Consider the equation (x - p) (x - 6) + 1 = 0 having integral coefficients. If the equation has integral roots, then what values can p have ?

Options

A. A)4 or 8
B. B)5 or 10
C. C)6 or 12
D. D)3 or 6

Answer

To solve the question step by step, we will analyze the given quadratic equation and find the possible values of \( p \) such that the equation has integral roots.

### Step 1: Write down the given equation
The given equation is:
\[ (x - p)(x - 6) + 1 = 0 \]

### Step 2: Expand the equation
We will expand the left-hand side:
\[ (x - p)(x - 6) = x^2 - 6x - px + 6p = x^2 - (p + 6)x + 6p \]
Adding 1 to both sides gives:
\[ x^2 - (p + 6)x + (6p + 1) = 0 \]

### Step 3: Identify the coefficients
Now, we can identify the coefficients of the quadratic equation:
- Coefficient of \( x^2 \) is \( 1 \)
- Coefficient of \( x \) is \( -(p + 6) \)
- Constant term is \( (6p + 1) \)

### Step 4: Use the condition for integral roots
For the quadratic equation to have integral roots, the discriminant must be a perfect square. The discriminant \( D \) is given by:
\[ D = b^2 - 4ac \]
where \( a = 1 \), \( b = -(p + 6) \), and \( c = (6p + 1) \).

Calculating the discriminant:
\[ D = (-(p + 6))^2 - 4 \cdot 1 \cdot (6p + 1) \]
\[ D = (p + 6)^2 - 4(6p + 1) \]

### Step 5: Simplify the discriminant
Now, let's simplify \( D \):
\[ D = (p^2 + 12p + 36) - (24p + 4) \]
\[ D = p^2 + 12p + 36 - 24p - 4 \]
\[ D = p^2 - 12p + 32 \]

### Step 6: Set the discriminant to be a perfect square
For \( D \) to be a perfect square, we can set:
\[ p^2 - 12p + 32 = k^2 \quad \text{for some integer } k \]
Rearranging gives:
\[ p^2 - 12p + (32 - k^2) = 0 \]

### Step 7: Calculate the discriminant of this new quadratic
The discriminant of this equation must also be a perfect square:
\[ D' = (-12)^2 - 4 \cdot 1 \cdot (32 - k^2) \]
\[ D' = 144 - 128 + 4k^2 \]
\[ D' = 16 + 4k^2 \]

### Step 8: Set \( D' \) to be a perfect square
Let \( D' = m^2 \) for some integer \( m \):
\[ m^2 - 4k^2 = 16 \]
This can be factored as:
\[ (m - 2k)(m + 2k) = 16 \]

### Step 9: Find integer pairs
The integer pairs that multiply to 16 are:
- (1, 16)
- (2, 8)
- (4, 4)
- (-1, -16)
- (-2, -8)
- (-4, -4)

### Step 10: Solve for \( p \)
From the pairs, we can find possible values for \( p \) by solving the equations derived from \( m - 2k \) and \( m + 2k \).

After solving, we find that the possible values for \( p \) are:
\[ p = 4 \quad \text{and} \quad p = 8 \]

### Conclusion
Thus, the values that \( p \) can have are \( 4 \) and \( 8 \).