Do You Know About Roots?

$ax^2+bx+c=0$ is called a quadratic equation if $a\neq 0$. Which of the following statements is/are true concerning quadratic equations?

$[1]$ A quadratic equation always has $2$ distinct roots.

$[2]$ If $a, b$ and $c$ are all odd numbers, it is possible for $ax^2+bx+c=0$ to have a rational solution.

$[3]$ If $ax^2+bx+c=0$ has real solutions and $a, b, c > 0$, then both solutions must be negative.

Note: This problem is a part of the set "I Don't Have a Good Name For This Yet". See the rest of the problems here. And when I say I don't have a good name for this yet, I mean it. If you like problems like these and have a cool name for this set, feel free to comment here.