To determine the domain of the expression involving the square root, $\sqrt{x^2 - 7x - 8}$, we need to find the values of $x$ for which the expression inside the square root is non-negative (greater than or equal to zero), since the square root of a negative number is undefined in the real number system.

Step 1: Set the expression inside the square root to be non-negative

$x^2 - 7x - 8 \geq 0$

Step 2: Solve the inequality

To solve this inequality, we first find the roots of the quadratic equation by setting it equal to zero:

$x^2 - 7x - 8 = 0$

We can solve this using the quadratic formula:

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

where $a = 1$, $b = -7$, and $c = -8$. Plugging in these values:

$x = \frac{-(-7) \pm \sqrt{(-7)^2 - 4(1)(-8)}}{2(1)} = \frac{7 \pm \sqrt{49 + 32}}{2} = \frac{7 \pm \sqrt{81}}{2} = \frac{7 \pm 9}{2}$

This gives us two solutions:

$x = \frac{7 + 9}{2} = 8 \quad \text{and} \quad x = \frac{7 - 9}{2} = -1$

Step 3: Test intervals around the roots

The roots split the number line into three intervals:

1. $x < -1$
2. $-1 \leq x \leq 8$
3. $x > 8$

To determine where the expression $x^2 - 7x - 8$ is non-negative, test a point from each interval.

1. For $x < -1$, choose $x = -2$: $(-2)^2 - 7(-2) - 8 = 4 + 14 - 8 = 10 \, (\text{positive})$

2. For $-1 \leq x \leq 8$, choose $x = 0$: $0^2 - 7(0) - 8 = -8 \, (\text{negative})$

3. For $x > 8$, choose $x = 9$: $9^2 - 7(9) - 8 = 81 - 63 - 8 = 10 \, (\text{positive})$

Step 4: Write the domain in interval notation

The expression is non-negative in the intervals $x \leq -1$ and $x \geq 8$. Therefore, the domain is:

$(-\infty, -1] \cup [8, \infty)$

Final Answer

The domain of $\sqrt{x^2 - 7x - 8}$ is:

$\boxed{(-\infty, -1] \cup [8, \infty)}$

Would you like further details, or do you have any questions? Here are some related problems:

1. What is the domain of $\sqrt{3x - 2}$?
2. Solve the inequality $x^2 - 4x - 12 \geq 0$ and express the domain in interval notation.
3. How do you determine the domain of a rational function?
4. Find the roots of the equation $x^2 - 5x + 6 = 0$.
5. Solve $\sqrt{4x - x^2} \geq 0$.

Tip: When finding domains involving square roots, always ensure the expression inside is non-negative to keep it defined in the real number system.