If the roots of the equation `x^(2) - nx + m = 0` differ by 1. then.

Options

A. `n^(2) - 4m - 1 = 0`
B. `n^(2) + 4m - 1 = 0`
C. `m^(2) + 4n + 1 = 0`
D. `m^(2) - 4n - 1 = 0`

Answer

To solve the problem, we need to find the relationship between \( n \) and \( m \) given that the roots of the quadratic equation \( x^2 - nx + m = 0 \) differ by 1.

### Step-by-Step Solution:

1. **Identify the Roots**:
Let the roots of the equation be \( \alpha \) and \( \beta \). According to the problem, the roots differ by 1, so we can express this as:
\[ \alpha - \beta = 1 \quad \text{(1)} \]

2. **Sum of the Roots**:
From Vieta's formulas, we know that the sum of the roots \( \alpha + \beta \) is equal to the coefficient of \( x \) (which is \( -n \)) divided by the coefficient of \( x^2 \) (which is 1):
\[ \alpha + \beta = n \quad \text{(2)} \]

3. **Product of the Roots**:
Again, from Vieta's formulas, the product of the roots \( \alpha \beta \) is equal to the constant term \( m \) divided by the coefficient of \( x^2 \):
\[ \alpha \beta = m \quad \text{(3)} \]

4. **Express \( \beta \) in terms of \( \alpha \)**:
From equation (1), we can express \( \beta \) as:
\[ \beta = \alpha - 1 \]

5. **Substitute \( \beta \) into the Sum of Roots**:
Substitute \( \beta \) from the previous step into equation (2):
\[ \alpha + (\alpha - 1) = n \]
Simplifying this gives:
\[ 2\alpha - 1 = n \]
Therefore,
\[ 2\alpha = n + 1 \quad \Rightarrow \quad \alpha = \frac{n + 1}{2} \quad \text{(4)} \]

6. **Substitute \( \alpha \) into the Product of Roots**:
Now substitute \( \alpha \) into equation (3):
\[ \alpha \beta = m \]
Using \( \beta = \alpha - 1 \):
\[ \alpha(\alpha - 1) = m \]
Substitute \( \alpha \) from equation (4):
\[ \left(\frac{n + 1}{2}\right)\left(\frac{n + 1}{2} - 1\right) = m \]
Simplifying the expression:
\[ \left(\frac{n + 1}{2}\right)\left(\frac{n + 1 - 2}{2}\right) = m \]
\[ \left(\frac{n + 1}{2}\right)\left(\frac{n - 1}{2}\right) = m \]
\[ \frac{(n + 1)(n - 1)}{4} = m \]
\[ \frac{n^2 - 1}{4} = m \quad \text{(5)} \]

7. **Rearranging the Equation**:
Rearranging equation (5) gives:
\[ n^2 - 4m - 1 = 0 \]

### Final Result:
Thus, we conclude that:
\[ n^2 - 4m - 1 = 0 \]