Quadratic Equation Solver
Solve any quadratic equation and visualize its graph instantly. Get roots, vertex, and step-by-step solution in one place.
x₂ = 1.0000
Understanding Quadratic Equations
A quadratic equation is a second-degree polynomial equation of the form ax² + bx + c = 0, where a ≠ 0. These equations are fundamental in algebra and appear in various domains such as physics (projectile motion, optics), engineering (parabolic structures), economics (profit maximization), and computer science (algorithm analysis).
The graph of a quadratic equation is a parabola, which opens upward if a > 0 and downward if a < 0. The vertex represents the maximum or minimum point, and the axis of symmetry passes through the vertex. The roots (solutions) are the x-values where the parabola intersects the x-axis.
Quadratic equations can have two real roots, one real root (repeated), or two complex roots, depending on the discriminant (Δ = b² - 4ac). This tool helps you analyze all these cases instantly.
How This Tool Works
The Quadratic Equation Solver takes your input coefficients (a, b, c) and applies the quadratic formula to find the roots. It calculates the discriminant to determine the nature of the roots (real or complex), computes the vertex, and generates a graph of the parabola. The tool also provides a step-by-step breakdown of the solution, making it easy to follow the math behind the results.
The graph visualization helps you understand how the coefficients affect the shape and position of the parabola. The tool is designed for students, teachers, engineers, and anyone who needs quick, accurate solutions to quadratic equations.
Key Features
- Instant Solution: Get roots, vertex, discriminant, and graph as soon as you enter coefficients.
- Step-by-Step Explanation: See each step of the quadratic formula, including handling of complex roots.
- Graph Visualization: Interactive graph shows the parabola, vertex, axis of symmetry, and roots.
- Handles All Cases: Supports equations with two real roots, one real root, or complex roots.
- Responsive Design: Works perfectly on desktop (85% width) and mobile (95% width).
- Copy Results: Easily copy the solution for homework, reports, or sharing.
- Educational Content: Includes a comprehensive guide to quadratic equations and their applications.
- Privacy Friendly: All calculations are performed in your browser; no data is sent anywhere.
This tool is ideal for students, teachers, engineers, and anyone who needs to solve or understand quadratic equations quickly and visually.
Common Use Cases
- Education: Students and teachers use this tool to solve homework problems, check answers, and visualize concepts.
- Engineering: Engineers analyze parabolic structures, projectile motion, and optimization problems.
- Physics: Used in kinematics (projectile motion), optics (parabolic mirrors), and more.
- Economics: Model profit maximization, cost minimization, and supply-demand curves.
- Computer Science: Analyze algorithms with quadratic time complexity, graphics, and simulations.
- Everyday Math: Anyone needing to solve or understand quadratic equations for practical problems.
Whether you are a student, professional, or enthusiast, this solver adapts to your needs and provides clear, actionable results.
Step-by-Step Guide
- Enter Coefficients: Input values for a, b, and c. Remember, a ≠ 0 for a quadratic equation.
- Review the Equation: The formatted equation is shown for clarity.
- Analyze the Results: See the discriminant, vertex, axis of symmetry, and roots.
- View the Graph: The parabola, vertex, and roots are visualized for better understanding.
- Show Steps: Toggle the step-by-step solution to see the math behind the answer.
- Copy Results: Use the copy button to save or share your results.
- Try Examples: Use the example button to load a sample equation and see how it works.
For best results, use the tool in a modern browser. If you enter invalid coefficients (like a = 0), the tool will alert you. All calculations are instant and private.