write the domain of expression using interval notation square root of x squared -7x -8

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.

Find the domain of the expression √(x² - 7x - 8) using interval notation. This solution involves solving a quadratic inequality to determine the valid x-values where the expression inside the square root is non-negative. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation