if the difference of the roots of the equation ` x^(2)-px +q=0` is unity.

Options

A. `p^(2)+4q=1`
B. `p^(2)-4q=1`
C. `p^(2)-4q^(2)=(1+2q)^(2)`
D. `4p^(2)+q^(2)= (1 +2p)^(2) `

Answer

To solve the problem, we need to find the relationship between the coefficients \( p \) and \( q \) of the quadratic equation \( x^2 - px + q = 0 \) given that the difference of its roots is unity.

### Step-by-Step Solution:

1. **Identify the Roots**:
Let the roots of the equation be \( \alpha \) and \( \beta \). According to the problem, the difference of the roots is given as:
\[ \alpha - \beta = 1 \]

2. **Use the Relationship of Roots**:
From Vieta's formulas, we know:
- The sum of the roots \( \alpha + \beta = p \)
- The product of the roots \( \alpha \beta = q \)

3. **Express \( \alpha \) in terms of \( \beta \)**:
Since \( \alpha - \beta = 1 \), we can express \( \alpha \) as:
\[ \alpha = \beta + 1 \]

4. **Substitute \( \alpha \) into the Sum of Roots**:
Substitute \( \alpha \) in the sum of the roots:
\[ (\beta + 1) + \beta = p \]
Simplifying this gives:
\[ 2\beta + 1 = p \quad \Rightarrow \quad 2\beta = p - 1 \quad \Rightarrow \quad \beta = \frac{p - 1}{2} \]

5. **Find \( \alpha \)**:
Substitute \( \beta \) back to find \( \alpha \):
\[ \alpha = \beta + 1 = \frac{p - 1}{2} + 1 = \frac{p - 1 + 2}{2} = \frac{p + 1}{2} \]

6. **Calculate the Product of Roots**:
Now, we can calculate the product of the roots:
\[ \alpha \beta = \left(\frac{p + 1}{2}\right) \left(\frac{p - 1}{2}\right) = \frac{(p + 1)(p - 1)}{4} = \frac{p^2 - 1}{4} \]

7. **Set the Product Equal to \( q \)**:
Since \( \alpha \beta = q \), we have:
\[ q = \frac{p^2 - 1}{4} \]

8. **Rearranging the Equation**:
To express this in a standard form, we can multiply both sides by 4:
\[ 4q = p^2 - 1 \]
Rearranging gives:
\[ p^2 - 4q = 1 \]

### Final Result:
Thus, the relationship between \( p \) and \( q \) is:
\[ p^2 - 4q = 1 \]