(ii) If `a-b=3 and a^2 + b^2 = 29`, find `ab`.

Answer

To solve the problem, we are given two equations:

1. \( a - b = 3 \)
2. \( a^2 + b^2 = 29 \)

We need to find the value of \( ab \).

### Step-by-Step Solution:

**Step 1: Use the identity for \( (a - b)^2 \)**

We know the identity:
\[ (a - b)^2 = a^2 + b^2 - 2ab \]

**Step 2: Substitute the known values into the identity**

From the problem, we have:
- \( a - b = 3 \)
- \( a^2 + b^2 = 29 \)

Substituting these values into the identity:
\[ (3)^2 = 29 - 2ab \]

**Step 3: Calculate \( (3)^2 \)**

Calculating the left side:
\[ 9 = 29 - 2ab \]

**Step 4: Rearrange the equation to isolate \( ab \)**

Now, we will rearrange the equation:
\[ 9 - 29 = -2ab \]
\[ -20 = -2ab \]

**Step 5: Divide both sides by -2**

To find \( ab \), divide both sides by -2:
\[ ab = \frac{-20}{-2} = 10 \]

### Final Answer:
Thus, the value of \( ab \) is \( 10 \).

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