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 find the value of \( h \) given that \( \alpha, \beta \) are the roots of the equation \( ax^2 + bx + c = 0 \) and \( \alpha + h, \beta + h \) are the roots of the equation \( px^2 + qx + r = 0 \), we can follow these steps:

### Step 1: Use Vieta's Formulas
From Vieta's formulas for the first equation \( ax^2 + bx + c = 0 \):
- The sum of the roots \( \alpha + \beta = -\frac{b}{a} \)
- The product of the roots \( \alpha \beta = \frac{c}{a} \)

### Step 2: Set Up the Second Equation
For the second equation \( px^2 + qx + r = 0 \), where the roots are \( \alpha + h \) and \( \beta + h \):
- The sum of the roots \( (\alpha + h) + (\beta + h) = -\frac{q}{p} \)

### Step 3: Express the Sum of the Roots
We can express the sum of the roots from the second equation:
\[ (\alpha + h) + (\beta + h) = \alpha + \beta + 2h \]
Substituting the value of \( \alpha + \beta \) from Step 1:
\[ -\frac{b}{a} + 2h = -\frac{q}{p} \]

### Step 4: Rearrange to Solve for \( h \)
Now, we rearrange the equation to isolate \( h \):
\[ 2h = -\frac{q}{p} + \frac{b}{a} \]
\[ h = \frac{1}{2} \left( \frac{b}{a} - \frac{q}{p} \right) \]

### Final Answer
Thus, the value of \( h \) is:
\[ h = \frac{1}{2} \left( \frac{b}{a} - \frac{q}{p} \right) \]