Let a, b, c `in R ` be such that a + b + c `lt 0, a - b + c lt 0 and c gt 0`. If `alpha and beta` are roots of the equation `ax^(2) + bx + c = 0 ` , then value of `[alpha] + [beta]` is

Options

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

Answer

To solve the problem step by step, we will analyze the given conditions and use properties of quadratic equations.

### Step 1: Understand the Given Conditions
We are given three inequalities involving \( a, b, c \):
1. \( a + b + c < 0 \)
2. \( a - b + c < 0 \)
3. \( c > 0 \)

### Step 2: Analyze the Quadratic Function
The quadratic equation is given by:
\[ f(x) = ax^2 + bx + c \]
We need to evaluate \( f(1) \) and \( f(-1) \):
- \( f(1) = a(1)^2 + b(1) + c = a + b + c \)
- \( f(-1) = a(-1)^2 + b(-1) + c = a - b + c \)

From the conditions, we know:
- \( f(1) < 0 \) (since \( a + b + c < 0 \))
- \( f(-1) < 0 \) (since \( a - b + c < 0 \))

### Step 3: Evaluate \( f(0) \)
Next, we evaluate \( f(0) \):
\[ f(0) = c \]
Since \( c > 0 \), we have \( f(0) > 0 \).

### Step 4: Graphical Interpretation
Since \( f(0) > 0 \), \( f(1) < 0 \), and \( f(-1) < 0 \), we can visualize the graph of the quadratic function:
- The graph is a downward-opening parabola (since \( a < 0 \) is implied by the conditions).
- The parabola crosses the x-axis at two points, which are the roots \( \alpha \) and \( \beta \).

### Step 5: Determine the Intervals for Roots
From the graphical interpretation:
- Since \( f(0) > 0 \) and \( f(1) < 0 \), one root \( \alpha \) must be between \( 0 \) and \( 1 \).
- Since \( f(-1) < 0 \) and \( f(0) > 0 \), the other root \( \beta \) must be between \( -1 \) and \( 0 \).

### Step 6: Find the Greatest Integer Values
Now we need to find the greatest integer values of \( \alpha \) and \( \beta \):
- Since \( \alpha \) is between \( 0 \) and \( 1 \), the greatest integer of \( \alpha \) is:
\[ [\alpha] = 0 \]
- Since \( \beta \) is between \( -1 \) and \( 0 \), the greatest integer of \( \beta \) is:
\[ [\beta] = -1 \]

### Step 7: Calculate the Final Result
We need to find the sum of the greatest integers:
\[ [\alpha] + [\beta] = 0 + (-1) = -1 \]

### Final Answer
The value of \( [\alpha] + [\beta] \) is:
\[ \boxed{-1} \]