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

Options

A. 1
B. 2
C. `-1`
D. 0

Answer

To solve the problem, we need to find the value of \( a^2 - ax \) given that

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

### Step 1: Rationalize the denominator

To simplify \( a \), we can multiply the numerator and the denominator by the conjugate of the denominator:

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

### Step 2: Simplify the numerator and denominator

The numerator becomes:

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

The denominator simplifies to:

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

Thus, we have:

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

### Step 3: Find \( a^2 \)

Now, we need to calculate \( a^2 \):

\[ a^2 = \left(\frac{x + \sqrt{x^2 - 4}}{2}\right)^2 = \frac{(x + \sqrt{x^2 - 4})^2}{4}. \]

Expanding the square:

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

Thus,

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

### Step 4: Calculate \( ax \)

Next, we calculate \( ax \):

\[ ax = \left(\frac{x + \sqrt{x^2 - 4}}{2}\right)x = \frac{x^2 + x\sqrt{x^2 - 4}}{2}. \]

### Step 5: Find \( a^2 - ax \)

Now we can find \( a^2 - ax \):

\[ a^2 - ax = \left(\frac{x^2 - 2 + x\sqrt{x^2 - 4}}{2}\right) - \left(\frac{x^2 + x\sqrt{x^2 - 4}}{2}\right). \]

Combining the fractions:

\[ a^2 - ax = \frac{x^2 - 2 + x\sqrt{x^2 - 4} - x^2 - x\sqrt{x^2 - 4}}{2} = \frac{-2}{2} = -1. \]

### Final Answer

The value of \( a^2 - ax \) is

\[ \boxed{-1}. \]