`(sqrt(3 + x) + sqrt(3 - x))/(sqrt( 3 + x) - sqrt( 3 - x))` = 2 then x is equal to

Options

A. (a) `(5)/(12)`
B. (b) `(12)/(5)`
C. (c) `(5)/(7)`
D. (d) `(7)/(5)`

Answer

To solve the equation

\[ \frac{\sqrt{3 + x} + \sqrt{3 - x}}{\sqrt{3 + x} - \sqrt{3 - x}} = 2, \]

we will follow these steps:

### Step 1: Cross-multiply the equation
We can start by cross-multiplying to eliminate the fraction:

\[ \sqrt{3 + x} + \sqrt{3 - x} = 2(\sqrt{3 + x} - \sqrt{3 - x}). \]

### Step 2: Expand the right-hand side
Expanding the right-hand side gives:

\[ \sqrt{3 + x} + \sqrt{3 - x} = 2\sqrt{3 + x} - 2\sqrt{3 - x}. \]

### Step 3: Rearrange the equation
Now, we can rearrange the equation to group similar terms:

\[ \sqrt{3 - x} + 2\sqrt{3 - x} = 2\sqrt{3 + x} - \sqrt{3 + x}. \]

This simplifies to:

\[ 3\sqrt{3 - x} = \sqrt{3 + x}. \]

### Step 4: Square both sides
Next, we square both sides to eliminate the square roots:

\[ (3\sqrt{3 - x})^2 = (\sqrt{3 + x})^2. \]

This results in:

\[ 9(3 - x) = 3 + x. \]

### Step 5: Distribute and simplify
Distributing the left-hand side gives:

\[ 27 - 9x = 3 + x. \]

### Step 6: Move all terms involving \(x\) to one side
Now, we can move all terms involving \(x\) to one side:

\[ 27 - 3 = 9x + x, \]

which simplifies to:

\[ 24 = 10x. \]

### Step 7: Solve for \(x\)
Dividing both sides by 10 gives:

\[ x = \frac{24}{10} = \frac{12}{5}. \]

### Conclusion
Thus, the value of \(x\) is

\[ \boxed{\frac{12}{5}}. \]