Quadratic Equation Solver

Solve ax² + bx + c = 0 with detailed step-by-step explanations

ax² + bx + c = 0

Enter the coefficients a, b, and c to solve your quadratic equation

Cannot be zero
Quick examples (click to try):

🧮 Fraction Form Quadratic Solver

Solve equations with fractional coefficients: (1/2)x² + (3/4)x - (5/3) = 0

Supports fractions like 2/3, decimals, integers, or -3/4
Fraction examples:

Understanding Quadratic Equations - Complete Guide

March 27, 2026 12 min read Quadratic Solver v2.0

📖 What is a Quadratic Equation?

ax² + bx + c = 0

Where a, b, and c are numbers, and a ≠ 0

A quadratic equation is a second-degree polynomial equation in a single variable x. The name "quadratic" comes from "quadratus," the Latin word for square. These equations appear in physics, engineering, economics, and many real-world applications including projectile motion, area optimization, and financial modeling.

🔧 The Quadratic Formula

x = [-b ± √(b² - 4ac)] / 2a

This formula provides the solutions (roots) for any quadratic equation. The expression under the square root, b² - 4ac, is called the discriminant (Δ). It determines the nature of the solutions:

  • Δ > 0: Two distinct real roots
  • Δ = 0: One real root (double root)
  • Δ < 0: Two complex conjugate roots

🎯 Our Calculator Features

What makes our quadratic solver special:
  • Exact Fraction Results - Converts decimal roots to exact fractions when possible (supports denominators up to 1000)
  • Fraction Input Support - Enter coefficients as fractions like 1/2, -3/4, or decimals like 0.75
  • Step-by-Step Solutions - Shows complete work including discriminant calculation and formula application
  • Complex Number Support - Handles negative discriminants with proper imaginary unit (i) notation
  • Real-Time Validation - Checks for invalid inputs and provides helpful error messages

📐 How to Use This Calculator

  1. Standard Mode: Enter numeric values for a, b, and c in the first calculator
  2. Fraction Mode: Enter fractions (e.g., 1/2, -3/4) or decimals in the second calculator
  3. Click "Solve" to get step-by-step solutions with exact results
  4. Use quick example buttons to test common equation types

💡 Real-World Applications

Physics - Projectile Motion

h(t) = -16t² + v₀t + h₀ - Quadratic equations model the trajectory of objects under gravity.

Economics - Profit Maximization

Revenue and cost functions often form quadratic relationships, where the vertex represents maximum profit.

⚠️ Common Mistakes to Avoid

  • Forgetting a ≠ 0: If a = 0, the equation is linear, not quadratic
  • Sign errors: Pay attention to negative signs when substituting into the formula
  • Incorrect discriminant calculation: Remember b² - 4ac, not b² - 4a + c
  • Fraction simplification: Our calculator automatically simplifies fraction results

📊 Example Problems Solved by Our Calculator

Example 1: 2x² + 4x - 6 = 0
Solution: x = 1, x = -3
Example 2: (1/2)x² + (1/3)x - 5/6 = 0
Solution: x = 1, x = -5/3 ≈ -1.6667
Example 3: x² + 2x + 5 = 0
Solution: x = -1 ± 2i (complex roots)
Pro Tip: For equations with rational roots, our calculator will display both decimal and exact fraction forms. This is especially useful for algebra homework where teachers expect simplified fraction answers!