# Quadratic Equation Calculator

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

> Canonical page: https://elysiatools.com/en/tools/quadratic-equation-calculator

- **Category:** Math & Numbers

- **Keywords:** quadratic equation, discriminant, vertex, roots, parabola, math

## 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.

## Inputs

- **Input Format** (select)
- **Coefficient a / Equation** (text): 1 or x^2 - 3x + 2 = 0
- **Coefficient b** (number)
- **Coefficient c** (number)
- **Decimal Places** (number)
- **Show Steps** (checkbox)

## 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

- Select your preferred input format: individual coefficients (a, b, c) or a complete equation string.
- Enter the equation or coefficients into the designated input fields.
- Adjust the decimal places and toggle the 'Show Steps' option if you need a detailed breakdown of the math.
- View 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.

## Frequently asked questions

### 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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