If a is the minimum root of the equation `x^(2) - 3|x|- 2 = 0` , then value of -1/a is

Options

A. `(sqrt(17) - 3) //4`
B. `(sqrt(17) +3) //4`
C. 2
D. `-3`

Answer

To solve the equation \( x^2 - 3|x| - 2 = 0 \) and find the value of \(-\frac{1}{a}\) where \( a \) is the minimum root, we will follow these steps:

### Step 1: Break down the equation based on the modulus function
The equation involves the absolute value function \( |x| \). We will consider two cases based on the definition of the absolute value.

1. **Case 1:** \( x \geq 0 \)
Here, \( |x| = x \). The equation becomes:
\[ x^2 - 3x - 2 = 0 \]

2. **Case 2:** \( x < 0 \)
Here, \( |x| = -x \). The equation becomes:
\[ x^2 + 3x - 2 = 0 \]

### Step 2: Solve the equations for both cases

**For Case 1:**
The equation is \( x^2 - 3x - 2 = 0 \).
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
- Here, \( a = 1, b = -3, c = -2 \).
- Calculate the discriminant:
\[ b^2 - 4ac = (-3)^2 - 4 \cdot 1 \cdot (-2) = 9 + 8 = 17 \]
- Now, applying the quadratic formula:
\[ x = \frac{3 \pm \sqrt{17}}{2} \]
- The roots are:
\[ x_1 = \frac{3 + \sqrt{17}}{2}, \quad x_2 = \frac{3 - \sqrt{17}}{2} \]

**For Case 2:**
The equation is \( x^2 + 3x - 2 = 0 \).
Using the quadratic formula:
- Here, \( a = 1, b = 3, c = -2 \).
- Calculate the discriminant:
\[ b^2 - 4ac = 3^2 - 4 \cdot 1 \cdot (-2) = 9 + 8 = 17 \]
- Now, applying the quadratic formula:
\[ x = \frac{-3 \pm \sqrt{17}}{2} \]
- The roots are:
\[ x_3 = \frac{-3 + \sqrt{17}}{2}, \quad x_4 = \frac{-3 - \sqrt{17}}{2} \]

### Step 3: Identify the minimum root
Now we have four roots:
- From Case 1: \( x_1 = \frac{3 + \sqrt{17}}{2}, \quad x_2 = \frac{3 - \sqrt{17}}{2} \)
- From Case 2: \( x_3 = \frac{-3 + \sqrt{17}}{2}, \quad x_4 = \frac{-3 - \sqrt{17}}{2} \)

The minimum root among these is \( x_4 = \frac{-3 - \sqrt{17}}{2} \).

### Step 4: Find \(-\frac{1}{a}\)
Let \( a = x_4 = \frac{-3 - \sqrt{17}}{2} \). We need to find \(-\frac{1}{a}\):
\[ -\frac{1}{a} = -\frac{1}{\frac{-3 - \sqrt{17}}{2}} = \frac{2}{3 + \sqrt{17}} \]

### Step 5: Rationalize the denominator
To rationalize the denominator:
\[ -\frac{1}{a} = \frac{2(3 - \sqrt{17})}{(3 + \sqrt{17})(3 - \sqrt{17})} = \frac{2(3 - \sqrt{17})}{9 - 17} = \frac{2(3 - \sqrt{17})}{-8} = -\frac{1}{4}(3 - \sqrt{17}) \]

### Final Answer
Thus, the value of \(-\frac{1}{a}\) is:
\[ \frac{\sqrt{17} - 3}{4} \]

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