Any equation of the form ax² + bx + c = 0, with a ≠ 0, is solved by the quadratic formula. It is derived by completing the square on the general form, so it works for every case rather than only the ones that factorise cleanly.
The discriminant tells you the answer before you compute it
The expression under the square root, Δ = b² − 4ac, decides the character of the solution:
- Δ > 0 — two distinct real roots. The parabola crosses the x-axis twice.
- Δ = 0 — one repeated root. The parabola touches the axis at its vertex.
- Δ < 0 — two complex conjugate roots. The parabola never crosses the axis.
If a, b and c are integers and Δ is a perfect square, the roots are rational and the expression factorises over the integers. That is a fast way to check whether factoring is worth attempting.
Vieta’s relations
The roots are tied to the coefficients directly, without solving:
These are useful both as a check on your arithmetic and as a shortcut in competition problems where you need a symmetric function of the roots but not the roots themselves.
The vertex
The parabola is symmetric about x = −b / 2a, which is the average of the two roots. Substituting back gives the vertex height. If a is positive the parabola opens upward and the vertex is a minimum; if a is negative it opens downward and the vertex is a maximum. This is the standard route to optimisation problems in introductory calculus courses, before derivatives are available.
A numerical trap worth knowing
When b² is much larger than 4ac, one of the two roots is computed as the difference of two nearly equal numbers, and floating-point precision collapses. The stable approach is to compute the well-conditioned root first:
This matters in physics simulations and ray tracers, where a quadratic is solved millions of times with wildly different coefficient magnitudes.
Where quadratics turn up
- Projectile height as a function of time, which is why time of flight is a quadratic solve.
- Ray–sphere intersection in graphics: the discriminant tells you whether the ray hits.
- Equilibrium concentrations in chemistry, where an ICE table produces a quadratic in x.
- The characteristic equation of a second-order circuit or a mass–spring–damper system, where the discriminant separates overdamped, critically damped and underdamped behaviour.