What is the quadratic formula?

x = (−b ± √(b² − 4ac)) / 2a. It gives both roots of ax² + bx + c = 0; the ± produces the two solutions.

What does the discriminant tell me?

The discriminant is b² − 4ac. If it is positive there are two real roots, if zero there is one repeated real root, and if negative there are two complex-conjugate roots.

What happens if a = 0?

Then it is not quadratic but linear: bx + c = 0 with the single solution x = −c/b. The tool detects this and solves the linear case instead.

What are the vertex and axis of symmetry?

The parabola y = ax² + bx + c has its turning point (vertex) at x = −b/2a, and that vertical line is the axis of symmetry. The vertex y-value is the minimum or maximum of the parabola.

Quadratic Equation Solver

A quadratic equation ax² + bx + c = 0 is solved by the quadratic formula x = (−b ± √(b² − 4ac)) / 2a. The part under the root, the discriminant b² − 4ac, decides what the roots look like: positive gives two distinct real roots, zero gives one repeated real root, and negative gives a complex-conjugate pair. Enter the three coefficients and this tool returns the roots (real or complex), the discriminant and what it means, the parabola's vertex and axis of symmetry, and the sum (−b/a) and product (c/a) of the roots. If a = 0 it falls back to the linear case bx + c = 0, and it flags the degenerate no-solution and all-numbers cases too.

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Coefficients

abc

Equation: x² −5x + 6 = 0

Roots

x₁ = 3, x₂ = 2

Two distinct real roots

Discriminant

Axis of symmetry

x = 2.5

Vertex

(2.5, -0.25)

Sum / product

5 / 6

x = (−b ± √(b² − 4ac)) / 2a. The discriminant b² − 4ac decides real vs. complex roots. Set a = 0 for a linear equation.

How to use

Enter coefficients a, b and c from ax² + bx + c = 0 (a may be negative or fractional).
Read the roots — two real, one repeated, or a complex conjugate pair.
Check the discriminant, vertex, axis of symmetry, and the sum and product of the roots.

Frequently asked questions

What is the quadratic formula?: x = (−b ± √(b² − 4ac)) / 2a. It gives both roots of ax² + bx + c = 0; the ± produces the two solutions.
What does the discriminant tell me?: The discriminant is b² − 4ac. If it is positive there are two real roots, if zero there is one repeated real root, and if negative there are two complex-conjugate roots.
What happens if a = 0?: Then it is not quadratic but linear: bx + c = 0 with the single solution x = −c/b. The tool detects this and solves the linear case instead.
What are the vertex and axis of symmetry?: The parabola y = ax² + bx + c has its turning point (vertex) at x = −b/2a, and that vertical line is the axis of symmetry. The vertex y-value is the minimum or maximum of the parabola.

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