The vertex says where a model is best. The roots say where it is worth zero — for a profit model, the two break-even outputs, with profit living strictly between them. Before solving for either, one expression announces how many there are: $\Delta = b^2 - 4ac$, the discriminant, written with the Greek capital delta. Read aloud: delta equals b-squared minus four a c. Positive Δ means two real roots, zero means exactly one (the model grazes zero and turns), negative means none at all — no real number squares to a negative, so a firm with a negative Δ never breaks even at any output whatsoever.

Then the formula that never stalls: $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ — x equals minus b, plus or minus the square root of b-squared minus four a c, all over two a. The ± is two answers on one line, and Δ is exactly what sits under that radical. Two disciplines pay for the whole lesson: the entire crown, $-b$ and the radical together, sits over $2a$ — all of it; and with a POSITIVE, the plus branch digs out the larger root and the minus branch the smaller.

One habit for the paper: a break-even condition often arrives with a negative leading coefficient, because profit models peak. Multiply through by −1 first — it changes none of the roots and it puts the branches back in the order your instincts expect. And watch c's sign: with c negative, $-4ac$ turns positive and Δ grows.