If ` ( sqrt( x + 4) + sqrt( x - 4))/( sqrt( x + 4) - sqrt( x - 4)) = 2` then x is equal to

Options

A. `2.4`
B. `3.2`
C. 4
D. 5

Answer

To solve the equation

\[ \frac{\sqrt{x + 4} + \sqrt{x - 4}}{\sqrt{x + 4} - \sqrt{x - 4}} = 2, \]

we will follow these steps:

### Step 1: Cross-Multiply
We start by cross-multiplying to eliminate the fraction:

\[ \sqrt{x + 4} + \sqrt{x - 4} = 2(\sqrt{x + 4} - \sqrt{x - 4}). \]

### Step 2: Expand the Right Side
Now, we expand the right side:

\[ \sqrt{x + 4} + \sqrt{x - 4} = 2\sqrt{x + 4} - 2\sqrt{x - 4}. \]

### Step 3: Rearrange the Equation
Next, we rearrange the equation to group all terms involving square roots on one side:

\[ \sqrt{x - 4} + 2\sqrt{x - 4} = 2\sqrt{x + 4} - \sqrt{x + 4}. \]

This simplifies to:

\[ 3\sqrt{x - 4} = \sqrt{x + 4}. \]

### Step 4: Square Both Sides
Now, we square both sides to eliminate the square roots:

\[ (3\sqrt{x - 4})^2 = (\sqrt{x + 4})^2. \]

This gives us:

\[ 9(x - 4) = x + 4. \]

### Step 5: Distribute and Rearrange
Distributing on the left side:

\[ 9x - 36 = x + 4. \]

Now, we rearrange the equation to isolate \(x\):

\[ 9x - x = 4 + 36, \]

which simplifies to:

\[ 8x = 40. \]

### Step 6: Solve for x
Finally, we divide both sides by 8 to solve for \(x\):

\[ x = \frac{40}{8} = 5. \]

### Final Answer
Thus, the value of \(x\) is

\[ \boxed{5}. \]