If x+y=18 and xy=72,what is the value `(x)^2 +(y)^2`

Options

A. 120
B. 90
C. 180
D. cannot be deletmined

Answer

To find the value of \( x^2 + y^2 \) given the equations \( x + y = 18 \) and \( xy = 72 \), we can use the identity:

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

### Step-by-Step Solution:

1. **Identify the values from the equations**:
- From the first equation, we have \( x + y = 18 \).
- From the second equation, we have \( xy = 72 \).

2. **Substitute the values into the identity**:
- We will square the sum \( x + y \):
\[ (x + y)^2 = 18^2 = 324 \]

3. **Calculate \( 2xy \)**:
- Now calculate \( 2xy \):
\[ 2xy = 2 \times 72 = 144 \]

4. **Use the identity to find \( x^2 + y^2 \)**:
- Substitute the values into the identity:
\[ x^2 + y^2 = (x + y)^2 - 2xy = 324 - 144 \]

5. **Perform the subtraction**:
\[ x^2 + y^2 = 324 - 144 = 180 \]

6. **Conclusion**:
- Therefore, the value of \( x^2 + y^2 \) is \( 180 \).

### Final Answer:
\[ x^2 + y^2 = 180 \]