If `a ^(2) + b ^(2) = 74 and ab = 35,` then find a + b.

Options

A. 11
B. 12
C. 13
D. 14

Answer

To solve the problem where \( a^2 + b^2 = 74 \) and \( ab = 35 \), we need to find \( a + b \).

**Step 1: Use the identity for \( (a + b)^2 \)**
We know that:
\[ (a + b)^2 = a^2 + b^2 + 2ab \]

**Step 2: Substitute the known values into the identity**
From the problem, we have:
- \( a^2 + b^2 = 74 \)
- \( ab = 35 \)

Substituting these values into the identity gives us:
\[ (a + b)^2 = 74 + 2 \times 35 \]

**Step 3: Calculate \( 2ab \)**
Now, calculate \( 2 \times 35 \):
\[ 2 \times 35 = 70 \]

**Step 4: Add the values**
Now, add \( 74 \) and \( 70 \):
\[ (a + b)^2 = 74 + 70 = 144 \]

**Step 5: Take the square root**
To find \( a + b \), take the square root of both sides:
\[ a + b = \sqrt{144} \]

**Step 6: Calculate the square root**
Since \( \sqrt{144} = 12 \), we have:
\[ a + b = 12 \]

Thus, the final answer is:
\[ \boxed{12} \]

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