What is the quadratic formula?

The quadratic formula solves any equation of the form ax squared plus bx plus c equals zero. The two solutions are x equals negative b plus or minus the square root of the discriminant (b squared minus 4ac), all divided by 2a. The discriminant tells you how many real solutions exist: positive means two distinct real roots, zero means one repeated real root, and negative means no real roots (two complex roots).

What does the discriminant tell you?

The discriminant is b squared minus 4ac. If it is positive, the quadratic has two distinct real roots. If it equals zero, there is exactly one real root (a repeated root). If it is negative, there are no real roots and the solutions are complex (imaginary) numbers.

What is the vertex of a parabola?

The vertex is the highest or lowest point of the parabola. Its x-coordinate is negative b divided by 2a, which is also the axis of symmetry. The y-coordinate of the vertex is found by substituting that x back into the original equation.

Quadratic Formula Calculator

A quadratic equation has the standard form ax squared plus bx plus c equals zero, where a cannot be zero. These equations appear throughout algebra, physics, engineering, and finance, describing trajectories, areas, profit functions, and countless other relationships that follow a curved, parabolic path. The quadratic formula gives you both solutions at once: x equals negative b plus or minus the square root of the quantity b squared minus 4ac, all divided by 2a. The quantity under the square root, called the discriminant, carries important information: when it is positive there are two distinct real roots; when it equals zero there is exactly one repeated real root; and when it is negative there are no real roots and the solutions are complex numbers. This calculator solves any quadratic equation instantly. You enter the three coefficients a, b, and c, and it returns both roots (or reports if they are complex), the value of the discriminant, the vertex coordinates of the parabola, the axis of symmetry, and the step-by-step working so you can follow the method. Negative coefficients work as expected; simply enter a negative number. It is useful for students working through algebra or calculus, engineers solving for unknown values, and anyone who needs to find where a parabolic function crosses zero. Enter your coefficients to see all results update as you type.

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a (coefficient of x²)

b (coefficient of x)

c (constant)

x = 3 and x = 2

roots of the equation

Discriminant (Δ)1

Root typeTwo real roots

Vertex(2.5, −0.25)

Axis of symmetryx = 2.5

Working: x = (5 ± √1) / 2 = (5 ± 1) / 2. x₁ = 3, x₂ = 2.

How it works

The calculator computes the discriminant as b squared minus 4ac. When positive, it takes the square root and applies the formula to give two distinct real roots. When zero, both roots are the same repeated value of negative b over 2a. When negative, no real roots exist and the calculator notes that the solutions are complex. The vertex x-coordinate is negative b over 2a; the vertex y-coordinate is found by substituting this back into the original equation. The axis of symmetry is the vertical line through the vertex x-coordinate.

Worked example

With a = 1, b = -5, c = 6 the equation is x squared minus 5x plus 6 equals zero. The discriminant is (-5) squared minus 4(1)(6) = 25 minus 24 = 1. Because the discriminant is positive there are two real roots. Applying the formula: x = (5 plus or minus 1) divided by 2, giving x = 3 and x = 2. The vertex is at x = 2.5, y = 2.5 squared minus 5(2.5) plus 6 = -0.25.

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