Solve : `(sqrt(2+x) + sqrt(2 - x))/(sqrt(2 +x) - sqrt(2-x)) = `

Options

A. 2
B. 4
C. 6
D. 8

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 will rearrange the equation to group the square root terms:

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

### Step 4: Move Terms to One Side
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
Now we square both sides to eliminate the square roots:

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

This results in:

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

### Step 6: Expand and Rearrange
Expanding the left side gives:

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

Now, rearranging the equation gives:

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

which simplifies to:

\[ 6 = 5x. \]

### Step 7: Solve for x
Dividing both sides by 5 gives:

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

### Step 8: Check for Extraneous Solutions
We should check if this value satisfies the original equation. Substitute \(x = \frac{6}{5}\) back into the original equation and verify.

### Final Answer
Thus, the solution to the equation is:

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

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