If ` x^(2) + y^(2) + z^(2) = 14 and xy + yz + zx = 11 ` then the value of `( x + y+ z) ^(2)` is

Options

A. 16
B. 25
C. 36
D. 49

Answer

To find the value of \((x + y + z)^2\) given the equations \(x^2 + y^2 + z^2 = 14\) and \(xy + yz + zx = 11\), we can use the identity:

\[ (x + y + z)^2 = x^2 + y^2 + z^2 + 2(xy + yz + zx) \]

### Step-by-step solution:

1. **Identify the given values:**
- \(x^2 + y^2 + z^2 = 14\)
- \(xy + yz + zx = 11\)

2. **Substitute the values into the identity:**
\[ (x + y + z)^2 = x^2 + y^2 + z^2 + 2(xy + yz + zx) \]
Substitute the known values:
\[ (x + y + z)^2 = 14 + 2(11) \]

3. **Calculate \(2(xy + yz + zx)\):**
\[ 2(11) = 22 \]

4. **Add the results:**
\[ (x + y + z)^2 = 14 + 22 = 36 \]

5. **Final result:**
\[ (x + y + z)^2 = 36 \]

### Conclusion:
The value of \((x + y + z)^2\) is \(36\).