Quadratic Equation Solver
Solve quadratic equations (ax² + bx + c = 0) and find both real and complex roots.
Solve ax² + bx + c = 0
What Is the Quadratic Equation Solver?
The quadratic equation solver finds the roots of any equation in the form ax² + bx + c = 0 using the quadratic formula. It handles real roots, repeated roots, and complex (imaginary) roots, showing complete step-by-step solutions including discriminant analysis.
Formula
How to Use
Enter the coefficients a, b, and c from your quadratic equation. The coefficient 'a' must not be zero. Click Solve to find the roots with full workings.
Example Calculation
For x² − 5x + 6 = 0 (a=1, b=−5, c=6): Discriminant = 25 − 24 = 1. Roots: x₁ = (5+1)/2 = 3, x₂ = (5−1)/2 = 2.
Understanding Quadratic Equation
Quadratic equations appear throughout mathematics, physics, engineering, and economics. They describe parabolic trajectories, optimize areas and costs, and model many natural phenomena.
The quadratic formula, derived by completing the square, provides a universal method for solving any quadratic equation. The discriminant (b² − 4ac) is the key to understanding the nature of the solutions before computing them.
A positive discriminant means two real roots — the parabola crosses the x-axis at two points. A zero discriminant means one repeated root — the parabola just touches the x-axis. A negative discriminant means complex roots — the parabola does not reach the x-axis.
Our solver displays the complete solution process, making it ideal for students learning algebra and anyone who needs quick, verified solutions to quadratic equations.
Frequently Asked Questions
What is the discriminant?
The discriminant (Δ = b² − 4ac) determines the nature of the roots. If Δ > 0, there are two distinct real roots. If Δ = 0, there is one repeated root. If Δ < 0, there are two complex conjugate roots.
What are complex roots?
Complex roots involve imaginary numbers (containing i = √−1). They occur when the discriminant is negative, meaning the parabola does not cross the x-axis.
Can this solve higher-degree equations?
This tool is specifically for quadratic (degree 2) equations. Cubic and quartic equations require different methods.