(i) If `a+b = 8 and ab = 15`, find `a^2 + b^2`.

Answer

To solve the problem, we need to find the value of \( a^2 + b^2 \) given that \( a + b = 8 \) and \( ab = 15 \).

### Step-by-step Solution:

1. **Use the identity for \( a^2 + b^2 \)**:
We know the identity:
\[ a^2 + b^2 = (a + b)^2 - 2ab \]
This means we can express \( a^2 + b^2 \) in terms of \( a + b \) and \( ab \).

2. **Substitute the known values**:
From the problem, we have:
- \( a + b = 8 \)
- \( ab = 15 \)

Now, we can substitute these values into the identity:
\[ a^2 + b^2 = (8)^2 - 2(15) \]

3. **Calculate \( (a + b)^2 \)**:
\[ (8)^2 = 64 \]

4. **Calculate \( 2ab \)**:
\[ 2(15) = 30 \]

5. **Substitute back into the equation**:
Now we substitute these results back into the equation:
\[ a^2 + b^2 = 64 - 30 \]

6. **Perform the subtraction**:
\[ a^2 + b^2 = 34 \]

### Final Answer:
Thus, the value of \( a^2 + b^2 \) is \( 34 \).

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