IF the different between the roots of ` x^(2) -px+q=0` is 2, then the relation between p , and q is

Options

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

Answer

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

### Step-by-Step Solution:

1. **Identify the Roots**:
Let the roots of the quadratic equation \( x^2 - px + q = 0 \) be \( \alpha \) and \( \beta \).

2. **Use the Given Information**:
We know that the difference between the roots is given by:
\[ \alpha - \beta = 2 \]

3. **Use the Relationship of Roots**:
From Vieta's formulas, we have:
- The sum of the roots:
\[ \alpha + \beta = p \]
- The product of the roots:
\[ \alpha \beta = q \]

4. **Express the Difference of Roots**:
The difference of the roots can also be expressed using the formula:
\[ \alpha - \beta = \sqrt{(\alpha + \beta)^2 - 4\alpha \beta} \]
Substituting the values from Vieta's formulas:
\[ 2 = \sqrt{p^2 - 4q} \]

5. **Square Both Sides**:
To eliminate the square root, we square both sides:
\[ 2^2 = p^2 - 4q \]
This simplifies to:
\[ 4 = p^2 - 4q \]

6. **Rearrange the Equation**:
Rearranging the equation gives us:
\[ p^2 = 4q + 4 \]

7. **Final Form**:
We can factor out the right side:
\[ p^2 = 4(q + 1) \]

### Conclusion:
The relationship between \( p \) and \( q \) is:
\[ p^2 = 4(q + 1) \]