If `( x - 2) ( x - p) = x^(2) - ax + 6` then the value of `(a - p)` is

Options

A. 0
B. 1
C. 2
D. 3

Answer

To solve the equation \(( x - 2)( x - p) = x^{2} - ax + 6\) and find the value of \(a - p\), we will follow these steps:

### Step 1: Expand the left-hand side
We start by expanding the left-hand side of the equation:
\[ (x - 2)(x - p) = x^2 - px - 2x + 2p \]
This simplifies to:
\[ x^2 - (p + 2)x + 2p \]

### Step 2: Set the expanded form equal to the right-hand side
Now, we set the expanded left-hand side equal to the right-hand side:
\[ x^2 - (p + 2)x + 2p = x^2 - ax + 6 \]

### Step 3: Compare coefficients
Since the equations are equal for all \(x\), we can compare the coefficients of \(x\) and the constant terms on both sides.

1. **Comparing coefficients of \(x\)**:
\[ -(p + 2) = -a \implies p + 2 = a \]

2. **Comparing constant terms**:
\[ 2p = 6 \implies p = 3 \]

### Step 4: Substitute \(p\) back to find \(a\)
Now that we have \(p\), we can substitute it back into the equation we derived for \(a\):
\[ a = p + 2 = 3 + 2 = 5 \]

### Step 5: Calculate \(a - p\)
Now we can find \(a - p\):
\[ a - p = 5 - 3 = 2 \]

Thus, the value of \(a - p\) is \(2\).