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 \(x^2 + ax + 20 = 0\) with integer roots, we need to find the possible integer pairs that multiply to 20. The roots of the quadratic equation can be represented as \(\alpha\) and \(\beta\). According to Vieta's formulas, we know:

1. The sum of the roots \(\alpha + \beta = -a\)
2. The product of the roots \(\alpha \beta = 20\)

### Step 1: Find the integer pairs that multiply to 20
The integer pairs \((\alpha, \beta)\) that satisfy \(\alpha \beta = 20\) are:

- \( (1, 20) \)
- \( (2, 10) \)
- \( (4, 5) \)
- \( (-1, -20) \)
- \( (-2, -10) \)
- \( (-4, -5) \)

### Step 2: Calculate the sum of the roots for each pair
Now we will calculate \(\alpha + \beta\) for each pair:

1. For \( (1, 20) \):
\[ \alpha + \beta = 1 + 20 = 21 \quad \Rightarrow \quad a = -21 \]

2. For \( (2, 10) \):
\[ \alpha + \beta = 2 + 10 = 12 \quad \Rightarrow \quad a = -12 \]

3. For \( (4, 5) \):
\[ \alpha + \beta = 4 + 5 = 9 \quad \Rightarrow \quad a = -9 \]

4. For \( (-1, -20) \):
\[ \alpha + \beta = -1 - 20 = -21 \quad \Rightarrow \quad a = 21 \]

5. For \( (-2, -10) \):
\[ \alpha + \beta = -2 - 10 = -12 \quad \Rightarrow \quad a = 12 \]

6. For \( (-4, -5) \):
\[ \alpha + \beta = -4 - 5 = -9 \quad \Rightarrow \quad a = 9 \]

### Step 3: List all possible values of \(a\)
From the calculations, the possible values of \(a\) are:
- \(-21\)
- \(-12\)
- \(-9\)
- \(21\)
- \(12\)
- \(9\)

### Step 4: Sum all possible values of \(a\)
Now, we sum all the possible values of \(a\):

\[ -21 + (-12) + (-9) + 21 + 12 + 9 \]

Calculating this step-by-step:

1. \(-21 + 21 = 0\)
2. \(-12 + 12 = 0\)
3. \(-9 + 9 = 0\)

Thus, the total sum is:

\[ 0 + 0 + 0 = 0 \]

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

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