If `a + b ne 0` and the roots of `x^(2) - px + q = 0` differ by -1, then `p^(2) - 4q` equals :

Options

A. `-1`
B. 0
C. 1
D. 2

Answer

To solve the problem, we need to find the value of \( p^2 - 4q \) given that the roots of the quadratic equation \( x^2 - px + q = 0 \) differ by -1. Let's denote the roots as \( \alpha \) and \( \beta \).

### Step 1: Understanding the relationship between the roots
Since the roots differ by -1, we can express this relationship as:
\[ \alpha - \beta = -1 \quad \text{or} \quad \alpha = \beta - 1 \]

### Step 2: Using Vieta's formulas
From Vieta's formulas, we know:
- The sum of the roots \( \alpha + \beta = p \)
- The product of the roots \( \alpha \beta = q \)

### Step 3: Expressing the sum of the roots
Substituting \( \alpha = \beta - 1 \) into the sum of the roots:
\[ (\beta - 1) + \beta = p \]
This simplifies to:
\[ 2\beta - 1 = p \quad \Rightarrow \quad 2\beta = p + 1 \quad \Rightarrow \quad \beta = \frac{p + 1}{2} \]

### Step 4: Finding \( \alpha \)
Now substituting back to find \( \alpha \):
\[ \alpha = \beta - 1 = \frac{p + 1}{2} - 1 = \frac{p - 1}{2} \]

### Step 5: Finding the product of the roots
Now we can calculate the product of the roots:
\[ \alpha \beta = \left(\frac{p - 1}{2}\right) \left(\frac{p + 1}{2}\right) \]
This can be simplified as:
\[ \alpha \beta = \frac{(p - 1)(p + 1)}{4} = \frac{p^2 - 1}{4} \]

### Step 6: Setting the product equal to \( q \)
From Vieta's, we have:
\[ q = \alpha \beta = \frac{p^2 - 1}{4} \]

### Step 7: Finding \( p^2 - 4q \)
Now we can substitute \( q \) into the expression \( p^2 - 4q \):
\[ p^2 - 4q = p^2 - 4 \left(\frac{p^2 - 1}{4}\right) \]
This simplifies to:
\[ p^2 - (p^2 - 1) = p^2 - p^2 + 1 = 1 \]

### Conclusion
Thus, the value of \( p^2 - 4q \) is:
\[ \boxed{1} \]