Under the formula's square root lives $\Delta = b^2 - 4ac$ — the discriminant, written with the Greek capital delta. Read aloud: delta equals b-squared minus four a c. Its SIGN answers a question before the formula is even asked: how many real roots are coming. Positive — two, one per branch of the ±. Zero — exactly one, because plus zero and minus zero land on the same spot. Negative — none at all: no real number squares to a negative.

On the graph that reads: crosses the x-axis twice, touches it once and turns, or floats clear of it entirely. One sign trap pays for the whole lesson: with c negative, $-4ac$ turns POSITIVE and the discriminant grows. Compute Δ before solving anything and you know how the story ends before it starts.