Suppose `alpha, beta ` are the roots of `x^(2) - px + 2 . 89 = 0 and gamma ` be a root of `x^(2) + px + 2. 89 = 0`, then `(gamma + alpha) (gamma + beta)` is equal to _______

Answer

To solve the problem, we need to find the value of \((\gamma + \alpha)(\gamma + \beta)\) given that \(\alpha\) and \(\beta\) are the roots of the equation \(x^2 - px + 2.89 = 0\) and \(\gamma\) is a root of the equation \(x^2 + px + 2.89 = 0\).

### Step-by-step Solution:

1. **Identify the Roots**:
- For the equation \(x^2 - px + 2.89 = 0\):
- The sum of the roots \(\alpha + \beta = p\) (from the formula \(-b/a\)).
- The product of the roots \(\alpha \beta = 2.89\) (from the formula \(c/a\)).

2. **Express \((\gamma + \alpha)(\gamma + \beta)\)**:
- We can expand this expression:
\[ (\gamma + \alpha)(\gamma + \beta) = \gamma^2 + \gamma\beta + \gamma\alpha + \alpha\beta \]

3. **Substitute Known Values**:
- We know:
- \(\alpha + \beta = p\)
- \(\alpha \beta = 2.89\)
- Thus, we can rewrite the expression as:
\[ \gamma^2 + \gamma(\alpha + \beta) + \alpha\beta = \gamma^2 + \gamma p + 2.89 \]

4. **Use the Equation for \(\gamma\)**:
- Since \(\gamma\) is a root of the equation \(x^2 + px + 2.89 = 0\), we know:
\[ \gamma^2 + p\gamma + 2.89 = 0 \]
- This implies:
\[ \gamma^2 + p\gamma + 2.89 = 0 \implies \gamma^2 + p\gamma + 2.89 = 0 \]

5. **Conclusion**:
- From the above, we can conclude that:
\[ \gamma^2 + p\gamma + 2.89 = 0 \implies (\gamma + \alpha)(\gamma + \beta) = 0 \]
- Therefore, the value of \((\gamma + \alpha)(\gamma + \beta)\) is:
\[ \boxed{0} \]