If `a = (sqrt(3) - sqrt(2))/(sqrt(3) + sqrt(2)) and b = (sqrt(3) + sqrt(2))/(sqrt(3) - sqrt(2))` then value of `sqrt(3a^2 - 5ab + 3b^2)` is

Options

A. `5`
B. `17`
C. `sqrt(17)`
D. `17sqrt(17)`

Answer

To solve the problem, we need to find the value of \( \sqrt{3a^2 - 5ab + 3b^2} \) given \( a = \frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} + \sqrt{2}} \) and \( b = \frac{\sqrt{3} + \sqrt{2}}{\sqrt{3} - \sqrt{2}} \).

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

First, we will rationalize \( a \) and \( b \).

**For \( a \)**:
\[ a = \frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} + \sqrt{2}} \cdot \frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} - \sqrt{2}} = \frac{(\sqrt{3} - \sqrt{2})^2}{(\sqrt{3})^2 - (\sqrt{2})^2} \]
Calculating the numerator:
\[ (\sqrt{3} - \sqrt{2})^2 = 3 - 2\sqrt{6} + 2 = 5 - 2\sqrt{6} \]
Calculating the denominator:
\[ 3 - 2 = 1 \]
Thus,
\[ a = 5 - 2\sqrt{6} \]

**For \( b \)**:
\[ b = \frac{\sqrt{3} + \sqrt{2}}{\sqrt{3} - \sqrt{2}} \cdot \frac{\sqrt{3} + \sqrt{2}}{\sqrt{3} + \sqrt{2}} = \frac{(\sqrt{3} + \sqrt{2})^2}{(\sqrt{3})^2 - (\sqrt{2})^2} \]
Calculating the numerator:
\[ (\sqrt{3} + \sqrt{2})^2 = 3 + 2\sqrt{6} + 2 = 5 + 2\sqrt{6} \]
Calculating the denominator:
\[ 3 - 2 = 1 \]
Thus,
\[ b = 5 + 2\sqrt{6} \]

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

**Calculating \( a^2 \)**:
\[ a^2 = (5 - 2\sqrt{6})^2 = 25 - 20\sqrt{6} + 24 = 49 - 20\sqrt{6} \]

**Calculating \( b^2 \)**:
\[ b^2 = (5 + 2\sqrt{6})^2 = 25 + 20\sqrt{6} + 24 = 49 + 20\sqrt{6} \]

**Calculating \( ab \)**:
\[ ab = (5 - 2\sqrt{6})(5 + 2\sqrt{6}) = 25 - (2\sqrt{6})^2 = 25 - 24 = 1 \]

### Step 3: Substitute into the expression

Now we substitute \( a^2 \), \( b^2 \), and \( ab \) into the expression \( 3a^2 - 5ab + 3b^2 \):
\[ 3a^2 = 3(49 - 20\sqrt{6}) = 147 - 60\sqrt{6} \]
\[ 3b^2 = 3(49 + 20\sqrt{6}) = 147 + 60\sqrt{6} \]
\[ 5ab = 5 \cdot 1 = 5 \]

Now, substituting these into the expression:
\[ 3a^2 - 5ab + 3b^2 = (147 - 60\sqrt{6}) - 5 + (147 + 60\sqrt{6}) \]
Combining like terms:
\[ = 147 + 147 - 5 + (-60\sqrt{6} + 60\sqrt{6}) = 294 - 5 = 289 \]

### Step 4: Take the square root

Finally, we find:
\[ \sqrt{3a^2 - 5ab + 3b^2} = \sqrt{289} = 17 \]

### Final Answer
The value of \( \sqrt{3a^2 - 5ab + 3b^2} \) is \( 17 \).

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