Consider a quadratic equation `ax^2 + bx + c = 0` having roots `alpha, beta`. If `4a + 2b + c > 0,a-b+c < 0 and 4a - 2b + C > 0` then `|[alpha] + [beta]|` can be {where [] is greatest integer}

Options

A. `-2`
B. `-1`
C. 0
D. 1

Explanation

To solve the problem step-by-step, we will analyze the given conditions and determine the possible values for the roots of the quadratic equation \( ax^2 + bx + c = 0 \).

### Step 1: Understand the conditions
We have the following inequalities:
1. \( 4a + 2b + c > 0 \)
2. \( a - b + c < 0 \)
3. \( 4a - 2b + c > 0 \)

These inequalities will help us understand the behavior of the quadratic function \( f(x) = ax^2 + bx + c \).

### Step 2: Evaluate \( f(2) \)
From the first condition, we can evaluate:
\[ f(2) = a(2^2) + b(2) + c = 4a + 2b + c > 0 \]
This means that the value of the quadratic function at \( x = 2 \) is positive.

### Step 3: Evaluate \( f(-1) \)
From the second condition, we can evaluate:
\[ f(-1) = a(-1^2) + b(-1) + c = a - b + c < 0 \]
This means that the value of the quadratic function at \( x = -1 \) is negative.

### Step 4: Evaluate \( f(-2) \)
From the third condition, we can evaluate:
\[ f(-2) = a(-2^2) + b(-2) + c = 4a - 2b + c > 0 \]
This means that the value of the quadratic function at \( x = -2 \) is positive.

### Step 5: Analyze the graph
Now, we can analyze the behavior of the quadratic function based on the values we calculated:
- \( f(2) > 0 \) indicates that the curve is above the x-axis at \( x = 2 \).
- \( f(-1) < 0 \) indicates that the curve is below the x-axis at \( x = -1 \).
- \( f(-2) > 0 \) indicates that the curve is above the x-axis at \( x = -2 \).

### Step 6: Determine the roots
Since the function is positive at \( x = -2 \) and \( x = 2 \), and negative at \( x = -1 \), we can conclude that:
- One root \( \alpha \) is between \( -2 \) and \( -1 \).
- The other root \( \beta \) is between \( -1 \) and \( 2 \).

### Step 7: Find the greatest integer values
- The greatest integer less than or equal to \( \alpha \) (which lies between \( -2 \) and \( -1 \)) is \( -2 \).
- The greatest integer less than or equal to \( \beta \) (which lies between \( -1 \) and \( 2 \)) can be \( -1 \), \( 0 \), or \( 1 \).

### Step 8: Calculate \( |[\alpha] + [\beta]| \)
We can summarize the possible cases for \( |[\alpha] + [\beta]| \):
1. If \( [\alpha] = -2 \) and \( [\beta] = -1 \):
\[ | -2 + (-1) | = | -3 | = 3 \]
2. If \( [\alpha] = -2 \) and \( [\beta] = 0 \):
\[ | -2 + 0 | = | -2 | = 2 \]
3. If \( [\alpha] = -2 \) and \( [\beta] = 1 \):
\[ | -2 + 1 | = | -1 | = 1 \]

### Conclusion
The possible values for \( |[\alpha] + [\beta]| \) are \( 3, 2, \) and \( 1 \).