Quadratic Equation Solver

What Is a Quadratic Equation Solver?

This free quadratic equation solver finds the roots, discriminant, vertex, and axis of symmetry for any equation in the form ax² + bx + c = 0. It handles all three possible cases — two distinct real roots, one repeated root, or complex (imaginary) roots — with a clear explanation of each result. In short, this quadratic equation solver does in seconds what takes several minutes of careful hand calculation.

Whether you are a student checking algebra homework, a teacher demonstrating the quadratic formula, or an engineer solving a physics problem, this quadratic equation solver shows every property of the equation in one place.

How Does This Quadratic Equation Solver Work?

This quadratic equation solver applies the quadratic formula to find the roots, then calculates the vertex and axis of symmetry from the same coefficients.

For example, for x² − 5x + 6 = 0 (a=1, b=−5, c=6): Δ = 25 − 24 = 1. x = (5 ± 1) ÷ 2. Therefore x₁ = 3 and x₂ = 2. This quadratic equation solver shows all of these results plus the vertex and axis of symmetry instantly.

How to Use This Quadratic Equation Solver

1. Enter coefficient a — the number multiplied by x². Cannot be 0 (that would make it a linear equation).
2. Enter coefficient b — the number multiplied by x. Can be 0 for equations like x² + 4 = 0.
3. Enter coefficient c — the constant term. Can be 0 for equations like x² + 3x = 0.
4. Click Solve — the quadratic equation solver shows roots, discriminant, vertex, nature, and axis of symmetry.

Understanding the Discriminant in This Quadratic Equation Solver

The discriminant (Δ = b² − 4ac) is the key to understanding what type of roots your equation has. This quadratic equation solver calculates it first and uses it to determine the nature of the solution.

• Δ > 0: Two distinct real roots. The parabola crosses the x-axis at two different points.
• Δ = 0: One repeated real root. The parabola touches the x-axis at exactly one point — the vertex.
• Δ < 0: Two complex (imaginary) roots. The parabola does not cross the x-axis at all. This quadratic equation solver expresses these as p ± qi.

What Is the Vertex and Why Does This Quadratic Equation Solver Show It?

The vertex is the turning point of the parabola — the maximum point if the parabola opens downward (a < 0) or the minimum point if it opens upward (a > 0). This quadratic equation solver shows the vertex coordinates because they are essential for graphing the parabola and for optimisation problems in physics, economics, and engineering.

Is This Quadratic Equation Solver Accurate?

This quadratic equation solver uses JavaScript’s double-precision floating-point arithmetic, accurate to approximately 15 significant digits. Results are displayed to 6 decimal places. According to Wolfram MathWorld’s quadratic formula reference, the quadratic formula is the complete and exact algebraic solution for any second-degree polynomial equation with real or complex coefficients.

Quadratic Equation Solver — Frequently Asked Questions