If the difference between the roots of `x^(2) + ax + b = 0` is equal to the difference between the roots of `x^(2)+px+q= 0` then `a^(2)-p^(2)`=

Options

A. a) b-q
B. b) 2(p-q)
C. c) 4(b-q)
D. d) 8(b+q)

Answer

To solve the problem, we need to find the value of \( a^2 - p^2 \) given that the difference between the roots of the equations \( x^2 + ax + b = 0 \) and \( x^2 + px + q = 0 \) is the same.

### Step-by-Step Solution:

1. **Identify the Roots**:
For the first equation \( x^2 + ax + b = 0 \), let the roots be \( \alpha \) and \( \beta \).
By Vieta's formulas:
- The sum of the roots \( \alpha + \beta = -a \)
- The product of the roots \( \alpha \beta = b \)

2. **Difference of Roots**:
The difference between the roots \( \alpha \) and \( \beta \) can be expressed as:
\[ \alpha - \beta = \sqrt{(\alpha + \beta)^2 - 4\alpha\beta} = \sqrt{(-a)^2 - 4b} = \sqrt{a^2 - 4b} \]

3. **For the Second Equation**:
For the second equation \( x^2 + px + q = 0 \), let the roots be \( \theta \) and \( \gamma \).
Again, by Vieta's formulas:
- The sum of the roots \( \theta + \gamma = -p \)
- The product of the roots \( \theta \gamma = q \)

4. **Difference of Roots for the Second Equation**:
The difference between the roots \( \theta \) and \( \gamma \) can be expressed as:
\[ \theta - \gamma = \sqrt{(\theta + \gamma)^2 - 4\theta\gamma} = \sqrt{(-p)^2 - 4q} = \sqrt{p^2 - 4q} \]

5. **Set the Differences Equal**:
According to the problem, the differences are equal:
\[ \sqrt{a^2 - 4b} = \sqrt{p^2 - 4q} \]

6. **Square Both Sides**:
Squaring both sides gives:
\[ a^2 - 4b = p^2 - 4q \]

7. **Rearranging the Equation**:
Rearranging the equation to isolate \( a^2 - p^2 \):
\[ a^2 - p^2 = 4b - 4q \]

8. **Factor Out the Common Term**:
Factoring out the common term gives:
\[ a^2 - p^2 = 4(b - q) \]

### Final Result:
Thus, the value of \( a^2 - p^2 \) is:
\[ \boxed{4(b - q)} \]