`1/((p+q+x))=1/p+1/q+1/x` solve for by factorization method .

Answer

To solve the equation \( \frac{1}{p + q + x} = \frac{1}{p} + \frac{1}{q} + \frac{1}{x} \) using the factorization method, we can follow these steps:

### Step 1: Find a common denominator for the right side
The right side of the equation is \( \frac{1}{p} + \frac{1}{q} + \frac{1}{x} \). The common denominator for these fractions is \( pqx \).

### Step 2: Rewrite the right side with the common denominator
We can express the right side as:
\[ \frac{qx + px + pq}{pqx} \]
So, the equation becomes:
\[ \frac{1}{p + q + x} = \frac{qx + px + pq}{pqx} \]

### Step 3: Cross-multiply to eliminate the fractions
Cross-multiplying gives us:
\[ pqx = (p + q + x)(qx + px + pq) \]

### Step 4: Expand the right side
Expanding the right side:
\[ pqx = (p + q + x)(qx + px + pq) \]
Distributing gives:
\[ pqx = (p + q)qx + (p + q)px + (p + q)pq + x(qx + px + pq) \]

### Step 5: Rearranging the equation
Rearranging the equation leads to:
\[ pqx - (p + q)qx - (p + q)px - (p + q)pq - x(qx + px + pq) = 0 \]

### Step 6: Factor out common terms
We can factor out \( x \) from the left side:
\[ x(pq - (p + q)q - (p + q)p - (p + q)pq) = 0 \]

### Step 7: Set the equation to zero
This gives us a quadratic equation in \( x \):
\[ x^2 + (p + q)x + pq = 0 \]

### Step 8: Solve the quadratic equation
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1, b = (p + q), c = pq \):
\[ x = \frac{-(p + q) \pm \sqrt{(p + q)^2 - 4(1)(pq)}}{2(1)} \]
This simplifies to:
\[ x = \frac{-(p + q) \pm \sqrt{p^2 + 2pq + q^2 - 4pq}}{2} \]
\[ x = \frac{-(p + q) \pm \sqrt{(p - q)^2}}{2} \]
Thus, we have two potential solutions:
\[ x = -p \quad \text{or} \quad x = -q \]

### Final Solution
The solutions for \( x \) are:
\[ x = -p \quad \text{and} \quad x = -q \]