Negative)

- **Interval 3: \( (-6, \infty) \)**  
  Pick \( x = -5 \):  
  \( x^2 + 15x + 54 = 25 - 75 + 54 = 4 \) (Positive)

### Step 4: Conclusion
The quadratic expression is negative only in the interval \( (-9, -6) \).

### Final Answer
The solution set is:  
\[
(-9, -6)
\]

Would you like further details or have any questions?

### Related Questions:
1. How do you determine the sign of a quadratic expression without testing intervals?
2. What are the steps to graphically represent the solution of a quadratic inequality?
3. How do we solve inequalities when the quadratic has no real roots?
4. How can the discriminant be used to predict the nature of the solution set for quadratic inequalities?
5. Can quadratic inequalities have multiple solution intervals?

**Tip:** Always check the intervals around the roots to determine where the quadratic expression is positive or negative.

This lesson teaches how to solve the quadratic inequality x^2 + 15x + 54 < 0 using the quadratic formula. Learn about testing intervals and finding the correct solution set. Perfect for students in Grades 9-10 studying quadratic inequalities.

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