If ` x^(2) + y^(2) + 1 = 2x` then the value of ` x^(3) + y^(5)` is

Options

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

Answer

To solve the equation \( x^2 + y^2 + 1 = 2x \) and find the value of \( x^3 + y^5 \), we can follow these steps:

### Step 1: Rearranging the Equation
Start with the given equation:
\[ x^2 + y^2 + 1 = 2x \]
Rearranging gives:
\[ x^2 - 2x + y^2 + 1 = 0 \]

### Step 2: Completing the Square
Next, we complete the square for the \( x \) terms:
\[ (x^2 - 2x + 1) + y^2 = 0 \]
This simplifies to:
\[ (x - 1)^2 + y^2 = 0 \]

### Step 3: Analyzing the Equation
Since both \( (x - 1)^2 \) and \( y^2 \) are squares, they are both non-negative. The only way their sum can equal zero is if both terms are zero:
\[ (x - 1)^2 = 0 \quad \text{and} \quad y^2 = 0 \]

### Step 4: Solving for \( x \) and \( y \)
From \( (x - 1)^2 = 0 \):
\[ x - 1 = 0 \implies x = 1 \]
From \( y^2 = 0 \):
\[ y = 0 \]

### Step 5: Finding \( x^3 + y^5 \)
Now substitute \( x = 1 \) and \( y = 0 \) into \( x^3 + y^5 \):
\[ x^3 + y^5 = 1^3 + 0^5 = 1 + 0 = 1 \]

### Final Answer
Thus, the value of \( x^3 + y^5 \) is:
\[ \boxed{1} \]