Discriminating

Below are several statements about the quadratic equation

$$ax^2 + bx + c = 0,$$
where $a$, $b$ and $c$ are allowed to be any real numbers except that $a$ is not $0$.

For each statement, decide whether it is ALWAYS true, SOMETIMES true, or NEVER true.



To show that a statement is ALWAYS true, we need to give a proof.

To show that a statement is NEVER true, we need to give a proof.

To show that a statement is SOMETIMES true, we need to give an example
when it is true and an example when it is false.  If you want a harder
challenge, can you determine exactly when it is and when it is not
true?


You might want to print and cut out the statements (downloadable from
the link here), so that you can sort them into piles.

(1) If $a < 0$, then the equation has no real roots

(2) If $b^2 - 4ac = 0$, then the equation has one repeated real root.

(3) If the equation has no real roots, then the equation $ax^2 + bx - c = 0$ has two distinct real roots.

(4) If $\frac{b^2}{a} < 4c$, then the equation has two distinct real roots.

(5) If $b = 0$, then the equation has one repeated real root.

(6) The equation has three real roots.

(7) If $c = 0$, then the equation has no real roots.

(8) The equation has the same number of real roots as $ax^2 - bx + c = 0$.

(9) If the equation has two distinct real roots, then $ac < \frac{b^2}{4}$.

(10) If $c > 0$, then the equation has two distinct real roots.

(11) The equation has the same number of real roots as the equation $cx^2 + bx + a = 0$.

(12) If the equation has no real roots, then the equation $-ax^2 - bx - c = 0$ has two distinct real roots.