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
We start by cross-multiplying to eliminate the fraction:

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

### Step 2: Expand the Right Side
Expanding the right side gives us:

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

### Step 3: Rearrange the Equation
Now, we rearrange the equation to isolate the square root terms:

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

This simplifies to:

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

### Step 4: Move Terms to One Side
Next, we move all terms involving \(\sqrt{2+x}\) to one side:

\[ \sqrt{2+x} - 2\sqrt{2+x} + 3\sqrt{2-x} = 0. \]

This simplifies to:

\[ - \sqrt{2+x} + 3\sqrt{2-x} = 0. \]

### Step 5: Isolate One Square Root
Now, we isolate \(\sqrt{2+x}\):

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

### Step 6: Square Both Sides
Next, we square both sides to eliminate the square roots:

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

This gives us:

\[ 9(2-x) = 2+x. \]

### Step 7: Expand and Rearrange
Expanding the left side:

\[ 18 - 9x = 2 + x. \]

Now, rearranging gives:

\[ 18 - 2 = 9x + x, \]
\[ 16 = 10x. \]

### Step 8: Solve for x
Finally, we solve for \(x\):

\[ x = \frac{16}{10} = \frac{8}{5}. \]

Thus, the value of \(x\) is

\[ \boxed{\frac{8}{5}}. \]

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