It is given that the equation `x^2 + ax + 20 = 0` has integer roots. What is the sum of all possible values of a ?

Answer

To solve the equation \(x^2 + ax + 20 = 0\) for integer roots, we will follow these steps:

### Step 1: Identify the roots
Let the roots of the equation be \(p\) and \(q\). According to Vieta's formulas, we have:
- The sum of the roots \(p + q = -a\)
- The product of the roots \(pq = 20\)

### Step 2: Factor pairs of 20
Next, we need to find all pairs of integers \((p, q)\) such that their product is 20. The possible factor pairs of 20 are:
1. \( (1, 20) \)
2. \( (2, 10) \)
3. \( (4, 5) \)
4. \( (-1, -20) \)
5. \( (-2, -10) \)
6. \( (-4, -5) \)

### Step 3: Calculate the sum of the roots for each pair
Now we will calculate \(a\) for each of these pairs using the formula \(a = -(p + q)\):

1. For \( (1, 20) \):
\[ p + q = 1 + 20 = 21 \implies a = -21 \]

2. For \( (2, 10) \):
\[ p + q = 2 + 10 = 12 \implies a = -12 \]

3. For \( (4, 5) \):
\[ p + q = 4 + 5 = 9 \implies a = -9 \]

4. For \( (-1, -20) \):
\[ p + q = -1 - 20 = -21 \implies a = 21 \]

5. For \( (-2, -10) \):
\[ p + q = -2 - 10 = -12 \implies a = 12 \]

6. For \( (-4, -5) \):
\[ p + q = -4 - 5 = -9 \implies a = 9 \]

### Step 4: List all possible values of \(a\)
The possible values of \(a\) from the calculations above are:
- \(-21, -12, -9, 21, 12, 9\)

### Step 5: Calculate the sum of all possible values of \(a\)
Now, we will sum all these values:
\[ -21 + (-12) + (-9) + 21 + 12 + 9 = 0 \]

### Conclusion
The sum of all possible values of \(a\) is \(0\).

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