If `alpha and beta` are the roots of the equation `x^(2) +px +p^(2) +q=0`, then the value of `alpha^(2) + alpha beta +beta^(2)+q=`

Options

A. 0
B. 1
C. q
D. 2q

Answer

To solve the problem, we need to find the value of \( \alpha^2 + \alpha \beta + \beta^2 + q \) given that \( \alpha \) and \( \beta \) are the roots of the equation \( x^2 + px + p^2 + q = 0 \).

### Step-by-Step Solution:

1. **Identify the coefficients of the quadratic equation:**
The given quadratic equation is \( x^2 + px + (p^2 + q) = 0 \).
Here, \( a = 1 \), \( b = p \), and \( c = p^2 + q \).

2. **Use Vieta's formulas:**
From Vieta's formulas, we know:
- The sum of the roots \( \alpha + \beta = -\frac{b}{a} = -p \).
- The product of the roots \( \alpha \beta = \frac{c}{a} = p^2 + q \).

3. **Calculate \( \alpha^2 + \beta^2 \):**
We can express \( \alpha^2 + \beta^2 \) in terms of \( \alpha + \beta \) and \( \alpha \beta \):
\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \]
Substituting the values we found:
\[ \alpha^2 + \beta^2 = (-p)^2 - 2(p^2 + q) \]
Simplifying this gives:
\[ \alpha^2 + \beta^2 = p^2 - 2(p^2 + q) = p^2 - 2p^2 - 2q = -p^2 - 2q \]

4. **Combine with \( \alpha \beta \) and \( q \):**
Now, we need to find \( \alpha^2 + \alpha \beta + \beta^2 + q \):
\[ \alpha^2 + \alpha \beta + \beta^2 + q = (-p^2 - 2q) + (p^2 + q) + q \]
Simplifying this expression:
\[ = -p^2 - 2q + p^2 + q + q = -p^2 + 0 = -p^2 \]

5. **Final Result:**
Therefore, the value of \( \alpha^2 + \alpha \beta + \beta^2 + q \) is:
\[ \boxed{0} \]