If `alpha, beta` are the roots of `ax^(2)+bx+c=0`, `alpha beta=3` and a, b, c are in A.P., then `alpha+beta` =

Options

A. -4
B. -1
C. 4
D. -2

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the given information
We know that \( \alpha \) and \( \beta \) are the roots of the quadratic equation \( ax^2 + bx + c = 0 \). We are given:
- \( \alpha \beta = 3 \)
- \( a, b, c \) are in Arithmetic Progression (A.P.)

### Step 2: Use Vieta's Formulas
From Vieta's formulas, we know:
- The sum of the roots \( \alpha + \beta = -\frac{b}{a} \)
- The product of the roots \( \alpha \beta = \frac{c}{a} \)

Since we are given \( \alpha \beta = 3 \), we can write:
\[ \frac{c}{a} = 3 \implies c = 3a \]

### Step 3: Apply the A.P. condition
Since \( a, b, c \) are in A.P., we can express this condition as:
\[ b = \frac{a + c}{2} \]
Substituting \( c = 3a \) into the A.P. condition gives:
\[ b = \frac{a + 3a}{2} = \frac{4a}{2} = 2a \]

### Step 4: Find \( \alpha + \beta \)
Now, substituting \( b = 2a \) into the formula for the sum of the roots:
\[ \alpha + \beta = -\frac{b}{a} = -\frac{2a}{a} = -2 \]

### Conclusion
Thus, the value of \( \alpha + \beta \) is:
\[ \alpha + \beta = -2 \]

### Final Answer
The answer is \( -2 \).

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