# Quadratic & Linear Equation Solver

Solve quadratic equations ax² + bx + c = 0 and linear equations bx + c = 0 with detailed steps, discriminant analysis, vertex calculation, and axis of symmetry.

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

- **Category:** Math & Numbers

- **Keywords:** quadratic, linear, equation, solver, mathematics, roots, discriminant, parabola, vertex

## Overview

The Quadratic & Linear Equation Solver is a precise mathematical tool designed to calculate the roots of quadratic equations (ax² + bx + c = 0) and linear equations (bx + c = 0) while providing comprehensive analytical insights.

## Inputs

- **Input Format** (select)
- **Coefficient a / Equation** (text): Enter coefficient a or complete equation (e.g., 2x² + 3x - 5 = 0)
- **Coefficient b (x term)** (number): Enter coefficient b
- **Coefficient c (constant term)** (number): Enter coefficient c
- **Decimal Precision** (number): Number of decimal places
- **Show Solution Steps** (checkbox)
- **Calculate Vertex** (checkbox)
- **Calculate Axis of Symmetry** (checkbox)
- **Show Complete Formula** (checkbox)

## When to use

- When you need to find the roots of a quadratic equation quickly and accurately.
- When you require a step-by-step breakdown of the quadratic formula application for educational purposes.
- When you need to determine the vertex and axis of symmetry for graphing a parabola.

## How it works

- Select your input format, choosing either individual coefficients (a, b, c) or a complete equation string.
- Enter your values into the designated fields and adjust the decimal precision if necessary.
- Enable optional features like 'Show Solution Steps' or 'Calculate Vertex' to receive detailed mathematical analysis.
- Submit the data to instantly view the roots, discriminant analysis, and geometric properties of the equation.

## Use cases

- High school and college students verifying their algebra homework.
- Teachers creating educational materials with detailed solution steps.
- Engineers or designers calculating parabolic trajectories or structural curves.

## Frequently asked questions

### Can this tool solve linear equations?

Yes, by setting the 'a' coefficient to 0, the tool effectively solves linear equations of the form bx + c = 0.

### What does the discriminant analysis tell me?

The discriminant (b² - 4ac) indicates the nature of the roots: positive means two real roots, zero means one real root, and negative means two complex roots.

### Can I adjust the number of decimal places in the result?

Yes, you can specify your desired decimal precision in the settings, ranging from 0 to 10 places.

### Does the tool provide the vertex of the parabola?

Yes, if you enable the 'Calculate Vertex' option, the tool will output the (h, k) coordinates of the parabola's vertex.

### Is it possible to see the steps taken to reach the solution?

Yes, by checking the 'Show Solution Steps' box, the tool will display the algebraic process used to derive the roots.

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