Root(s) of the equatio `9x^(2) - 18|x|+5 = 0` belonging to the domain of definition of the function `f(x) = log(x^(2) - x - 2)`, is (are)

Options

A. `(-5)/(3),(-1)/(3)`
B. `5/3, 1/3`
C. `(-5)/(3)`
D. `(-1)/(3)`

Answer

To solve the equation \(9x^2 - 18|x| + 5 = 0\) and find the roots that belong to the domain of the function \(f(x) = \log(x^2 - x - 2)\), we will follow these steps:

### Step 1: Determine the domain of the function \(f(x)\)

The logarithmic function \(f(x) = \log(x^2 - x - 2)\) is defined when the argument is positive:
\[ x^2 - x - 2 > 0 \]
We can factor the quadratic:
\[ x^2 - x - 2 = (x - 2)(x + 1) \]
Setting the factors greater than zero gives:
\[ (x - 2)(x + 1) > 0 \]
### Step 2: Find the critical points

The critical points are found by setting each factor to zero:
\[ x - 2 = 0 \quad \Rightarrow \quad x = 2 \]
\[ x + 1 = 0 \quad \Rightarrow \quad x = -1 \]

### Step 3: Test intervals around the critical points

We will test the intervals determined by the critical points:
1. \( (-\infty, -1) \)
2. \( (-1, 2) \)
3. \( (2, \infty) \)

- For \(x < -1\) (e.g., \(x = -2\)):
\[ (-2 - 2)(-2 + 1) = (-4)(-1) > 0 \quad \text{(True)} \]
- For \(-1 < x < 2\) (e.g., \(x = 0\)):
\[ (0 - 2)(0 + 1) = (-2)(1) < 0 \quad \text{(False)} \]
- For \(x > 2\) (e.g., \(x = 3\)):
\[ (3 - 2)(3 + 1) = (1)(4) > 0 \quad \text{(True)} \]

### Step 4: Conclusion for the domain

The function \(f(x)\) is defined for:
\[ (-\infty, -1) \cup (2, \infty) \]

### Step 5: Solve the equation \(9x^2 - 18|x| + 5 = 0\)

We will consider two cases based on the absolute value.

#### Case 1: \(x \geq 0\) (thus \(|x| = x\))

The equation becomes:
\[ 9x^2 - 18x + 5 = 0 \]
Using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{18 \pm \sqrt{(-18)^2 - 4 \cdot 9 \cdot 5}}{2 \cdot 9} \]
Calculating the discriminant:
\[ = \frac{18 \pm \sqrt{324 - 180}}{18} = \frac{18 \pm \sqrt{144}}{18} = \frac{18 \pm 12}{18} \]
Thus, we have:
\[ x = \frac{30}{18} = \frac{5}{3} \quad \text{and} \quad x = \frac{6}{18} = \frac{1}{3} \]

#### Case 2: \(x < 0\) (thus \(|x| = -x\))

The equation becomes:
\[ 9x^2 + 18x + 5 = 0 \]
Using the quadratic formula:
\[ x = \frac{-18 \pm \sqrt{(18)^2 - 4 \cdot 9 \cdot 5}}{2 \cdot 9} \]
Calculating the discriminant:
\[ = \frac{-18 \pm \sqrt{324 - 180}}{18} = \frac{-18 \pm \sqrt{144}}{18} = \frac{-18 \pm 12}{18} \]
Thus, we have:
\[ x = \frac{-6}{18} = -\frac{1}{3} \quad \text{and} \quad x = \frac{-30}{18} = -\frac{5}{3} \]

### Step 6: Check which roots belong to the domain

The roots we found are:
1. \(x = \frac{5}{3}\) (not in the domain)
2. \(x = \frac{1}{3}\) (not in the domain)
3. \(x = -\frac{1}{3}\) (not in the domain)
4. \(x = -\frac{5}{3}\) (in the domain)

### Final Answer

The root of the equation \(9x^2 - 18|x| + 5 = 0\) that belongs to the domain of the function \(f(x) = \log(x^2 - x - 2)\) is:
\[ \boxed{-\frac{5}{3}} \]