Discriminant of a Quadratic

The discriminant is the part of the quadratic formula that lives under the square root, and its sign alone answers "how many real solutions?" without solving anything. Positive means two distinct real roots (the parabola cuts the x-axis twice), zero means one repeated root (the parabola just kisses the axis at its vertex), and negative means no real roots at all (the parabola floats entirely above or below). Worked example: 3x² − 4x − 2 = 0 gives Δ = (−4)² − 4(3)(−2) = 16 + 24 = 40 — positive but not a perfect square, so two real irrational roots.

That last test is the practical one for students: Δ a perfect square (and a, b, c whole numbers) means the quadratic factors neatly over the integers, so you can skip the formula entirely. Engineers read the same quantity as a stability check — the discriminant of a system's characteristic equation decides whether a damped oscillator returns smoothly to rest or overshoots and rings. The common trap is sign bookkeeping: with c negative, −4ac becomes an addition, which is why Δ = 40 above rather than −8. Square b first, then subtract, and keep the minus sign attached to c.