One degree higher than linear — now the graph is a curve, a parabola, and it can cross the x-axis twice, touch it once, or miss it entirely.
A quadratic equation in one variable has the general form:
Its graph is a parabola — a smooth U-shaped (or ∩-shaped, if a is negative) curve. Because it bends, it can cross the x-axis at up to two points, giving up to two real roots. Whether it has two, one, or zero real roots depends on a single quantity called the discriminant:
If D > 0, two distinct real roots. If D = 0, one repeated real root (the parabola just touches the axis). If D < 0, no real roots — the parabola never reaches the x-axis at all.
Switch to Interactive Grapher to change a, b, c and watch the parabola — and the discriminant case — update live.