Find the integral values of `a` for which `(a+2)x^2+2(a+1)x+a=0` will have both roots integers

Options

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

Answer

To solve the problem of finding the integral values of \( a \) for which the quadratic equation

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

has both roots as integers, we can follow these steps:

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

### Step 2: Use Vieta's formulas
According to Vieta's formulas, for a quadratic equation \( Ax^2 + Bx + C = 0 \):
- The sum of the roots \( \alpha + \beta = -\frac{B}{A} \)
- The product of the roots \( \alpha \beta = \frac{C}{A} \)

Substituting the coefficients, we have:
- Sum of the roots:
\[ \alpha + \beta = -\frac{2(a + 1)}{a + 2} \]
- Product of the roots:
\[ \alpha \beta = \frac{a}{a + 2} \]

### Step 3: Express the roots in terms of integers
Since both roots \( \alpha \) and \( \beta \) are integers, we can denote them as \( \alpha \) and \( \beta \). From Vieta's formulas, we can express:
- \( \alpha + \beta = -\frac{2(a + 1)}{a + 2} \)
- \( \alpha \beta = \frac{a}{a + 2} \)

### Step 4: Set conditions for integer roots
For \( \alpha \beta \) to be an integer, \( a \) must be such that \( a + 2 \) divides \( a \). This gives us the condition:
\[ \frac{a}{a + 2} \text{ must be an integer.} \]

### Step 5: Analyze the divisibility condition
This implies that \( a \) must be a multiple of \( a + 2 \). Let \( k = a + 2 \), then \( a = k - 2 \). Substituting this into \( \alpha \beta \):
\[ \alpha \beta = \frac{k - 2}{k} = 1 - \frac{2}{k} \]
For \( \alpha \beta \) to be an integer, \( k \) must divide \( 2 \). The divisors of \( 2 \) are \( \pm 1, \pm 2 \).

### Step 6: Find possible values of \( k \)
1. If \( k = 1 \):
\[ a + 2 = 1 \implies a = -1 \]
2. If \( k = -1 \):
\[ a + 2 = -1 \implies a = -3 \]
3. If \( k = 2 \):
\[ a + 2 = 2 \implies a = 0 \]
4. If \( k = -2 \):
\[ a + 2 = -2 \implies a = -4 \]

### Step 7: List the integral values of \( a \)
The integral values of \( a \) for which the roots are integers are:
\[ a = -1, -3, 0, -4 \]

### Conclusion
Thus, the integral values of \( a \) for which the quadratic equation has both roots as integers are \( -1, -3, 0, -4 \).