If `alpha, beta` are the roots of `ax^(2) + bx + c = 0 and alpha + h, beta + h` are the roots of `px^(2) + qx + r = 0`, then h =

Options

A. `-1/2(a/b - p/q)`
B. `(b/a - q/p)`
C. `1/2(b/a - q/p)`
D. none of these

Explanation

To solve the problem, we need to find the value of \( h \) given that \( \alpha \) and \( \beta \) are the roots of the equation \( ax^2 + bx + c = 0 \) and \( \alpha + h \) and \( \beta + h \) are the roots of the equation \( px^2 + qx + r = 0 \).

### Step-by-Step Solution:

1. **Identify the roots of the first equation**:
The roots of the equation \( ax^2 + bx + c = 0 \) are \( \alpha \) and \( \beta \).
- The sum of the roots \( \alpha + \beta = -\frac{b}{a} \).
- The product of the roots \( \alpha \beta = \frac{c}{a} \).

2. **Identify the roots of the second equation**:
The roots of the equation \( px^2 + qx + r = 0 \) are \( \alpha + h \) and \( \beta + h \).
- The sum of the roots \( (\alpha + h) + (\beta + h) = -\frac{q}{p} \).
- This simplifies to \( \alpha + \beta + 2h = -\frac{q}{p} \).

3. **Substitute the sum of the roots from the first equation**:
We know from step 1 that \( \alpha + \beta = -\frac{b}{a} \).
- Substitute this into the equation from step 2:
\[ -\frac{b}{a} + 2h = -\frac{q}{p} \]

4. **Solve for \( 2h \)**:
Rearranging the equation gives:
\[ 2h = -\frac{q}{p} + \frac{b}{a} \]

5. **Solve for \( h \)**:
Divide both sides by 2 to isolate \( h \):
\[ h = \frac{1}{2}\left(\frac{b}{a} - \frac{q}{p}\right) \]

### Final Answer:
\[ h = \frac{1}{2}\left(\frac{b}{a} - \frac{q}{p}\right) \]