Let p and q be the roots of the quadratic equation ` x^(2) - (alpha - 2) x -alpha - 1 = 0`. What is the minimum possible value of ` p^(2) + q^(2)` ?

Options

A. 0
B. 3
C. 4
D. 5

Explanation

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

### Step 1: Identify the coefficients
The quadratic equation is given as:
\[ x^2 - (\alpha - 2)x - (\alpha + 1) = 0 \]
From this, we can identify:
- \( a = 1 \)
- \( b = -(\alpha - 2) \)
- \( c = -(\alpha + 1) \)

### Step 2: Use Vieta's Formulas
According to Vieta's formulas:
- The sum of the roots \( p + q = -\frac{b}{a} = \alpha - 2 \)
- The product of the roots \( pq = \frac{c}{a} = -(\alpha + 1) \)

### Step 3: Express \( p^2 + q^2 \)
We know that:
\[ p^2 + q^2 = (p + q)^2 - 2pq \]
Substituting the values from Vieta's formulas:
\[ p^2 + q^2 = (\alpha - 2)^2 - 2(-(\alpha + 1)) \]
This simplifies to:
\[ p^2 + q^2 = (\alpha - 2)^2 + 2(\alpha + 1) \]

### Step 4: Expand and simplify
Now, let's expand and simplify:
\[ p^2 + q^2 = (\alpha^2 - 4\alpha + 4) + (2\alpha + 2) \]
Combining like terms:
\[ p^2 + q^2 = \alpha^2 - 4\alpha + 4 + 2\alpha + 2 \]
\[ p^2 + q^2 = \alpha^2 - 2\alpha + 6 \]

### Step 5: Find the minimum value
To find the minimum value of \( p^2 + q^2 \), we can complete the square:
\[ p^2 + q^2 = (\alpha^2 - 2\alpha + 1) + 5 = (\alpha - 1)^2 + 5 \]
The minimum value occurs when \( (\alpha - 1)^2 = 0 \), which is when \( \alpha = 1 \).

### Step 6: Calculate the minimum value
Substituting \( \alpha = 1 \):
\[ p^2 + q^2 = 0 + 5 = 5 \]

Thus, the minimum possible value of \( p^2 + q^2 \) is:
\[ \boxed{5} \]