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Quadratic Equation Calculator

· Last updated：2026-04-21

Solve quadratic equations, classify the roots, and return discriminant, vertex, and factor information

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Written by the Elysia Tools team
Math502

Tool usage guide

Learn when to use this tool, what it supports, and how real users apply it.

Key facts

Category
Math & Numbers
Input types
select, text, number, checkbox
Output type
json
Sample coverage
4
API ready
Yes

Overview

The Quadratic Equation Calculator is a mathematical utility designed to solve quadratic equations instantly. By inputting either individual coefficients or a complete equation string, users can calculate real or complex roots, determine the discriminant, and find the vertex of the corresponding parabola. It also provides step-by-step solutions and factor information, making it an ideal tool for students, educators, and engineers who need quick, accurate algebraic analysis.

When to use

  • When you need to quickly find the roots of a quadratic equation without manual calculation.
  • When analyzing the geometric properties of a parabola, such as its vertex and axis of symmetry.
  • When verifying homework answers or generating step-by-step mathematical proofs for algebraic problems.

How it works

  1. 1Select your preferred input format: individual coefficients (a, b, c) or a complete equation string.
  2. 2Enter the equation or coefficients into the designated input fields.
  3. 3Adjust the decimal places and toggle the 'Show Steps' option if you need a detailed breakdown of the math.
  4. 4View the calculated roots, discriminant, vertex coordinates, and step-by-step solution in the structured JSON output.

Use cases

  • Students checking their algebra homework and reviewing step-by-step solutions for quadratic formulas.
  • Math teachers generating answer keys with detailed discriminant and vertex breakdowns for exams.
  • Engineers and physicists calculating parabolic trajectories or optimizing quadratic cost functions.

Examples

1. Solving a standard quadratic equation

High School Student

Background

A student is working on algebra homework and needs to find the roots of a polynomial while understanding how the answer was derived.

Problem

Needs to solve x^2 - 3x + 2 = 0 and review the mathematical steps.

How to use

Select the 'Complete Equation' format, enter the equation string, and enable the 'Show Steps' checkbox.

Input Format: equation, Equation: x^2 - 3x + 2 = 0, Show Steps: true

Outcome

The tool returns the roots [1, 2], the discriminant (1), the vertex coordinates, and a step-by-step factorization.

2. Analyzing a parabolic trajectory

Physics Student

Background

Analyzing projectile motion where the height is modeled by a quadratic function with specific decimal coefficients.

Problem

Needs to find the maximum height (vertex) and landing point (positive root) using coefficients a=-4.9, b=20, c=1.5.

How to use

Select 'Individual Coefficients', input the values for a, b, and c, and set the decimal places to 4 for high precision.

FAQ

Can I input a full equation instead of just coefficients?

Yes, you can select the 'Complete Equation' input format and type the equation directly, such as x^2 - 3x + 2 = 0.

Does this tool support complex roots?

Yes, if the discriminant is negative, the calculator will compute and return the complex roots of the equation.

Can I see the steps used to solve the equation?

Absolutely. Check the 'Show Steps' box before calculating to include a step-by-step breakdown in the results.

How do I change the precision of the results?

You can adjust the 'Decimal Places' setting to round the roots and vertex coordinates to your desired precision, up to 10 decimal places.

What information is included in the output?

The JSON output includes the roots, discriminant, vertex coordinates, factor information, and optional step-by-step solving procedures.

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Input Format: coefficients, a: -4.9, b: 20, c: 1.5, Decimal Places: 4

Outcome

The JSON output provides the precise vertex coordinates representing the maximum height, along with the roots representing the launch and landing times.