If `alpha, beta` are the roots of `x^(2)+ax+12=0` such that `alpha-beta=1`, then a =

Options

A. 7
B. -7
C. `+-1`
D. `+-7`

Answer

To solve the problem, we need to find the value of \( a \) given that \( \alpha \) and \( \beta \) are the roots of the quadratic equation \( x^2 + ax + 12 = 0 \) and that \( \alpha - \beta = 1 \).

### Step-by-Step Solution:

1. **Identify the coefficients of the quadratic equation**:
The given equation is \( x^2 + ax + 12 = 0 \). Here, we can identify:
- \( a = 1 \) (coefficient of \( x^2 \))
- \( b = a \) (coefficient of \( x \))
- \( c = 12 \) (constant term)

2. **Use Vieta's formulas**:
According to Vieta's formulas, for a quadratic equation \( ax^2 + bx + c = 0 \):
- The sum of the roots \( \alpha + \beta = -\frac{b}{a} = -a \)
- The product of the roots \( \alpha \beta = \frac{c}{a} = 12 \)

3. **Express \( \alpha + \beta \) and \( \alpha - \beta \)**:
We know from the problem that \( \alpha - \beta = 1 \). We can express \( \alpha + \beta \) and \( \alpha - \beta \) in terms of \( \alpha \) and \( \beta \):
- Let \( S = \alpha + \beta \)
- Let \( D = \alpha - \beta \)

From the given information:
- \( D = 1 \)

We can express \( \alpha \) and \( \beta \) in terms of \( S \) and \( D \):
\[ \alpha = \frac{S + D}{2} = \frac{S + 1}{2} \]
\[ \beta = \frac{S - D}{2} = \frac{S - 1}{2} \]

4. **Find the product \( \alpha \beta \)**:
Using the expressions for \( \alpha \) and \( \beta \):
\[ \alpha \beta = \left(\frac{S + 1}{2}\right) \left(\frac{S - 1}{2}\right) = \frac{(S + 1)(S - 1)}{4} = \frac{S^2 - 1}{4} \]

According to Vieta's formulas, we also know that:
\[ \alpha \beta = 12 \]

Therefore, we can set up the equation:
\[ \frac{S^2 - 1}{4} = 12 \]

5. **Solve for \( S \)**:
Multiply both sides by 4:
\[ S^2 - 1 = 48 \]
\[ S^2 = 49 \]
\[ S = \pm 7 \]

6. **Find the value of \( a \)**:
Recall that \( S = -a \). Therefore:
- If \( S = 7 \), then \( a = -7 \).
- If \( S = -7 \), then \( a = 7 \).

Thus, \( a = \pm 7 \).

### Final Answer:
The value of \( a \) is \( \pm 7 \).