If a+b=7 and ab=10, then find a−b.

Options

A. 7
B. 1
C. 2
D. 3

Answer

To solve the problem, we need to find the value of \( a - b \) given the equations \( a + b = 7 \) and \( ab = 10 \).

### Step-by-step Solution:

1. **Write down the given equations:**
\[ a + b = 7 \quad \text{(1)} \]
\[ ab = 10 \quad \text{(2)} \]

2. **Square the first equation:**
\[ (a + b)^2 = 7^2 \]
Expanding the left side:
\[ a^2 + 2ab + b^2 = 49 \quad \text{(3)} \]

3. **Substitute the value of \( ab \) from equation (2) into equation (3):**
\[ a^2 + 2(10) + b^2 = 49 \]
Simplifying:
\[ a^2 + 20 + b^2 = 49 \]

4. **Rearrange to find \( a^2 + b^2 \):**
\[ a^2 + b^2 = 49 - 20 \]
\[ a^2 + b^2 = 29 \quad \text{(4)} \]

5. **Use the identity for \( a - b \):**
\[ a - b = \sqrt{(a + b)^2 - 4ab} \]
Substitute the values from equations (1) and (2):
\[ a - b = \sqrt{(7)^2 - 4(10)} \]
\[ a - b = \sqrt{49 - 40} \]
\[ a - b = \sqrt{9} \]

6. **Find the final value of \( a - b \):**
\[ a - b = \pm 3 \]

### Final Answer:
Thus, the value of \( a - b \) is \( \pm 3 \).