Enter the coefficients of ax² + bx + c = 0 to get the discriminant, both roots (real or complex), the vertex, and the axis of symmetry of the parabola.
Any equation of the form ax² + bx + c = 0 (with a ≠ 0) can be solved directly with the quadratic formula — no factoring guesswork required. This calculator applies the formula, classifies the roots using the discriminant, and locates the vertex of the parabola y = ax² + bx + c, which is the same curve that describes a projectile's trajectory or a parabolic reflector.
For ax² + bx + c = 0, the roots are x = (−b ± √(b² − 4ac)) / (2a). The term under the square root, Δ = b² − 4ac, is the discriminant, and its sign alone tells you what kind of roots to expect before you even finish the calculation.
If Δ > 0, the equation has two distinct real roots — the parabola crosses the x-axis twice. If Δ = 0, there is exactly one repeated real root — the parabola is tangent to the x-axis at its vertex. If Δ < 0, the square root of a negative number appears, giving two complex conjugate roots of the form p ± qi — the parabola never touches the x-axis at all.
The vertex — the minimum point if a > 0, or the maximum point if a < 0 — sits at x = −b/(2a). Substituting that x back into the equation gives the vertex's y-coordinate, c − b²/(4a). The vertical line through the vertex, x = −b/(2a), is the axis of symmetry: the parabola is a mirror image of itself across that line.
Take x² − 5x + 6 = 0 (a = 1, b = −5, c = 6). The discriminant is (−5)² − 4(1)(6) = 25 − 24 = 1, which is positive, so there are two real roots: x = (5 ± 1)/2, giving x₁ = 3 and x₂ = 2. The vertex sits at x = −(−5)/(2·1) = 2.5, with y = 6 − 25/4 = −0.25.
If a = 0 the equation isn't quadratic anymore — it reduces to the linear equation bx + c = 0, which the quadratic formula can't handle (it would require dividing by zero). This calculator requires a ≠ 0.
Complex roots mean the parabola y = ax² + bx + c never crosses the x-axis — there is no real value of x that makes the expression zero. In engineering contexts (like the characteristic equation of a damped oscillator), complex roots typically indicate oscillatory behavior rather than a simple real decay.
The vertex sits exactly halfway between the two roots when real roots exist, because the roots are symmetric about the axis of symmetry. Averaging the quadratic-formula roots, [(−b+√Δ)/(2a) + (−b−√Δ)/(2a)]/2, the √Δ terms cancel and leave exactly −b/(2a) — which also works when the roots are complex.
Yes — the height equation for a projectile, h(t) = h₀ + v₀t − ½gt², is a quadratic in time. Setting h(t) = 0 and solving with this calculator (using a = −g/2, b = v₀, c = h₀) gives the time the projectile lands, which is exactly what the Projectile Motion Calculator does under the hood for its landing-time computation.