Quadratic Equation Solver

Author: Henrick Yau
- 30 September 2026
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Calculators

Solve quadratic equations in the form ax² + bx + c = 0. Enter the coefficients a, b, and c to find the roots, discriminant, vertex, and graph the parabola.

Equation Coefficients

1x² + 0x + 0 = 0
Cannot be zero for quadratic equation
Linear term coefficient
Constant term

Display Options

Understanding Quadratic Equations

A quadratic equation is a polynomial equation of degree 2 in the form ax² + bx + c = 0, where a ≠ 0. These equations describe parabolic curves and have many real-world applications.

Key Concepts
  • Discriminant (Δ): b² - 4ac determines the nature of roots
  • Vertex: The highest or lowest point of the parabola at (-b/2a, f(-b/2a))
  • Axis of Symmetry: Vertical line x = -b/2a that divides the parabola
  • Roots/Zeros: x-values where the parabola crosses the x-axis
  • Y-intercept: Point where parabola crosses y-axis at (0, c)
Solution Methods
  • Quadratic Formula: x = (-b ± √(b² - 4ac)) / 2a
  • Factoring: Express as (px + q)(rx + s) = 0
  • Completing the Square: Rewrite as (x + h)² = k
  • Graphing: Find x-intercepts of the parabola
Types of Solutions
  • Two Real Roots: Δ > 0, parabola crosses x-axis twice
  • One Real Root: Δ = 0, parabola touches x-axis (perfect square)
  • Complex Roots: Δ < 0, parabola doesn't cross x-axis

Note: This calculator provides exact and approximate solutions for quadratic equations. For educational purposes, review the solution steps to understand the mathematical process. Complex solutions are displayed in a + bi form.

Solving ax² + bx + c = 0 with the Quadratic Formula

The Quadratic Equation Calculator is a handy tool for tackling equations in the standard form ax² + bx + c = 0. You simply key in the coefficients (a, b, and c) to get quick and accurate solutions. This calculator supports various solving methods and visual aids, making it great for students, teachers, and anyone dealing with quadratic functions.

Quadratic Formula:
x = (-b ± √(b² - 4ac)) / 2a

Why Use This Quadratic Solver?

This calculator lets you:

Whether you're doing homework or brushing up on concepts, this quadratic formula solver serves as a dependable math problem solver and learning aid.

How to Solve a Quadratic Equation

Just follow these straightforward steps:

  • Key in the values for coefficients a, b, and c.
  • Pick a solving method: Quadratic Formula, Factoring, or Completing the Square.
  • Decide if you want to display solution steps and the graph.
  • Set decimal precision and graph range as needed.
  • Hit Solve Equation to view the results.

What You Get: Roots, Discriminant, Vertex, and Graph

The output includes:

  • Roots: The x-values where the equation equals zero (real or complex).
  • Discriminant: Shows the nature of the roots.
  • Vertex: The turning point of the parabola.
  • Graph: A visual plot of the parabola, highlighting key features.
  • Equation Properties: Domain, range, intercepts, and line of symmetry.

Learning Made Easy with Step-by-Step Solutions

Figuring out how to solve quadratic equations becomes simpler when you can follow each stage. This tool breaks down every part of the solution, acting as a math solver tool and quadratic formula solution guide. It's especially handy for those studying algebra or gearing up for exams.

Common Questions About Solving Quadratics with This Tool

  • Can this tool handle all quadratic equations?
    Yes, as long as coefficient a ≠ 0, it can solve any quadratic equation, even those with complex roots.
  • Does it show all solution steps?
    Yes, you can opt to view the full step-by-step breakdown for the chosen solving method.
  • Can I graph the equation?
    Absolutely. Turn on the graph option to see a detailed plot of the parabola with labelled features.
  • What if the equation can’t be factored?
    If factoring isn't possible, the calculator automatically switches to the quadratic formula.

Other Useful Math Tools

This calculator pairs well with other tools that support learning and equation solving:

Going Beyond the Formula: Understanding Quadratics

The Quadratic Equation Calculator is more than just a formula tool—it's a practical resource for understanding, solving, and visualising quadratic equations. Whether you're learning to find roots of quadratic functions or just need a fast answer, this quadratic formula calculator is a reliable companion.