If ` x^(2) + y^(2) + z^(2) = 2 ( x + z -1 )` then the value of : ` x ^(3) + y^(3) + z^(3) = ` ?

Options

A. 2
B. 0
C. `-1`
D. 1

Answer

To solve the equation \( x^2 + y^2 + z^2 = 2(x + z - 1) \) and find the value of \( x^3 + y^3 + z^3 \), we can follow these steps:

### Step 1: Rearrange the given equation
We start with the equation:
\[ x^2 + y^2 + z^2 = 2(x + z - 1) \]
Distributing the 2 on the right side gives:
\[ x^2 + y^2 + z^2 = 2x + 2z - 2 \]

### Step 2: Move all terms to one side
Rearranging the equation, we have:
\[ x^2 - 2x + y^2 + z^2 - 2z + 2 = 0 \]

### Step 3: Complete the square for \(x\) and \(z\)
We can complete the square for \(x\) and \(z\):
- For \(x^2 - 2x\), we can write it as:
\[ (x - 1)^2 - 1 \]
- For \(z^2 - 2z\), we can write it as:
\[ (z - 1)^2 - 1 \]

Substituting these back into the equation gives:
\[ (x - 1)^2 - 1 + y^2 + (z - 1)^2 - 1 + 2 = 0 \]
This simplifies to:
\[ (x - 1)^2 + y^2 + (z - 1)^2 = 0 \]

### Step 4: Analyze the equation
Since the sum of squares equals zero, each term must be zero:
\[ (x - 1)^2 = 0, \quad y^2 = 0, \quad (z - 1)^2 = 0 \]
This implies:
\[ x - 1 = 0 \implies x = 1 \]
\[ y = 0 \]
\[ z - 1 = 0 \implies z = 1 \]

### Step 5: Calculate \(x^3 + y^3 + z^3\)
Now we substitute the values of \(x\), \(y\), and \(z\) into \(x^3 + y^3 + z^3\):
\[ x^3 + y^3 + z^3 = 1^3 + 0^3 + 1^3 = 1 + 0 + 1 = 2 \]

Thus, the value of \(x^3 + y^3 + z^3\) is:
\[ \boxed{2} \]