If ` x = (4 sqrt(ab))/(sqrt a + sqrt b) `, then what is the value of `(x + 2sqrt a)/(x - 2 sqrta) + (x + 2 sqrtb)/(x - 2 sqrt b) ("when " a ne b)` ?

Options

A. 0
B. 2
C. 4
D. `((sqrt a + sqrt b))/(( sqrt a - sqrt b))`

Answer

To solve the problem, we start with the given expression for \( x \):

\[ x = \frac{4 \sqrt{ab}}{\sqrt{a} + \sqrt{b}} \]

We need to find the value of:

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

### Step 1: Substitute \( x \) into the expression

Substituting the value of \( x \) into the expression gives us:

\[ \frac{\frac{4 \sqrt{ab}}{\sqrt{a} + \sqrt{b}} + 2\sqrt{a}}{\frac{4 \sqrt{ab}}{\sqrt{a} + \sqrt{b}} - 2\sqrt{a}} + \frac{\frac{4 \sqrt{ab}}{\sqrt{a} + \sqrt{b}} + 2\sqrt{b}}{\frac{4 \sqrt{ab}}{\sqrt{a} + \sqrt{b}} - 2\sqrt{b}} \]

### Step 2: Simplify the first fraction

For the first fraction:

\[ \frac{\frac{4 \sqrt{ab}}{\sqrt{a} + \sqrt{b}} + 2\sqrt{a}}{\frac{4 \sqrt{ab}}{\sqrt{a} + \sqrt{b}} - 2\sqrt{a}} \]

Multiply the numerator and denominator by \( \sqrt{a} + \sqrt{b} \):

\[ = \frac{4 \sqrt{ab} + 2\sqrt{a}(\sqrt{a} + \sqrt{b})}{4 \sqrt{ab} - 2\sqrt{a}(\sqrt{a} + \sqrt{b})} \]

This simplifies to:

\[ = \frac{4 \sqrt{ab} + 2a + 2\sqrt{ab}}{4 \sqrt{ab} - 2a - 2\sqrt{ab}} = \frac{6 \sqrt{ab} + 2a}{2(2 \sqrt{ab} - a - \sqrt{ab})} \]

### Step 3: Simplify the second fraction

For the second fraction:

\[ \frac{\frac{4 \sqrt{ab}}{\sqrt{a} + \sqrt{b}} + 2\sqrt{b}}{\frac{4 \sqrt{ab}}{\sqrt{a} + \sqrt{b}} - 2\sqrt{b}} \]

Again, multiply the numerator and denominator by \( \sqrt{a} + \sqrt{b} \):

\[ = \frac{4 \sqrt{ab} + 2\sqrt{b}(\sqrt{a} + \sqrt{b})}{4 \sqrt{ab} - 2\sqrt{b}(\sqrt{a} + \sqrt{b})} \]

This simplifies to:

\[ = \frac{4 \sqrt{ab} + 2\sqrt{ab} + 2b}{4 \sqrt{ab} - 2\sqrt{ab} - 2b} = \frac{6 \sqrt{ab} + 2b}{2(2 \sqrt{ab} - b - \sqrt{ab})} \]

### Step 4: Combine the two fractions

Now we combine the two simplified fractions:

\[ \frac{6 \sqrt{ab} + 2a}{2(2 \sqrt{ab} - a - \sqrt{ab})} + \frac{6 \sqrt{ab} + 2b}{2(2 \sqrt{ab} - b - \sqrt{ab})} \]

### Step 5: Find a common denominator and simplify

The common denominator is:

\[ 2(2 \sqrt{ab} - a - \sqrt{ab})(2 \sqrt{ab} - b - \sqrt{ab}) \]

Combining the numerators leads to:

\[ \frac{(6 \sqrt{ab} + 2a)(2 \sqrt{ab} - b - \sqrt{ab}) + (6 \sqrt{ab} + 2b)(2 \sqrt{ab} - a - \sqrt{ab})}{\text{common denominator}} \]

### Final Result

After simplification, we find that the expression evaluates to:

\[ \text{Final Answer} = 2 \]