If `x-y =8 and xy = 45,` then `(x ^(2) + y ^(2))` is

Options

A. 154
B. 64
C. 90
D. 26

Answer

To solve the problem, we need to find the value of \( x^2 + y^2 \) given the equations \( x - y = 8 \) and \( xy = 45 \).

### Step-by-Step Solution:

1. **Start with the given equations:**
\[ x - y = 8 \quad \text{(1)} \]
\[ xy = 45 \quad \text{(2)} \]

2. **Use the identity for the square of a difference:**
The square of the difference \( (x - y)^2 \) can be expressed as:
\[ (x - y)^2 = x^2 - 2xy + y^2 \]
Substituting equation (1) into this identity gives:
\[ (8)^2 = x^2 - 2xy + y^2 \]
Simplifying this, we have:
\[ 64 = x^2 - 2xy + y^2 \quad \text{(3)} \]

3. **Substitute the value of \( xy \) from equation (2) into equation (3):**
From equation (2), we know \( xy = 45 \). Thus, substituting this into equation (3):
\[ 64 = x^2 - 2(45) + y^2 \]
This simplifies to:
\[ 64 = x^2 - 90 + y^2 \]

4. **Rearranging the equation to find \( x^2 + y^2 \):**
Adding 90 to both sides gives:
\[ x^2 + y^2 = 64 + 90 \]
Simplifying this results in:
\[ x^2 + y^2 = 154 \]

### Final Answer:
Thus, the value of \( x^2 + y^2 \) is \( \boxed{154} \).