Quadratic Vertex Word Problems

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Find all integers $a$ such that the polynomial $x^2+ax+a$ has integer roots. (Check all that apply).

Source: Lexington Math Tournament

Choose all correct answers

Option A: A

There exist solutions $a \ge 100$

Option B: B

There exist solutions $10 \le a < 100$

Option C: C

There exist solutions $1 \le a < 10$ .

Option D: D

There exist solutions $-10 < a \le 0$ .

Option E: E

There exist solutions $-100 \le a < -10$ .

Option F: F

There exist solutions $a < -100$ .

Option G: G

There exist more than $5$ solutions for $a$ .

Option H: H

There exist $3$ , $4$ or $5$ solutions for $a$ .

Option I: I

There exist $1$ or $2$ solutions for $a$ .

Option J: J

There exist no solutions for $a$ .

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