Find the roots of the equation
`1/(a+b+x)-1/x=1/a+1/b`?

Options

A. `a,-b`
B. `-a,b`
C. `a,b`
D. `-a,-b`

Answer

To find the roots of the equation

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

we will follow these steps:

### Step 1: Cross-multiply the equation
We start by cross-multiplying to eliminate the fractions. The left-hand side becomes:

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

The right-hand side is:

\[ \frac{1}{a} + \frac{1}{b} = \frac{b + a}{ab} \]

Setting both sides equal gives us:

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

### Step 2: Cross-multiply again
Cross-multiplying gives us:

\[ -(a+b) \cdot ab = (a+b) \cdot x(a+b+x) \]

### Step 3: Simplify the equation
Assuming \(a + b \neq 0\), we can divide both sides by \(a + b\):

\[ -ab = x(a+b+x) \]

### Step 4: Rearrange the equation
Rearranging gives us:

\[ x^2 + (a+b)x + ab = 0 \]

### Step 5: Factor the quadratic equation
The quadratic equation can be factored as:

\[ (x + a)(x + b) = 0 \]

### Step 6: Find the roots
Setting each factor to zero gives us the roots:

\[ x + a = 0 \quad \Rightarrow \quad x = -a \]
\[ x + b = 0 \quad \Rightarrow \quad x = -b \]

Thus, the roots of the equation are:

\[ x = -a \quad \text{and} \quad x = -b \]

### Final Answer
The roots of the equation are \(x = -a\) and \(x = -b\).

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