If `a ne 0` and the equation `ax ^(2)+bx+c=0` has two roots `alpha and beta` such thet `alpha lt -3 and beta gt 2.` Which of the following is always true ?

Options

A. `a (a+|b| +c) gt 0`
B. `a (a+|b| +c) lt 0`
C. `9a -3b + c gt 0`
D. `(9a -3b+c) (4a +2b +c) lt 0`

Answer

To solve the problem, we need to analyze the given quadratic equation \( ax^2 + bx + c = 0 \) with roots \( \alpha \) and \( \beta \) such that \( \alpha < -3 \) and \( \beta > 2 \). We will explore the implications of these conditions on the coefficients \( a \), \( b \), and \( c \).

### Step 1: Understanding the Roots
The roots of the quadratic equation \( ax^2 + bx + c = 0 \) can be expressed using Vieta's formulas:
- The sum of the roots \( \alpha + \beta = -\frac{b}{a} \)
- The product of the roots \( \alpha \beta = \frac{c}{a} \)

### Step 2: Analyzing the Conditions on Roots
Given the conditions \( \alpha < -3 \) and \( \beta > 2 \):
- Since \( \alpha < -3 \), we can say \( \alpha + \beta < -3 + 2 = -1 \).
- This implies that \( -\frac{b}{a} < -1 \), which leads to \( \frac{b}{a} > 1 \) or \( b > a \) (if \( a > 0 \)) or \( b < a \) (if \( a < 0 \)).

### Step 3: Analyzing the Product of the Roots
From the product of the roots:
- Since \( \alpha < -3 \) and \( \beta > 2 \), we have \( \alpha \beta < -3 \cdot 2 = -6 \).
- This implies \( \frac{c}{a} < -6 \), leading to \( c < -6a \) (if \( a > 0 \)) or \( c > -6a \) (if \( a < 0 \)).

### Step 4: Considering the Sign of \( a \)
We need to consider two cases based on the sign of \( a \):

**Case 1: \( a > 0 \)**
- The parabola opens upwards.
- The conditions imply \( b > a \) and \( c < -6a \).

**Case 2: \( a < 0 \)**
- The parabola opens downwards.
- The conditions imply \( b < a \) and \( c > -6a \).

### Step 5: Conclusion
Based on the analysis, we can conclude that the following statements are always true:
- If \( a > 0 \), then \( b > a \) and \( c < -6a \).
- If \( a < 0 \), then \( b < a \) and \( c > -6a \).

Thus, the correct answer among the options provided would be the one that captures the relationship between \( a \), \( b \), and \( c \) based on the conditions of the roots.