If `alpha and beta` are the roots of equation `x^(2) - q(1 + x) - r = 0` then what is the value of `(1 + alpha) (1 + beta)` ?

Options

A. 1 - r
B. q - r
C. 1 + r
D. q + r

Answer

To find the value of \((1 + \alpha)(1 + \beta)\) where \(\alpha\) and \(\beta\) are the roots of the equation \(x^2 - q(1 + x) - r = 0\), we can follow these steps:

### Step 1: Rewrite the equation
The given equation is:
\[ x^2 - q(1 + x) - r = 0 \]
We can expand this equation:
\[ x^2 - q - qx - r = 0 \]
Rearranging gives us:
\[ x^2 - qx - (r + q) = 0 \]

### Step 2: Identify coefficients
From the standard form of a quadratic equation \(ax^2 + bx + c = 0\), we can identify:
- \(a = 1\)
- \(b = -q\)
- \(c = -(r + q)\)

### Step 3: Use Vieta's formulas
According to Vieta's formulas, for the roots \(\alpha\) and \(\beta\):
- The sum of the roots \(\alpha + \beta = -\frac{b}{a} = -\frac{-q}{1} = q\)
- The product of the roots \(\alpha \beta = \frac{c}{a} = -\frac{-(r + q)}{1} = r + q\)

### Step 4: Calculate \((1 + \alpha)(1 + \beta)\)
We can expand \((1 + \alpha)(1 + \beta)\):
\[ (1 + \alpha)(1 + \beta) = 1 + \alpha + \beta + \alpha\beta \]
Substituting the values from Vieta's formulas:
\[ = 1 + (q) + (r + q) \]
This simplifies to:
\[ = 1 + q + r + q = 1 + 2q + r \]

### Final Answer
Thus, the value of \((1 + \alpha)(1 + \beta)\) is:
\[ \boxed{1 + 2q + r} \]