To solve the inequality \( x^2 - 2x - 15 > 0 \), we first need to find the roots of the corresponding quadratic equation \( x^2 - 2x - 15 = 0 \).
We can factor the quadratic:
\[ x^2 - 2x - 15 = (x - 5)(x + 3) \]
Now, we set each factor to zero to find the roots:
- \( x - 5 = 0 \) gives \( x = 5 \)
- \( x + 3 = 0 \) gives \( x = -3 \)
The roots are \( x = -3 \) and \( x = 5 \). These roots divide the number line into three intervals:
- \( (-\infty, -3) \)
- \( (-3, 5) \)
- \( (5, \infty) \)
Next, we will test each interval to determine where the expression \( (x - 5)(x + 3) > 0 \).
Test Interval \( (-\infty, -3) \): Choose \( x = -4 \): \[ (-4 - 5)(-4 + 3) = (-9)(-1) = 9 > 0 \] So, this interval satisfies the inequality.
Test Interval \( (-3, 5) \): Choose \( x = 0 \): \[ (0 - 5)(0 + 3) = (-5)(3) = -15 < 0 \] So, this interval does not satisfy the inequality.
Test Interval \( (5, \infty) \): Choose \( x = 6 \): \[ (6 - 5)(6 + 3) = (1)(9) = 9 > 0 \] So, this interval satisfies the inequality.
Now we combine our results:
The solution to the inequality \( x^2 - 2x - 15 > 0 \) is: \[ x \in (-\infty, -3) \cup (5, \infty) \]
Thus, the final answer is: \[ \boxed{(-\infty, -3) \cup (5, \infty)} \]