Package of problems

## Problem

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 MUST, MAY or CAN’T is the correct word to use in the statement.

To say that something MUST be the case, we need it to be true in all cases; we will need to give a convincing explanation (a proof) of why this must be always true.

To show that something CAN’T be the case, we likewise need to give a convincing explanation (a proof) of why.

To show that something MAY be the case, 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 download cards with the statements on them, so that you can sort them into piles.

If $a < 0$, then $ax^2+bx+c=0$
MUST / MAY / CAN’T have real roots.

If $b^2 - 4ac = 0$, then $ax^2+bx+c=0$
MUST / MAY / CAN’T have one repeated real root.

If $ax^2+bx+c=0$ has no real roots, then $ax^2 + bx - c = 0$
MUST / MAY / CAN’T have two distinct real roots.

If $\frac{b^2}{a} < 4c$, then $ax^2+bx+c=0$
MUST / MAY / CAN’T have two distinct real roots.

If $b = 0$, then $ax^2+bx+c=0$
MUST / MAY / CAN’T have one repeated real root.

The equation $ax^2+bx+c=0$
MUST / MAY / CAN’T have three real roots.

If $c = 0$, then $ax^2+bx+c=0$
MUST / MAY / CAN’T have real roots.

The equation $ax^2+bx+c=0$
MUST / MAY / CAN’T have the same number of real roots as $ax^2 - bx + c = 0$.

If $ax^2+bx+c=0$ has two distinct real roots, then we
MUST / MAY / CAN’T have $ac < \frac{b^2}{4}$.

If $c > 0$, then $ax^2+bx+c=0$
MUST / MAY / CAN’T have two distinct real roots.

The equation $ax^2+bx+c=0$
MUST / MAY / CAN’T have the same number of real roots as $cx^2 + bx + a = 0$.

If $ax^2+bx+c=0$ has no real roots, then $-ax^2 - bx - c = 0$
MUST / MAY / CAN’T have two distinct real roots.