Find all the integral values of a for which the quadratic equation `(x - a) (x - 10) + 1 = 0` has integral roots.

Answer

To find all the integral values of \( a \) for which the quadratic equation

\[ (x - a)(x - 10) + 1 = 0 \]

has integral roots, we can follow these steps:

### Step 1: Expand the equation
Start by expanding the left-hand side of the equation:

\[ (x - a)(x - 10) + 1 = 0 \]

Expanding gives:

\[ x^2 - 10x - ax + 10a + 1 = 0 \]

Combining like terms results in:

\[ x^2 - (a + 10)x + (10a + 1) = 0 \]

### Step 2: Identify coefficients
From the standard form of a quadratic equation \( ax^2 + bx + c = 0 \), we identify:

- \( A = 1 \)
- \( B = -(a + 10) \)
- \( C = 10a + 1 \)

### Step 3: Calculate the discriminant
The discriminant \( D \) of a quadratic equation is given by:

\[ D = B^2 - 4AC \]

Substituting the values we found:

\[ D = (-(a + 10))^2 - 4(1)(10a + 1) \]

This simplifies to:

\[ D = (a + 10)^2 - 40a - 4 \]

### Step 4: Expand and simplify the discriminant
Expanding \( (a + 10)^2 \):

\[ D = a^2 + 20a + 100 - 40a - 4 \]

Combining like terms gives:

\[ D = a^2 - 20a + 96 \]

### Step 5: Set the discriminant to be a perfect square
For the quadratic equation to have integral roots, the discriminant must be a perfect square. Let’s set:

\[ D = k^2 \quad \text{for some integer } k \]

This leads to the equation:

\[ a^2 - 20a + 96 - k^2 = 0 \]

### Step 6: Solve for \( a \)
The above is a quadratic in \( a \). The discriminant of this new quadratic must also be a perfect square:

\[ (-20)^2 - 4(1)(96 - k^2) \geq 0 \]

Calculating this gives:

\[ 400 - 384 + 4k^2 \geq 0 \]

Simplifying yields:

\[ 16 + 4k^2 \geq 0 \]

This is always true for any integer \( k \).

### Step 7: Factor the quadratic
We can rewrite the quadratic \( a^2 - 20a + (96 - k^2) = 0 \) and find its roots using the quadratic formula:

\[ a = \frac{20 \pm \sqrt{D}}{2} \]

Substituting \( D = k^2 \):

\[ a = 10 \pm \sqrt{k^2 - 4} \]

### Step 8: Find integral values
For \( a \) to be an integer, \( \sqrt{k^2 - 4} \) must also be an integer. Thus, \( k^2 - 4 \) must be a perfect square. Let’s set:

\[ k^2 - 4 = m^2 \quad \text{for some integer } m \]

This leads to:

\[ (k - m)(k + m) = 4 \]

The pairs of factors of 4 are \( (1, 4), (2, 2), (-1, -4), (-2, -2) \). Solving these gives possible values for \( k \).

### Step 9: Calculate \( a \) for each \( k \)
1. For \( k = 2 \):
- \( a = 10 \pm 0 \) → \( a = 10 \)
2. For \( k = 0 \):
- \( a = 10 \pm 2 \) → \( a = 12, 8 \)

Thus, the integral values of \( a \) that satisfy the condition are:

\[ \boxed{8, 10, 12} \]