What should be added to `a/(a - b) + b/(a + b)` to get 1?

Options

A. `(-2ab)/(a^2 + b^2)`
B. `(2ab)/(a^2 - b^2)`
C. `(2ab)/(b^2 - a^2)`
D. `(-2ab)/(b^2 - a^2)`

Answer

To solve the problem of what should be added to \( \frac{a}{a - b} + \frac{b}{a + b} \) to get 1, we can follow these steps:

### Step 1: Combine the fractions
We start with the expression \( \frac{a}{a - b} + \frac{b}{a + b} \). To combine these fractions, we need a common denominator, which is \( (a - b)(a + b) \).

\[ \frac{a}{a - b} + \frac{b}{a + b} = \frac{a(a + b) + b(a - b)}{(a - b)(a + b)} \]

### Step 2: Simplify the numerator
Now, we simplify the numerator:

\[ a(a + b) + b(a - b) = a^2 + ab + ab - b^2 = a^2 + 2ab - b^2 \]

So, we have:

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

### Step 3: Set up the equation
We want to find what should be added to this expression to make it equal to 1:

\[ \frac{a^2 + 2ab - b^2}{(a - b)(a + b)} + x = 1 \]

### Step 4: Isolate x
Rearranging the equation gives us:

\[ x = 1 - \frac{a^2 + 2ab - b^2}{(a - b)(a + b)} \]

### Step 5: Express 1 with a common denominator
Now, we express 1 with the common denominator:

\[ 1 = \frac{(a - b)(a + b)}{(a - b)(a + b)} \]

### Step 6: Combine the fractions
Now we can combine the fractions:

\[ x = \frac{(a - b)(a + b) - (a^2 + 2ab - b^2)}{(a - b)(a + b)} \]

### Step 7: Simplify the numerator
Now we simplify the numerator:

\[ (a - b)(a + b) = a^2 - b^2 \]

Thus,

\[ x = \frac{(a^2 - b^2) - (a^2 + 2ab - b^2)}{(a - b)(a + b)} = \frac{a^2 - b^2 - a^2 - 2ab + b^2}{(a - b)(a + b)} \]

This simplifies to:

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

### Step 8: Factor out the negative sign
We can factor out the negative sign:

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

### Final Answer
Thus, the value that should be added to \( \frac{a}{a - b} + \frac{b}{a + b} \) to get 1 is:

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

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