Advanced Quadratic Equation Solver

Instantly solve any quadratic equation with real-time graphing, step-by-step solutions, and history tracking. The ultimate tool for students and professionals.

Enter Coefficients

ax² + bx + c = 0

Coefficient a (x²)

Coefficient b (x)

Constant term c

Enter any real numbers for a, b, and c. Coefficient 'a' cannot be zero.

Solution

Enter coefficients and click "Solve Equation" to see the solution

Equation Information

Discriminant

Root Type

Vertex

Axis of Symmetry

The graph shows the quadratic function f(x) = ax² + bx + c. Roots are marked where the parabola intersects the x-axis.

About Quadratic Equation Solver

What is a Quadratic Equation?

A quadratic equation is a second-degree polynomial equation in a single variable x, with the general form: ax² + bx + c = 0 where a, b, and c are coefficients, and a ≠ 0.

Quadratic equations appear in various fields including physics, engineering, economics, and computer science. They describe parabolic paths, optimal solutions, and many natural phenomena.

Quadratic Formula

x = [-b ± √(b² - 4ac)] / (2a)

Advanced Features

Interactive Graphing: Visualize the parabola with zoom controls and root markers
Step-by-Step Solutions: Detailed breakdown of the solving process
Solution History: Track your previous equations and solutions
Fraction Mode: View solutions as fractions for exact representation
Random Examples: Generate random quadratic equations to practice

Frequently Asked Questions

What is the discriminant in a quadratic equation?

The discriminant (D = b² - 4ac) determines the nature of the roots:

D > 0: Two distinct real roots
D = 0: Exactly one real root (repeated root)
D < 0: Two complex conjugate roots

How can I interpret the graph of a quadratic equation?

The graph of a quadratic equation is a parabola. The roots are the x-values where the parabola intersects the x-axis. The vertex represents the minimum or maximum point of the function, and the axis of symmetry is a vertical line through the vertex.

What practical applications do quadratic equations have?

Quadratic equations are used in physics for projectile motion, in engineering for structural design, in economics for profit maximization, in computer graphics for curve rendering, and in many other fields where relationships between variables follow a parabolic pattern.