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We are asked to find the domain of the function:

\[
g(x) = \sqrt{x^2 - 6x - 27}
\]

### Key Idea:
For a square root function, the expression inside the square root must be non-negative (i.e., \( \geq 0 \)) because the square root of a negative number is not defined in the set of real numbers.

Thus, we need to solve the inequality:

\[
x^2 - 6x - 27 \geq 0
\]

### Step 1: Solve the inequality

We start by finding the roots of the quadratic equation:

\[
x^2 - 6x - 27 = 0
\]

To solve this quadratic equation, we use the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

For \(x^2 - 6x - 27 = 0\), we have:

- \(a = 1\)
- \(b = -6\)
- \(c = -27\)

Substituting into the quadratic formula:

\[
x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(1)(-27)}}{2(1)}
\]

\[
x = \frac{6 \pm \sqrt{36 + 108}}{2}
\]

\[
x = \frac{6 \pm \sqrt{144}}{2}
\]

\[
x = \frac{6 \pm 12}{2}
\]

Thus, the solutions are:

\[
x = \frac{6 + 12}{2} = 9 \quad \text{and} \quad x = \frac{6 - 12}{2} = -3
\]

### Step 2: Analyze the intervals

Now that we have the roots \(x = 9\) and \(x = -3\), we need to check the sign of \(x^2 - 6x - 27\) in the intervals determined by these roots: \( (-\infty, -3) \), \( (-3, 9) \), and \( (9, \infty) \).

- For \(x \in (-\infty, -3)\), pick \(x = -4\):

\[
(-4)^2 - 6(-4) - 27 = 16 + 24 - 27 = 13 > 0
\]

- For \(x \in (-3, 9)\), pick \(x = 0\):

\[
0^2 - 6(0) - 27 = -27 < 0
\]

- For \(x \in (9, \infty)\), pick \(x = 10\):

\[
10^2 - 6(10) - 27 = 100 - 60 - 27 = 13 > 0
\]

### Step 3: Conclusion on the domain

The quadratic expression is non-negative on the intervals \( (-\infty, -3] \cup [9, \infty) \). Therefore, the domain of the function is:

\[
(-\infty, -3] \cup [9, \infty)
\]

Would you like a further breakdown of this process or more details on solving inequalities?

### Related Questions:
1. What is the domain of \( f(x) = \sqrt{x^2 - 4x - 12} \)?
2. How can we find the domain of a function involving cube roots?
3. Can a function with an even-degree radical ever have a domain of all real numbers?
4. How do you apply the quadratic formula to different types of quadratic functions?
5. What other methods can be used to solve quadratic inequalities?

### Tip:
Always check the intervals determined by the roots of a quadratic expression when solving for where it is positive or non-negative. This will ensure you capture the correct domain.

Find the domain of the function g(x) = sqrt(x^2 - 6x - 27). (Enter your answer using interval notation.)

Find the domain of the function g(x) = sqrt(x^2 - 6x - 27) using interval notation. This solution explains how to solve a quadratic inequality and apply the quadratic formula to find the domain of a square root function.

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