If x+y=20 and xy=84 then `(x)^2+(y)^2=?`

Options

A. 232
B. 400
C. 128
D. can not be determined

Answer

To solve the problem, we need to find the value of \( x^2 + y^2 \) given that \( x + y = 20 \) and \( xy = 84 \).

### Step-by-Step Solution:

1. **Use the identity for the square of a sum**:
We know that:
\[ (x + y)^2 = x^2 + y^2 + 2xy \]
This can be rearranged to find \( x^2 + y^2 \):
\[ x^2 + y^2 = (x + y)^2 - 2xy \]

2. **Substitute the known values**:
From the problem, we know:
- \( x + y = 20 \)
- \( xy = 84 \)

Now, we can substitute these values into the rearranged formula:
\[ x^2 + y^2 = (20)^2 - 2(84) \]

3. **Calculate \( (20)^2 \)**:
\[ (20)^2 = 400 \]

4. **Calculate \( 2 \times 84 \)**:
\[ 2 \times 84 = 168 \]

5. **Substitute back into the equation**:
Now we substitute these results back into the equation:
\[ x^2 + y^2 = 400 - 168 \]

6. **Perform the subtraction**:
\[ 400 - 168 = 232 \]

7. **Final result**:
Therefore, the value of \( x^2 + y^2 \) is:
\[ x^2 + y^2 = 232 \]

### Conclusion:
The answer is \( 232 \).

---