If `( sqrt(2 + x ) + sqrt (2 - x))/( sqrt (2 + x) - sqrt 2 - x)= 2,` the value of x is

Options

A. `4/5`
B. `3/5`
C. `8/5`
D. `1/5`

Answer

To solve the equation

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

we will follow these steps:

### Step 1: Cross-multiply to eliminate the fraction

Multiply both sides by the denominator:

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

### Step 2: Distribute on the right side

Distributing the 2 on the right side gives us:

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

### Step 3: Rearrange the equation

Rearranging the equation, we can bring all terms involving square roots to one side:

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

### Step 4: Isolate one of the square root terms

Subtract \(\sqrt{2 + x}\) from both sides:

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

This simplifies to:

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

### Step 5: Square both sides to eliminate the square root

Squaring both sides gives:

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

This results in:

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

### Step 6: Expand and simplify

Expanding the left side:

\[ 8 - 4x = 2 + x. \]

### Step 7: Rearrange to isolate x

Rearranging gives:

\[ 8 - 2 = x + 4x, \]
\[ 6 = 5x. \]

### Step 8: Solve for x

Dividing both sides by 5:

\[ x = \frac{6}{5}. \]

### Final Answer

Thus, the value of \(x\) is

\[ \boxed{\frac{6}{5}}. \]

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