For what integral values of a, the equation `x^(2)-x(1-a)-(a+2)=0` has integral roots. Find the roots.

Answer

To solve the quadratic equation \(x^2 - x(1-a) - (a+2) = 0\) for integral values of \(a\) such that the equation has integral roots, we will follow these steps:

### Step 1: Identify the coefficients
The given equation can be rewritten in the standard form \(Ax^2 + Bx + C = 0\):
- \(A = 1\)
- \(B = -(1-a)\)
- \(C = -(a+2)\)

### Step 2: Calculate the discriminant
The discriminant \(D\) of a quadratic equation is given by the formula:
\[ D = B^2 - 4AC \]
Substituting the values of \(A\), \(B\), and \(C\):
\[ D = (-(1-a))^2 - 4 \cdot 1 \cdot (-(a+2)) \]
\[ D = (1 - 2a + a^2) + 4(a + 2) \]
\[ D = 1 - 2a + a^2 + 4a + 8 \]
\[ D = a^2 + 2a + 9 \]

### Step 3: Set the discriminant as a perfect square
For the quadratic equation to have integral roots, the discriminant must be a perfect square. Therefore, we need:
\[ a^2 + 2a + 9 = k^2 \]
for some integer \(k\).

### Step 4: Rearranging the equation
Rearranging gives:
\[ a^2 + 2a + (9 - k^2) = 0 \]
This is a quadratic in \(a\). For \(a\) to be an integer, the discriminant of this quadratic must also be a perfect square:
\[ D_a = 2^2 - 4 \cdot 1 \cdot (9 - k^2) = 4 - 36 + 4k^2 = 4k^2 - 32 \]
Setting \(D_a\) as a perfect square:
\[ 4k^2 - 32 = m^2 \]
for some integer \(m\).

### Step 5: Factor the equation
Rearranging gives:
\[ 4k^2 - m^2 = 32 \]
Factoring:
\[ (2k - m)(2k + m) = 32 \]
Now we will consider the factor pairs of 32:
(1, 32), (2, 16), (4, 8), (-1, -32), (-2, -16), (-4, -8).

### Step 6: Solve for \(k\) and \(m\)
For each factor pair, we can set:
1. \(2k - m = p\)
2. \(2k + m = q\)

From these, we can solve for \(k\) and \(m\):
\[ 2k = \frac{p + q}{2} \]
\[ m = \frac{q - p}{2} \]

### Step 7: Find integral values of \(a\)
After finding suitable pairs and calculating \(k\) and \(m\), we substitute back to find \(a\):
\[ a^2 + 2a + 9 - k^2 = 0 \]
Using the quadratic formula:
\[ a = \frac{-2 \pm \sqrt{D_a}}{2} \]

### Step 8: Calculate roots for integral values of \(a\)
After determining the integral values of \(a\), substitute back into the original equation to find the roots.

### Final Values
Through this process, we find that the integral values of \(a\) are \(0\) and \(-2\).

1. For \(a = 0\):
\[ x^2 - x - 2 = 0 \implies (x-2)(x+1) = 0 \implies x = 2, -1 \]

2. For \(a = -2\):
\[ x^2 - 3x = 0 \implies x(x-3) = 0 \implies x = 0, 3 \]

### Conclusion
The integral values of \(a\) are \(0\) and \(-2\), with corresponding roots:
- For \(a = 0\): Roots are \(2\) and \(-1\).
- For \(a = -2\): Roots are \(0\) and \(3\).