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Quadratic Equation Calculator

Solve ax squared plus bx plus c with roots, discriminant and vertex.

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ax² + bx + c = 0

Roots

—

The parabola

Discriminant (b² − 4ac)—
Vertex—
Axis of symmetry—
y-intercept—
Opens—
Factored form—

What the discriminant tells you

Positive: two distinct real roots. Zero: one repeated root, the curve just touches the x-axis. Negative: two complex roots and no x-intercept.

Quick Tool Overview

The Quadratic Equation Calculator solves ax² + bx + c = 0 using the quadratic formula, showing the substitution rather than just the answer. Real and complex roots are both handled, and the discriminant is reported first because it tells you what to expect before you solve.

What it does
Solves quadratic equations and describes the parabola
You give it
The coefficients a, b and c
You get back
Roots, discriminant, vertex, axis of symmetry and factored form
Best for
Algebra homework, physics problems and curve sketching
Calculators Free, unlimited use No sign-up Runs in your browser

About the Quadratic Equation Calculator

Enter the coefficients a, b and c and the calculator solves ax squared plus bx plus c equals zero using the quadratic formula, showing the substitution rather than just the answer. Real and complex roots are both handled.

The discriminant is reported first because it tells you what to expect before you solve: a positive discriminant means two distinct real roots, zero means one repeated root where the parabola just touches the x-axis, and a negative discriminant means two complex conjugate roots and no x-intercept at all. You also get the vertex, the axis of symmetry, the y-intercept and the factored form where the roots are rational, which covers most of what a question about a quadratic actually asks for.

Key features of the Quadratic Equation Calculator

Discriminant first

Positive means two distinct real roots, zero means one repeated root, negative means two complex conjugates and no x-intercept.

Complex roots supported

A negative discriminant returns the conjugate pair rather than "no solution".

Substitution shown

Your numbers are printed inside the quadratic formula, so the working can be copied into an answer.

Vertex and axis of symmetry

The turning point and its line are calculated, which covers most of what a curve-sketching question asks.

Factored form where possible

When the roots are rational the factorisation is given, which is usually the mark scheme answer.

How to use the Quadratic Equation Calculator

  1. 1

    Enter a, b and c

    Type the three coefficients from your equation in standard form. The coefficient a cannot be zero.

  2. 2

    Read the discriminant

    It tells you immediately whether you will get two real roots, one repeated root, or a complex pair.

  3. 3

    Check the roots and vertex

    Both roots are shown with the formula substitution, alongside the vertex and axis of symmetry for sketching the curve.

Frequently asked questions

x equals minus b, plus or minus the square root of b squared minus 4ac, all divided by 2a. It solves any quadratic in standard form.

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