If `alpha, beta` be the roots of the quadratic equation `x^(2)-5x+k=0`, find the value of k such that `alpha-beta=1`:

Options

A. 2
B. 4
C. 6
D. 8

Answer

To solve the problem, we need to find the value of \( k \) in the quadratic equation \( x^2 - 5x + k = 0 \) such that the difference between the roots \( \alpha \) and \( \beta \) is equal to 1.

### Step-by-Step Solution:

1. **Identify the coefficients**:
The given quadratic equation is in the form \( ax^2 + bx + c = 0 \).
Here, \( a = 1 \), \( b = -5 \), and \( c = k \).

2. **Use the relationships for roots**:
From Vieta's formulas, we know:
- The sum of the roots \( \alpha + \beta = -\frac{b}{a} = -\frac{-5}{1} = 5 \).
- The product of the roots \( \alpha \beta = \frac{c}{a} = \frac{k}{1} = k \).

3. **Set up the equations**:
We have two equations based on the roots:
- Equation 1: \( \alpha + \beta = 5 \)
- Equation 2: \( \alpha - \beta = 1 \)

4. **Solve the equations**:
To eliminate \( \beta \), we can add the two equations:
\[ (\alpha + \beta) + (\alpha - \beta) = 5 + 1 \]
This simplifies to:
\[ 2\alpha = 6 \implies \alpha = 3 \]

5. **Find \( \beta \)**:
Substitute \( \alpha = 3 \) back into Equation 1:
\[ 3 + \beta = 5 \implies \beta = 5 - 3 = 2 \]

6. **Calculate \( k \)**:
Now, we can find \( k \) using the product of the roots:
\[ \alpha \beta = k \implies 3 \cdot 2 = k \implies k = 6 \]

### Final Answer:
The value of \( k \) is \( 6 \).