Square root of `2a - sqrt(3a^2 - 2ab - b^2) , (a gt b gt 0) ` is

Options

A. `1/(sqrt2) (sqrt(3a + b) + sqrt(a - b))`
B. `1/(sqrt(2)) (sqrt(3a - b) + sqrt(a + b))`
C. `1/(sqrt(2)) (sqrt(3a + b) - sqrt(a - b))`
D. `1/(sqrt2)(sqrt(3a - b) - sqrt(a + b))`

Answer

To solve the expression \( \sqrt{2a - \sqrt{3a^2 - 2ab - b^2}} \), where \( a > b > 0 \), we will simplify the expression step-by-step.

### Step 1: Simplify the expression under the square root
We start with the expression:
\[ \sqrt{3a^2 - 2ab - b^2} \]
This can be rearranged as:
\[ 3a^2 - 2ab - b^2 = (3a^2 - 2ab + b^2) - 2b^2 = (3a - b)^2 - 2b^2 \]

### Step 2: Rewrite the expression
Now, we can rewrite the original expression:
\[ \sqrt{2a - \sqrt{(3a - b)^2 - 2b^2}} \]

### Step 3: Substitute back into the original expression
Now, we substitute this back into the original expression:
\[ \sqrt{2a - \sqrt{(3a - b)^2 - 2b^2}} \]

### Step 4: Simplify further
To simplify further, we can assume that \( \sqrt{(3a - b)^2 - 2b^2} \) can be expressed in a simpler form. We can analyze it as:
\[ \sqrt{(3a - b)^2 - 2b^2} = \sqrt{(3a - b + \sqrt{2}b)(3a - b - \sqrt{2}b)} \]

### Step 5: Final simplification
Now, we can express the entire expression as:
\[ \sqrt{2a - \sqrt{(3a - b)^2 - 2b^2}} = \sqrt{2a - (3a - b)} \]
This simplifies to:
\[ \sqrt{b - a} \]

### Step 6: Final expression
Thus, we can conclude that:
\[ \sqrt{2a - \sqrt{3a^2 - 2ab - b^2}} = \frac{1}{\sqrt{2}} \cdot \sqrt{3a + b} + \sqrt{a - b} \]