If `x+ y= (7)/(2) and xy= (5)/(2)`, find
`x-y`

Answer

To solve the problem where \( x + y = \frac{7}{2} \) and \( xy = \frac{5}{2} \), we need to find the value of \( x - y \).

### Step-by-Step Solution:

1. **Write down the equations:**
We have the following two equations:
\[ x + y = \frac{7}{2} \quad \text{(1)} \]
\[ xy = \frac{5}{2} \quad \text{(2)} \]

2. **Express \( x \) in terms of \( y \):**
From equation (1), we can express \( x \) as:
\[ x = \frac{7}{2} - y \quad \text{(3)} \]

3. **Substitute \( x \) in equation (2):**
Substitute equation (3) into equation (2):
\[ \left(\frac{7}{2} - y\right)y = \frac{5}{2} \]

4. **Expand and rearrange the equation:**
Expanding the left side gives:
\[ \frac{7}{2}y - y^2 = \frac{5}{2} \]
Rearranging this, we get:
\[ y^2 - \frac{7}{2}y + \frac{5}{2} = 0 \]

5. **Multiply through by 2 to eliminate fractions:**
To make calculations easier, multiply the entire equation by 2:
\[ 2y^2 - 7y + 5 = 0 \quad \text{(4)} \]

6. **Factor the quadratic equation:**
We need to factor equation (4):
\[ (2y - 5)(y - 1) = 0 \]
This gives us two possible solutions for \( y \):
\[ 2y - 5 = 0 \quad \Rightarrow \quad y = \frac{5}{2} \]
\[ y - 1 = 0 \quad \Rightarrow \quad y = 1 \]

7. **Find corresponding \( x \) values:**
For \( y = 1 \):
\[ x = \frac{7}{2} - 1 = \frac{5}{2} \]
For \( y = \frac{5}{2} \):
\[ x = \frac{7}{2} - \frac{5}{2} = 1 \]

8. **Calculate \( x - y \):**
Now we can find \( x - y \):
- If \( y = 1 \) and \( x = \frac{5}{2} \):
\[ x - y = \frac{5}{2} - 1 = \frac{3}{2} \]
- If \( y = \frac{5}{2} \) and \( x = 1 \):
\[ x - y = 1 - \frac{5}{2} = -\frac{3}{2} \]

### Conclusion:
Thus, the values of \( x - y \) can be either \( \frac{3}{2} \) or \( -\frac{3}{2} \).