The Quadratic Formula
Any equation that can be rearranged into ax² + bx + c = 0 is a quadratic, and every one of them is solved by the same formula. It comes from completing the square on the general form, which is why it works whether or not the equation factors neatly.
Formula
Discriminant Δ = b² − 4ac
Vertex = (−b / 2a, c − b² / 4a)
The two roots are symmetric about −b/2a, which is why the vertex always sits exactly halfway between them. That symmetry is also a useful check: if you have found both roots, their average must equal −b/2a, and their product must equal c/a.
Reading the Discriminant
The sign of b² − 4ac decides the character of the solution before you calculate anything else, which is why it is worth evaluating first.
Δ > 0 — two distinct real roots
The parabola crosses the x-axis at two separate points. If Δ is also a perfect square and a, b, c are integers, the roots are rational and the equation factors with whole numbers.
Δ = 0 — one repeated root
The curve touches the axis at exactly one point, its vertex. The quadratic is a perfect square: a(x − r)².
Δ < 0 — a complex conjugate pair
No real value of x satisfies the equation and the parabola never meets the x-axis. The two roots share the same real part, −b/2a, and differ only in the sign of the imaginary part.
Where Quadratics Show Up
Projectile height against time is quadratic because gravity applies constant acceleration, so solving for the roots tells you when an object launched upward returns to the ground, and the vertex tells you its peak height and when it occurs.
The same shape appears wherever two linear quantities multiply: area against a single varying dimension, revenue against price when demand falls linearly with price, or braking distance against speed. In each case the vertex is the interesting point — the maximum area, the revenue-maximising price — and the roots mark where the quantity falls to zero.