Quadratic Equation & Parabola Solver

The discriminant Δ is the expression under the square root in the quadratic formula: Δ = b^2 - 4ac

Its sign determines the nature of the roots without needing the full formula: a positive discriminant (Δ > 0) gives two distinct real roots, where the parabola crosses the x-axis twice. A zero discriminant (Δ = 0) gives one repeated real root at x = -b/2a, where the parabola's vertex just touches the x-axis. A negative discriminant (Δ < 0) gives two complex conjugate roots, x = -b/2a ± i√(|Δ|)/2a and the parabola never touches the real x-axis.

Graphically, y = ax^2 + bx + c is a parabola, a symmetric U-shaped curve. If a > 0 it opens upward with a minimum point; if a < 0 it opens downward with a maximum point.

The vertex (h, k) is the parabola's turning point. In vertex form, y = a(x - h)^2 + k, where: h = -b/2a and k = c - b^2/4a = f(h)

The axis of symmetry is the vertical line x = h = -b/2a that the parabola is mirror-symmetric about.

Parabolas have a reflective property (incoming parallel rays reflect off the surface straight to a focus point), which is why the shape appears in satellite dishes, headlight reflectors, and solar concentrators.

Beyond the quadratic formula, factoring and completing the square are two other ways to solve a quadratic equation by hand.

Factoring rewrites ax^2 + bx + c = 0 as (px + q)(rx + s) = 0, then applies the zero product property: set each factor to zero and solve. It's fastest when the coefficients are small integers.

Completing the square transforms the equation into (x + h)^2 = k form, solvable by taking square roots directly: divide by a so the leading coefficient is 1, add (b/2a)^2 to both sides, factor the left side as a perfect square, then take square roots. This derivation is exactly how the quadratic formula itself is proven.

Enter coefficients a, b, and c for ax^2 + bx + c = 0. The calculator computes the discriminant, classifies the roots (two real, one repeated, or complex conjugate), finds the vertex, and plots the parabola.

Δ = (-5)^2 - 4(1)(6) = 25 - 24 = 1

Since Δ > 0, there are two distinct real roots:

x_1 = (5 + √(1))/2 = 3, x_2 = (5 - √(1))/2 = 2

Vertex: h = 5/2 = 2.5, k = (2.5)^2 - 5(2.5) + 6 = -0.25

So the parabola crosses the x-axis at x = 2 and x = 3, with its minimum at (2.5, -0.25), matching what the calculator returns for its default inputs.

A quadratic equation also describes the parabolic path of a projectile under gravity. The vertical position y(t) of an object launched with initial velocity v_0 at angle θ from height y_0 is: y(t) = -1/2gt^2 + (v_0 sinθ)t + y_0 where g ≈ 9.81 m/s^2.

Setting y(t) = 0 and solving with the quadratic formula gives the total time of flight. This calculator solves the abstract ax^2+bx+c=0 form. For a dedicated projectile-motion tool with these physics variables, see the physics calculators on the site.

To decompose a rational expression whose denominator uses roots like these, see the partial fractions calculator. For systems of two linear equations instead of a single quadratic, use the system of equations calculator.