256 Tools

Quadratic Equation Solver

Equation: ax² + bx + c = 0
a
b
c
Two real roots
x₁ = 3, x₂ = 2
Discriminant: 1
Vertex: (2.5, -0.25)

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A quadratic equation has the form ax² + bx + c = 0. Enter the coefficients a, b, and c and this solver finds the roots — real or complex — along with the discriminant that tells you how many real roots there are and the vertex of the parabola. Everything is calculated in your browser.

How to use

  1. Enter the coefficients a, b, and c.
  2. Read the roots, discriminant, and vertex below.
  3. Copy the roots if you need them elsewhere.

How it works

The solver computes the discriminant D = b² − 4ac. If D is positive there are two real roots, if zero there is one repeated root, and if negative the roots are complex. The roots come from the quadratic formula x = (−b ± √D) / (2a). The vertex of the parabola is at x = −b / (2a), with its y value found by substituting back. If a is zero the equation is linear, and it is solved as bx + c = 0 instead.

The quadratic formula x equals minus b plus or minus the square root of b squared minus 4ac, over 2a.x =−b ± √(b²−4ac)2a

Features

Real & complex roots

Handles all cases, including complex roots when the discriminant is negative.

Discriminant

See D = b² − 4ac and what it means for the roots.

Vertex

Get the parabola's turning point for graphing.

Private & instant

All math runs in your browser; nothing is uploaded.

When to use it

Homework

Check your answers to quadratic problems.

Graphing

Find roots and the vertex to sketch a parabola.

Physics

Solve motion equations that take a quadratic form.

Learning

See how the discriminant decides the type of roots.

Notes

  • If a = 0 the equation is linear, not quadratic.
  • A negative discriminant gives complex roots a ± bi.
  • Results are rounded for readability.
  • All calculations run on your device; nothing is uploaded.

FAQ

What is the discriminant?
It's b² − 4ac. Positive means two real roots, zero means one repeated root, and negative means two complex roots.
Does it handle complex roots?
Yes. When the discriminant is negative, the roots are shown in the form a ± bi.
What if a is 0?
Then it isn't quadratic. The tool solves the linear equation bx + c = 0 instead.
What is the vertex?
The highest or lowest point of the parabola, at x = −b / (2a), useful for graphing.
Is my data uploaded?
No. Everything is calculated in your browser and nothing leaves your device.

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