If ` a = sqrt(6) + sqrt(5) , b = sqrt(6) - sqrt(5)` then `2 a^(2) - 5 ab + 2b^(2) = `?

Options

A. 38
B. 39
C. 40
D. 41

Answer

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

### Step 1: Calculate \( a^2 \) and \( b^2 \)

1. **Calculate \( a^2 \)**:
\[ a^2 = (\sqrt{6} + \sqrt{5})^2 = (\sqrt{6})^2 + 2(\sqrt{6})(\sqrt{5}) + (\sqrt{5})^2 \]
\[ = 6 + 2\sqrt{30} + 5 = 11 + 2\sqrt{30} \]

2. **Calculate \( b^2 \)**:
\[ b^2 = (\sqrt{6} - \sqrt{5})^2 = (\sqrt{6})^2 - 2(\sqrt{6})(\sqrt{5}) + (\sqrt{5})^2 \]
\[ = 6 - 2\sqrt{30} + 5 = 11 - 2\sqrt{30} \]

### Step 2: Substitute \( a^2 \) and \( b^2 \) into the expression

Now we can substitute \( a^2 \) and \( b^2 \) into the expression \( 2a^2 - 5ab + 2b^2 \).

3. **Calculate \( 2a^2 \)**:
\[ 2a^2 = 2(11 + 2\sqrt{30}) = 22 + 4\sqrt{30} \]

4. **Calculate \( 2b^2 \)**:
\[ 2b^2 = 2(11 - 2\sqrt{30}) = 22 - 4\sqrt{30} \]

### Step 3: Calculate \( ab \)

5. **Calculate \( ab \)**:
\[ ab = (\sqrt{6} + \sqrt{5})(\sqrt{6} - \sqrt{5}) = (\sqrt{6})^2 - (\sqrt{5})^2 = 6 - 5 = 1 \]

### Step 4: Substitute \( ab \) into the expression

6. **Calculate \( -5ab \)**:
\[ -5ab = -5 \times 1 = -5 \]

### Step 5: Combine all parts

Now we combine all the parts we calculated:
\[ 2a^2 - 5ab + 2b^2 = (22 + 4\sqrt{30}) + (-5) + (22 - 4\sqrt{30}) \]
\[ = 22 + 4\sqrt{30} - 5 + 22 - 4\sqrt{30} \]
\[ = 22 + 22 - 5 + 4\sqrt{30} - 4\sqrt{30} \]
\[ = 44 - 5 = 39 \]

### Final Answer
The value of \( 2a^2 - 5ab + 2b^2 \) is \( \boxed{39} \).