If ` p^(3) - q^(3) = (p - q) {( p + q)^(2) - x p q }` then the value of x is

Options

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

Answer

To solve the equation \( p^3 - q^3 = (p - q) \{ (p + q)^2 - x pq \} \) and find the value of \( x \), we can follow these steps:

### Step 1: Use the formula for the difference of cubes
We know that the difference of cubes can be expressed as:
\[ p^3 - q^3 = (p - q)(p^2 + pq + q^2) \]
So, we can rewrite the left side of the equation using this formula:
\[ (p - q)(p^2 + pq + q^2) = (p - q) \{ (p + q)^2 - x pq \} \]

### Step 2: Cancel out \( (p - q) \) from both sides
Assuming \( p \neq q \), we can divide both sides by \( (p - q) \):
\[ p^2 + pq + q^2 = (p + q)^2 - x pq \]

### Step 3: Expand \( (p + q)^2 \)
Now, we expand \( (p + q)^2 \):
\[ (p + q)^2 = p^2 + 2pq + q^2 \]
Substituting this back into the equation gives us:
\[ p^2 + pq + q^2 = p^2 + 2pq + q^2 - x pq \]

### Step 4: Simplify the equation
Now we can simplify the equation:
\[ p^2 + pq + q^2 = p^2 + q^2 + (2 - x)pq \]
By cancelling \( p^2 + q^2 \) from both sides, we get:
\[ pq = (2 - x)pq \]

### Step 5: Factor out \( pq \)
Assuming \( pq \neq 0 \), we can divide both sides by \( pq \):
\[ 1 = 2 - x \]

### Step 6: Solve for \( x \)
Now, we solve for \( x \):
\[ x = 2 - 1 = 1 \]

Thus, the value of \( x \) is:
\[ \boxed{1} \]