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Quadratic Formula Calculator

Solve quadratic equations ax² + bx + c = 0 with step-by-step solutions. Perfect for students learning algebra and understanding discriminants.

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Enter Coefficients

Standard Form
ax² + bx + c = 0
Cannot be 0
Linear coefficient
Constant term
💡 Tip: You can use fractions like 3/4, decimals like 1.5, or integers like 2.

Solution

Step-by-Step Solution

Enter coefficients to see the step-by-step solution

Understanding Quadratic Equations

What is a Quadratic Equation?

A quadratic equation is any equation that can be written in the standard form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. The highest power of the variable is 2, making it a second-degree polynomial equation.

The Quadratic Formula

x = (-b ± √(b² - 4ac)) / (2a)

This formula gives you the solutions (roots) to any quadratic equation. The ± symbol means there are usually two solutions.

Real-World Applications

  • Physics: Calculating projectile motion and trajectory paths
  • Engineering: Designing parabolic structures like bridges and antennas
  • Economics: Optimizing profit and cost functions
  • Architecture: Creating curved designs and optimal shapes

The Discriminant

Discriminant = b² - 4ac
If > 0: Two real, distinct solutions
If = 0: One real solution (repeated root)
If < 0: Two complex solutions

Solving Methods

  • Quadratic Formula: Works for all quadratic equations
  • Factoring: When the equation factors nicely
  • Completing the Square: Useful for vertex form
  • Graphing: Visual method to find x-intercepts

Common Examples

x² - 5x + 6 = 0
Solutions: x = 2, x = 3
2x² + 4x - 6 = 0
Solutions: x = 1, x = -3

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