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

Options

A. `2{p^(2) - 2q + p'^(2) - 2q' - pp'}`
B. `2{p^(2) - 2q + p'^(2) - 2q' - qq'}`
C. `2{p^(2) - 2q - p'^(2) - 2q' + pp'}`
D. `2{p^(2) - 2q - p'^(2) - 2q' - qq'}`

Explanation

To solve the problem, we will follow these steps:

1. **Identify the given roots and their relationships**:
- The roots \( \alpha \) and \( \beta \) of the equation \( x^2 - px + q = 0 \) give us:
\[ \alpha + \beta = p \quad \text{(1)} \]
\[ \alpha \beta = q \quad \text{(2)} \]
- The roots \( \alpha' \) and \( \beta' \) of the equation \( x^2 - p'x + q' = 0 \) give us:
\[ \alpha' + \beta' = p' \quad \text{(3)} \]
\[ \alpha' \beta' = q' \quad \text{(4)} \]

2. **Set up the expression to evaluate**:
We need to find the value of:
\[ (\alpha - \alpha')^2 + (\beta - \alpha')^2 + (\alpha - \beta')^2 + (\beta - \beta')^2 \]

3. **Expand each term**:
- Expanding \( (\alpha - \alpha')^2 \):
\[ (\alpha - \alpha')^2 = \alpha^2 - 2\alpha\alpha' + \alpha'^2 \]
- Expanding \( (\beta - \alpha')^2 \):
\[ (\beta - \alpha')^2 = \beta^2 - 2\beta\alpha' + \alpha'^2 \]
- Expanding \( (\alpha - \beta')^2 \):
\[ (\alpha - \beta')^2 = \alpha^2 - 2\alpha\beta' + \beta'^2 \]
- Expanding \( (\beta - \beta')^2 \):
\[ (\beta - \beta')^2 = \beta^2 - 2\beta\beta' + \beta'^2 \]

4. **Combine all expanded terms**:
Combining all these expansions, we have:
\[ (\alpha - \alpha')^2 + (\beta - \alpha')^2 + (\alpha - \beta')^2 + (\beta - \beta')^2 = 2\alpha^2 + 2\beta^2 + 2\alpha'^2 + 2\beta'^2 - 2\alpha\alpha' - 2\beta\alpha' - 2\alpha\beta' - 2\beta\beta' \]

5. **Factor out the common terms**:
This can be simplified to:
\[ 2(\alpha^2 + \beta^2 + \alpha'^2 + \beta'^2) - 2(\alpha + \beta)(\alpha' + \beta') = 2(\alpha^2 + \beta^2 + \alpha'^2 + \beta'^2 - (\alpha + \beta)(\alpha' + \beta')) \]

6. **Use the relationships from steps 1-4**:
We know:
\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta = p^2 - 2q \quad \text{(from (1) and (2))} \]
\[ \alpha'^2 + \beta'^2 = (\alpha' + \beta')^2 - 2\alpha'\beta' = (p')^2 - 2q' \quad \text{(from (3) and (4))} \]

7. **Substitute back into the expression**:
Substituting these values back, we get:
\[ 2\left((p^2 - 2q) + (p'^2 - 2q')\right) - 2(p)(p') \]
Simplifying gives:
\[ 2p^2 - 4q + 2p'^2 - 4q' - 2pp' \]

8. **Final expression**:
Thus, the final value is:
\[ 2p^2 - 2q - 2q' + 2p'^2 - 2pp' \]

9. **Conclusion**:
The final result is:
\[ 2(p^2 - 2q + p'^2 - 2q' - pp') \]