If `alpha, beta` are the roots of the equation `x^(2)+alphax + beta = 0` such that `alpha ne beta` and `||x-beta|-alpha|| lt alpha`, then

Options

A. inequality is satisfied by exactly two integral values of x
B. inequality is satisfied by all values of `x in (-4,-2)`
C. Roots of the equation are opposite in sign
D. `x^(2)+alphax+beta lt 0 AA x in [-1,0]`

Answer

To solve the problem step by step, we start with the given quadratic equation and the conditions provided.

### Step 1: Identify the roots and their relationships
Given the quadratic equation:
\[ x^2 + \alpha x + \beta = 0 \]
with roots \( \alpha \) and \( \beta \), we can use Vieta's formulas which state:
- The sum of the roots \( \alpha + \beta = -\alpha \)
- The product of the roots \( \alpha \beta = \beta \)

### Step 2: Solve for \( \alpha \) and \( \beta \)
From the sum of the roots:
\[ \alpha + \beta = -\alpha \]
This implies:
\[ \beta = -2\alpha \]

From the product of the roots:
\[ \alpha \beta = \beta \]
If \( \beta \neq 0 \) (since \( \alpha \neq \beta \)), we can divide both sides by \( \beta \):
\[ \alpha = 1 \]

Substituting \( \alpha = 1 \) into \( \beta = -2\alpha \):
\[ \beta = -2(1) = -2 \]

### Step 3: Substitute values into the inequality
Now we have:
\[ \alpha = 1, \quad \beta = -2 \]
We need to analyze the inequality:
\[ ||x - \beta| - \alpha| < \alpha \]
Substituting the values of \( \alpha \) and \( \beta \):
\[ ||x + 2| - 1| < 1 \]

### Step 4: Break down the absolute value inequality
We can break this down into two parts:
1. \( |x + 2| - 1 < 1 \)
2. \( |x + 2| - 1 > -1 \)

#### Part 1: Solve \( |x + 2| - 1 < 1 \)
This simplifies to:
\[ |x + 2| < 2 \]
This means:
\[ -2 < x + 2 < 2 \]
Subtracting 2 from all parts:
\[ -4 < x < 0 \]

#### Part 2: Solve \( |x + 2| - 1 > -1 \)
This simplifies to:
\[ |x + 2| > 0 \]
This is always true for \( x \in \mathbb{R} \) except when \( x + 2 = 0 \) (i.e., \( x = -2 \)).

### Step 5: Combine the results
From Part 1, we have:
\[ -4 < x < 0 \]
From Part 2, \( x \neq -2 \).

Thus, the solution set is:
\[ x \in (-4, -2) \cup (-2, 0) \]

### Step 6: Determine integral values in the solution set
The integral values in the interval \( (-4, 0) \) are:
- \( -3, -2, -1 \)

However, since \( x \neq -2 \), the valid integral values are:
- \( -3, -1 \)

### Final Answer
The inequality is satisfied by exactly two integral values: \( -3 \) and \( -1 \).