If `x=q+r +s, y = r +s -p and z= p+q +r` then the value of `x^(2) +y^(2) +z^(2) -2xy -2xz +2yz` is

Options

A. `p^(2)`
B. `q^(2)`
C. `r^(2)`
D. `s^(2)`

Answer

To solve the given problem, we need to find the value of the expression:

\[ E = x^2 + y^2 + z^2 - 2xy - 2xz + 2yz \]

where:
- \( x = q + r + s \)
- \( y = r + s - p \)
- \( z = p + q + r \)

### Step 1: Rewrite the expression using algebraic identities

We can recognize that the expression can be rewritten using the identity for the square of a sum:

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

Thus, we can express \( E \) as:

\[ E = (x - y - z)^2 \]

### Step 2: Calculate \( x - y - z \)

Now, we need to calculate \( x - y - z \):

\[ x - y - z = (q + r + s) - (r + s - p) - (p + q + r) \]

### Step 3: Simplify the expression

Now, let's simplify the expression:

1. Substitute the values of \( x, y, z \):

\[ x - y - z = (q + r + s) - (r + s - p) - (p + q + r) \]

2. Distributing the negative signs:

\[ = q + r + s - r - s + p - p - q - r \]

3. Combine like terms:

\[ = q - q + r - r + s - s + p - p = 0 \]

### Step 4: Substitute back into the expression for \( E \)

Now substituting back into our expression for \( E \):

\[ E = (x - y - z)^2 = 0^2 = 0 \]

### Conclusion

Thus, the value of the expression \( x^2 + y^2 + z^2 - 2xy - 2xz + 2yz \) is:

\[ \boxed{0} \]