If `alpha, beta ` are the roots of the quadratic equation `x ^(2)+ px+q=0 and gamma, delta` are the roots of `x ^(2)+px-r =0` then `(alpha- gamma ) (alpha -delta )` is equal to :

Options

A. `q+r`
B. `q-r`
C. `-(q+r)`
D. `-(p+q+r)`

Answer

To solve the problem, we need to find the value of \((\alpha - \gamma)(\alpha - \delta)\) given the roots of two quadratic equations. Let's break down the solution step by step.

### Step 1: Identify the roots and their relationships
We have two quadratic equations:
1. \(x^2 + px + q = 0\) with roots \(\alpha\) and \(\beta\).
2. \(x^2 + px - r = 0\) with roots \(\gamma\) and \(\delta\).

From Vieta's formulas, we know:
- For the first equation:
- \(\alpha + \beta = -p\) (sum of roots)
- \(\alpha \beta = q\) (product of roots)

- For the second equation:
- \(\gamma + \delta = -p\) (sum of roots)
- \(\gamma \delta = -r\) (product of roots)

### Step 2: Expand the expression
We need to find:
\[ (\alpha - \gamma)(\alpha - \delta) \]
Expanding this expression gives:
\[ \alpha^2 - \alpha(\gamma + \delta) + \gamma \delta \]

### Step 3: Substitute the known values
From Vieta's formulas, we know:
- \(\gamma + \delta = -p\)
- \(\gamma \delta = -r\)

Substituting these into the expanded expression:
\[ \alpha^2 - \alpha(-p) + (-r) = \alpha^2 + \alpha p - r \]

### Step 4: Relate \(\alpha^2 + \alpha p\) to \(q\)
Since \(\alpha\) is a root of the first equation \(x^2 + px + q = 0\), we can substitute \(\alpha\) into this equation:
\[ \alpha^2 + \alpha p = -q \]

### Step 5: Substitute back into the expression
Now substituting \(-q\) into our expression:
\[ \alpha^2 + \alpha p - r = -q - r \]

### Final Answer
Thus, we find:
\[ (\alpha - \gamma)(\alpha - \delta) = -q - r \]

### Conclusion
The final result is:
\[ (\alpha - \gamma)(\alpha - \delta) = -q - r \]