Determine the value of ` x^(2) + y^(2) ` when ` x^(3) - y ^(3) = 54 , x - y = 18 and xy = 2 `

Options

A. 0
B. 1
C. 2
D. 3

Answer

To determine the value of \( x^2 + y^2 \) given the equations \( x^3 - y^3 = 54 \), \( x - y = 18 \), and \( xy = 2 \), we can follow these steps:

### Step 1: Use the identity for the difference of cubes
We know that:
\[ x^3 - y^3 = (x - y)(x^2 + xy + y^2) \]
Substituting the known values into this identity, we have:
\[ 54 = (x - y)(x^2 + xy + y^2) \]
Given \( x - y = 18 \), we can substitute this into the equation:
\[ 54 = 18(x^2 + xy + y^2) \]

### Step 2: Simplify the equation
Now, we can simplify the equation:
\[ x^2 + xy + y^2 = \frac{54}{18} = 3 \]

### Step 3: Substitute \( xy \) into the equation
We know that \( xy = 2 \). Therefore, we can substitute this into the equation:
\[ x^2 + 2 + y^2 = 3 \]

### Step 4: Solve for \( x^2 + y^2 \)
Now, we can isolate \( x^2 + y^2 \):
\[ x^2 + y^2 = 3 - 2 = 1 \]

Thus, the value of \( x^2 + y^2 \) is \( 1 \).

### Final Answer
\[ \boxed{1} \]