What is the quadratic formula?

x = (-b ± √(b²-4ac)) / 2a. The discriminant Δ = b²-4ac tells the number of real roots: Δ>0 two roots, Δ=0 one root, Δ<0 complex roots.

What does the discriminant tell us?

Δ = b²-4ac: positive → 2 real roots, zero → 1 root (double root), negative → no real roots (complex roots).

How do I solve x²-3x+2=0?

Here a=1, b=-3, c=2. Δ = 9-8 = 1. x = (3 ± 1) / 2. x₁ = 2, x₂ = 1.

The vertex is the highest or lowest point of the parabola. x-coordinate: -b/(2a), y-coordinate is found by substituting x into the equation.

📐 Quadratic Formula Calculator

Name: Quadratic Formula
Author: Miikka

Solve quadratic equations ax²+bx+c=0. Enter coefficients a, b and c — the calculator shows the discriminant, roots x₁ and x₂, and the vertex coordinates of the parabola.

By Miikka · CalcBrew · Updated Jul 11, 2026

ax² + bx + c = 0

Formula: x = (-b ± √(b²-4ac)) / 2a

Enter coefficients

a (coefficient of x²)b (coefficient of x)c (constant)

Discriminant (Δ)

Δ = 1

Two real roots

Roots

x₁

x₂

Vertex x

1.5

Vertex y

-0.25

This quadratic equation solver finds the roots of ax²+bx+c=0 from the coefficients a, b and c you enter. It returns the discriminant, the roots x₁ and x₂, and the vertex of the parabola, and it also handles cases where the roots are complex.

How the calculator works and what it’s for▾

How the solver works

The tool applies the quadratic formula x = (−b ± √(b²−4ac)) / (2a). At its heart is the discriminant D = b²−4ac, which tells you what kind of roots the equation has.

When D is positive there are two distinct real roots. When D is zero there is one repeated root. When D is negative there are no real roots and the solutions are complex numbers instead.

What you enter and what you get

Enter three coefficients: a is the coefficient of the squared term, b the coefficient of the linear term, and c the constant. The value of a must be non-zero, otherwise the equation is not quadratic.

You get the value of the discriminant, the roots x₁ and x₂, and the coordinates of the parabola's vertex. The vertex shows where the curve reaches its minimum or maximum point.

Worked example

Take the equation x²−5x+6=0, where a=1, b=−5 and c=6. The discriminant is (−5)²−4·1·6 = 25−24 = 1, so there are two real roots.

The quadratic formula gives x = (5 ± 1) / 2, so the roots are x₁=3 and x₂=2. You can always check the answer by substituting the roots back into the equation to confirm it equals zero.

Who it is for

The solver suits high school and college math practice, checking homework, and physics or engineering calculations where quadratic equations come up regularly.

Before solving, make sure the equation is rearranged into the standard form ax²+bx+c=0. A common mistake is leaving a term on the wrong side of the equals sign or mixing up the signs of the coefficients.

🔄 Reviewed June 2026

Frequently asked questions

Cite this calculator

APACalcBrew. (2026). Quadratic Formula. https://calcbrew.com/quadratic-formula-calculator

MLA“Quadratic Formula.” CalcBrew, 2026, calcbrew.com/quadratic-formula-calculator.

Link to https://calcbrew.com/quadratic-formula-calculator. Results are estimates; cite the underlying formula or data source for academic work.

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