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

Options

A. `(1)/(2) ((b)/(a) - (q)/(p))`
B. `(r )/(2) (-(b)/(a) + (q)/(p))`
C. `(1)/(2) ((b)/(p) + (q)/(a))`
D. `(1)/(2) (-(b)/(a) + (q)/(a))`

Answer

To solve the problem step by step, we need to find the value of \( h \) given the roots of two quadratic equations.

### Step 1: Identify the roots of the first equation
The first equation is given as:
\[ ax^2 + bx + c = 0 \]
Let \( \alpha \) and \( \beta \) be the roots of this equation. By Vieta's formulas, we know:
\[ \alpha + \beta = -\frac{b}{a} \]

### Step 2: Identify the roots of the second equation
The second equation is given as:
\[ px^2 + qx + r = 0 \]
The roots of this equation are \( \alpha + h \) and \( \beta + h \).

### Step 3: Calculate the sum of the roots for the second equation
The sum of the roots \( \alpha + h \) and \( \beta + h \) can be expressed as:
\[ (\alpha + h) + (\beta + h) = \alpha + \beta + 2h \]
Substituting the value of \( \alpha + \beta \) from Step 1:
\[ \alpha + \beta + 2h = -\frac{b}{a} + 2h \]

### Step 4: Relate the sum of the roots to the coefficients of the second equation
According to Vieta's formulas for the second equation, the sum of the roots is also given by:
\[ -\frac{q}{p} \]
Setting the two expressions for the sum of the roots equal gives:
\[ -\frac{b}{a} + 2h = -\frac{q}{p} \]

### Step 5: Solve for \( h \)
Rearranging the equation to isolate \( h \):
\[ 2h = -\frac{q}{p} + \frac{b}{a} \]
Dividing both sides by 2:
\[ h = \frac{1}{2} \left( \frac{b}{a} - \frac{q}{p} \right) \]

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

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