For ` a gt b, if a + b = 5 and ab = 6` then the value of `(a^(2) - b^(2))` is

Options

A. 1
B. 3
C. 5
D. 7

Answer

To solve the problem, we need to find the value of \( a^2 - b^2 \) given that \( a + b = 5 \) and \( ab = 6 \), with the condition that \( a > b \).

### Step-by-Step Solution:

1. **Use the identity for \( a^2 - b^2 \)**:
\[ a^2 - b^2 = (a - b)(a + b) \]
We know \( a + b = 5 \), so we can rewrite the expression as:
\[ a^2 - b^2 = (a - b) \cdot 5 \]

**Hint**: Remember that \( a^2 - b^2 \) can be factored into \( (a - b)(a + b) \).

2. **Find \( a - b \)**:
To find \( a - b \), we can use the square of \( a - b \):
\[ (a - b)^2 = (a + b)^2 - 4ab \]
Substituting the known values:
\[ (a - b)^2 = 5^2 - 4 \cdot 6 \]
\[ (a - b)^2 = 25 - 24 = 1 \]
Taking the square root gives:
\[ a - b = 1 \quad \text{(since \( a > b \))} \]

**Hint**: Use the relationship between the squares to find \( a - b \) and remember to consider the condition \( a > b \).

3. **Substitute back to find \( a^2 - b^2 \)**:
Now we can substitute \( a - b = 1 \) back into our expression for \( a^2 - b^2 \):
\[ a^2 - b^2 = (a - b)(a + b) = 1 \cdot 5 = 5 \]

**Hint**: After finding \( a - b \), substitute it back into the expression for \( a^2 - b^2 \).

### Final Answer:
The value of \( a^2 - b^2 \) is \( 5 \).