Free to Use

Equation Solver

Solve linear, quadratic, and cubic equations online with step-by-step solutions. Perfect for students, teachers, and professionals who need detailed algebraic solutions with clear explanations.

Equation Solver Tool

Linear Equation: ax + b = 0

Coefficient a
Constant b

Quadratic Equation: ax² + bx + c = 0

Coefficient a
Coefficient b
Constant c

Cubic Equation: ax³ + bx² + cx + d = 0

Coefficient a
Coefficient b
Coefficient c
Constant d

General Equation Input

What is an Equation Solver?

An equation solver is a powerful mathematical tool that automatically finds the values of unknown variables (typically x) that satisfy given algebraic equations. Our online equation solver can handle linear, quadratic, and cubic equations, providing not just the final answers but also detailed step-by-step solutions to help you understand the solving process.

Whether you're a student learning algebra, a teacher preparing lessons, or a professional dealing with mathematical problems, our equation solver simplifies complex calculations and provides educational insights into algebraic problem-solving techniques.

Types of Equations We Can Solve

Linear Equations
ax + b = 0

Single-variable equations with the highest power of 1.

Example: 2x + 6 = 0 → x = -3
Quadratic Equations
ax² + bx + c = 0

Second-degree polynomial equations that may have two solutions.

Example: x² - 5x + 6 = 0 → x = 2, 3
Cubic Equations
ax³ + bx² + cx + d = 0

Third-degree polynomial equations with up to three solutions.

Example: x³ - 6x² + 11x - 6 = 0 → x = 1, 2, 3

How to Use the Equation Solver

Our equation solver is designed to be intuitive and easy to use. Follow these simple steps:

For Linear Equations (ax + b = 0):

  1. Select "Linear Equations" tab - Choose the appropriate equation type
  2. Enter coefficient 'a' - The number multiplying x
  3. Enter constant 'b' - The constant term
  4. Click "Solve Linear Equation" - Get instant solution with steps

For Quadratic Equations (ax² + bx + c = 0):

  1. Select "Quadratic Equations" tab - Switch to quadratic mode
  2. Enter coefficients a, b, c - Fill in all three values
  3. Solve automatically - The solver uses the quadratic formula
  4. View discriminant analysis - Understand the nature of solutions

For Cubic Equations (ax³ + bx² + cx + d = 0):

  1. Select "Cubic Equations" tab - Choose cubic equation mode
  2. Enter all coefficients - Input a, b, c, and d values
  3. Get complex solutions - May include real and complex roots
  4. Follow step-by-step process - Understand advanced solving methods

For General Form Input:

  1. Select "General Form" tab - Use natural equation input
  2. Type your equation - Enter in standard mathematical notation
  3. Automatic parsing - The solver identifies equation type
  4. Get comprehensive solution - Detailed steps and explanations

Mathematical Formulas and Methods

Our equation solver uses proven mathematical formulas and methods to find accurate solutions:

Linear Equation Formula

x = -b/a

For linear equations ax + b = 0, where a ≠ 0.

Quadratic Formula

x = (-b ± √(b² - 4ac)) / (2a)

The discriminant (b² - 4ac) determines the nature of solutions:

Cubic Equation Methods

For cubic equations, we use advanced methods including:

Solver Features

Our equation solver offers comprehensive features to enhance your mathematical problem-solving experience:

📝 Step-by-Step Solutions

Detailed explanations of each solving step, helping you understand the mathematical process and learn problem-solving techniques.

🔢 Multiple Equation Types

Support for linear, quadratic, and cubic equations with both coefficient input and general form parsing.

✅ Solution Verification

Automatic verification of solutions by substituting back into the original equation to ensure accuracy.

🧮 Complex Number Support

Handles complex solutions for quadratic and cubic equations when real solutions don't exist.

📊 Discriminant Analysis

For quadratic equations, provides discriminant analysis to predict the nature and number of solutions.

📱 Mobile Optimized

Fully responsive design that works perfectly on desktop, tablet, and mobile devices.

Example Problems and Solutions

Here are some example problems to demonstrate how our equation solver works:

Linear Equation Example

Solve: 3x + 12 = 0
Step 1: 3x + 12 = 0
Step 2: 3x = -12
Step 3: x = -12/3
Solution: x = -4

Quadratic Equation Example

Solve: x² - 5x + 6 = 0
Step 1: Identify a=1, b=-5, c=6
Step 2: Calculate discriminant: (-5)² - 4(1)(6) = 25 - 24 = 1
Step 3: Apply quadratic formula: x = (5 ± √1)/2
Solutions: x = 3, x = 2

Cubic Equation Example

Solve: x³ - 6x² + 11x - 6 = 0
Step 1: Try rational roots using rational root theorem
Step 2: Test x = 1: 1 - 6 + 11 - 6 = 0 ✓
Step 3: Factor out (x-1): (x-1)(x²-5x+6) = 0
Solutions: x = 1, x = 2, x = 3

Real-World Applications

Equation solving has numerous practical applications across various fields:

🎓 Education & Learning

Students can check homework answers, understand solving methods, and practice algebraic techniques with immediate feedback.

🔬 Science & Engineering

Solve equations in physics, chemistry, and engineering problems involving motion, rates, concentrations, and design parameters.

💰 Finance & Economics

Calculate break-even points, optimize profit functions, and solve economic models involving quadratic cost functions.

📐 Geometry & Design

Find dimensions, areas, and volumes in geometric problems, architectural design, and space optimization.

📊 Data Analysis

Fit polynomial models to data, find intersections of curves, and solve regression equations.

🎯 Optimization Problems

Find maximum and minimum values in business optimization, resource allocation, and efficiency problems.

Tips and Best Practices

Input Guidelines:

Understanding Solutions:

Important Note: While our solver handles most standard equations accurately, extremely large coefficients or very small numbers may be subject to floating-point precision limitations. For critical applications, verify results independently.

Frequently Asked Questions (FAQ)

What types of equations can this solver handle?
Our equation solver can handle linear equations (ax + b = 0), quadratic equations (ax² + bx + c = 0), and cubic equations (ax³ + bx² + cx + d = 0). It provides step-by-step solutions for all these equation types with both real and complex solutions when applicable.
How do I enter equations in general form?
Use the "General Form" tab and enter equations using standard mathematical notation. For example: "2x + 5 = 0", "x^2 - 4x + 3 = 0", or "x^3 + 2x^2 - x - 2 = 0". The solver will automatically parse and identify the equation type.
What does it mean when I get complex solutions?
Complex solutions occur when quadratic or cubic equations don't have real number solutions. These are expressed as a + bi, where i is the imaginary unit (√-1). Complex solutions are mathematically valid and important in advanced mathematics and engineering applications.
How accurate are the solutions provided?
Our solver uses precise mathematical algorithms and provides highly accurate solutions for most equations. However, due to floating-point arithmetic limitations, very large numbers or equations with extreme coefficients may have minor precision variations. Solutions are typically accurate to 10+ decimal places.
Can I solve equations with fractions or decimals?
Yes! You can enter coefficients as decimals (like 0.5, 2.75) or convert fractions to decimals before input. For example, if you have 1/2 as a coefficient, enter it as 0.5. The solver handles decimal coefficients accurately.
What is the discriminant and why is it important?
The discriminant (b² - 4ac) in quadratic equations tells us about the nature of solutions: positive means two distinct real solutions, zero means one repeated solution, and negative means two complex solutions. Our solver automatically calculates and explains the discriminant's significance.
Can I use this solver for homework help?
Absolutely! Our solver is designed to be educational, providing step-by-step solutions that help you understand the solving process. Use it to check your work, learn solving techniques, and understand where you might have made errors in manual calculations.
What should I do if the solver says "no solution"?
"No solution" typically occurs with linear equations where the coefficient of x is zero (like 0x + 5 = 0), making the equation either impossible (5 = 0) or an identity (0 = 0). Check your equation input and ensure it's properly formatted.
How do I interpret step-by-step solutions?
Our step-by-step solutions show each mathematical operation clearly, from the original equation to the final answer. Each step explains the mathematical principle being applied, helping you understand not just what to do, but why each step is necessary.
Is this solver suitable for advanced mathematics courses?
Yes! Our solver handles equations commonly found in algebra, pre-calculus, and calculus courses. It's suitable for high school through early college mathematics, providing the detailed explanations needed for advanced learning.

About This Equation Solver

Our equation solver was developed to bridge the gap between manual calculation and computer-assisted learning. It combines the power of algorithmic solving with educational step-by-step explanations, making it an ideal tool for both learning and practical problem-solving.

Why Choose Our Equation Solver?

Mathematical Accuracy

Our solver implements standard mathematical algorithms including the quadratic formula, Cardano's method for cubic equations, and numerical techniques for complex cases. All solutions are verified through substitution to ensure accuracy.

Privacy and Security

All calculations are performed locally in your browser. No equation data is transmitted to our servers, ensuring complete privacy of your mathematical work.

Disclaimer: While we strive to provide accurate solutions for all standard equations, this tool is designed for educational and general use. For critical engineering, scientific, or financial calculations, please verify results with appropriate professional software or consultation.