To solve the inequality \( x^2 - 3x - 15 \geq 3 \), follow these steps:

1. **Subtract 3 from both sides**:
   \[
   x^2 - 3x - 15 - 3 \geq 0
   \]
   Simplifying, we get:
   \[
   x^2 - 3x - 18 \geq 0
   \]

2. **Factor the quadratic expression**:
   We need to factor the quadratic expression \( x^2 - 3x - 18 \). However, this expression doesn't factor easily with integers, so we'll use the quadratic formula to find the roots.

   The quadratic formula is given by:
   \[
   x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
   \]
   Here, \( a = 1 \), \( b = -3 \), and \( c = -18 \).

3. **Calculate the discriminant**:
   \[
   \text{Discriminant} = (-3)^2 - 4(1)(-18) = 9 + 72 = 81
   \]
   Since the discriminant is 81, we have:
   \[
   x = \frac{3 \pm \sqrt{81}}{2} = \frac{3 \pm 9}{2}
   \]
   This gives us the roots:
   \[
   x = \frac{3 + 9}{2} = 6 \quad \text{and} \quad x = \frac{3 - 9}{2} = -3
   \]

4. **Determine the intervals**:
   The roots divide the number line into three intervals: \( (-\infty, -3) \), \( (-3, 6) \), and \( (6, \infty) \).

5. **Test the intervals**:
   - For \( x \in (-\infty, -3) \), choose \( x = -4 \):
     \[
     (-4)^2 - 3(-4) - 18 = 16 + 12 - 18 = 10 \quad (\text{positive})
     \]
   - For \( x \in (-3, 6) \), choose \( x = 0 \):
     \[
     0^2 - 3(0) - 18 = -18 \quad (\text{negative})
     \]
   - For \( x \in (6, \infty) \), choose \( x = 7 \):
     \[
     7^2 - 3(7) - 18 = 49 - 21 - 18 = 10 \quad (\text{positive})
     \]

6. **Conclusion**:
   The solution to the inequality is where the expression is non-negative. This occurs in the intervals \( (-\infty, -3] \) and \( [6, \infty) \).

   **Final Answer**:
   \[
   x \in (-\infty, -3] \cup [6, \infty)
   \]

Would you like more details or have any questions about this solution?

Here are five related questions:
1. How do you solve a quadratic inequality with different coefficients?
2. What does the discriminant tell you about the roots of a quadratic equation?
3. How can you graph the solution set of a quadratic inequality?
4. What are the applications of quadratic inequalities in real life?
5. How do you complete the square to solve a quadratic equation?

**Tip:** Always test intervals around the roots to determine where the inequality holds true.

Understand how to solve the quadratic inequality x^2 - 3x - 15 ≥ 3 with a step-by-step explanation. This solution involves key algebraic concepts such as the quadratic equation, discriminant calculation, and testing intervals. Ideal for students in Grades 9-11.

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