📐 Quadratic Equation Solver

Solve quadratic equations and analyze their properties

ax² + bx + c = 0
Enter coefficients a, b, and c to solve the quadratic equation
Coefficient a
x² +
Coefficient b
x +
Coefficient c
= 0
Discriminant Analysis
Discriminant (Δ)
16
Nature of Roots
Real & Distinct
Vertex
(0, -4)
Solution Steps
Parabola Properties
Example Equations
Simple Quadratic
x² - 4 = 0
Standard Form
x² - 5x + 6 = 0
Complex Roots
x² + x + 1 = 0
Perfect Square
x² - 6x + 9 = 0
Physics Problem
-4.9t² + 20t + 2 = 0

📐 Complete Guide to Using a Quadratic Equation Solver

Master quadratic equations with step-by-step solutions

A quadratic equation solver is an essential tool for solving second-degree polynomial equations of the form ax² + bx + c = 0. Whether you're working on algebra homework, physics problems, or engineering calculations, our online quadratic equation solver provides instant, accurate solutions with detailed steps right in your browser.

🎯 What is a Quadratic Equation Solver?

A quadratic equation solver is a tool that finds the roots (solutions) of quadratic equations using the quadratic formula. It calculates both real and complex solutions, shows the discriminant, provides step-by-step solutions, and can graph the parabola. This eliminates manual calculations and helps you understand the solution process.

💡 Why Use an Online Quadratic Equation Solver?

Our online solver offers several advantages: instant solutions with detailed steps, handles both real and complex roots, shows discriminant and vertex, provides visual graph, works on all devices, and requires no installation. Perfect for students, teachers, engineers, and anyone working with quadratic equations!

🏃 Common Uses for a Quadratic Equation Solver

📚
Algebra Homework
Solve quadratic equations quickly. Verify your manual calculations.
🔬
Physics Problems
Calculate projectile motion, acceleration, and trajectories.
🏗️
Engineering
Solve structural, electrical, and mechanical engineering problems.
📊
Data Analysis
Fit quadratic models to data. Analyze parabolic relationships.
🎓
Test Preparation
Practice for SAT, ACT, GRE. Check answers quickly.
👨‍🏫
Teaching
Create examples and verify solutions. Demonstrate concepts visually.

📖 How to Use Our Quadratic Equation Solver

Using our solver is simple:

  1. Enter Coefficient a: The coefficient of x² (cannot be 0)
  2. Enter Coefficient b: The coefficient of x
  3. Enter Coefficient c: The constant term
  4. Solve: Click to see solutions
  5. View Results: See roots, discriminant, vertex, and graph
  6. Try Examples: Use quick example buttons to learn
📝 Example: Simple Quadratic

Equation: x² - 4 = 0 (a=1, b=0, c=-4)

Solutions: x₁ = 2, x₂ = -2. Two real roots!

⚡ Pro Tips for Quadratic Equations

1
Check the Discriminant
b² - 4ac > 0: Two real roots. = 0: One real root. < 0: Two complex roots.
2
Verify Your Answer
Substitute solutions back into original equation. Both sides should equal zero.
3
Understand the Graph
Roots are x-intercepts. Vertex is the turning point. a > 0: opens up, a < 0: opens down.
4
Simplify First
Factor out common terms before solving. Makes calculations easier and cleaner.

🔑 Key Features

  • Complete Solutions: Both roots calculated with the quadratic formula.
  • Step-by-Step: Detailed solution process shown.
  • Discriminant Analysis: Understand the nature of roots.
  • Vertex Calculation: Find the parabola's turning point.
  • Visual Graph: See the parabola and roots plotted.
  • Complex Roots: Handles imaginary solutions.
  • Quick Examples: One-click example equations.
  • Mobile Friendly: Works perfectly on all devices.
  • No Installation: Works directly in your browser.
  • Always Free: No subscriptions or hidden fees.

💪 Benefits of Using a Quadratic Equation Solver

Having a reliable quadratic equation solver offers numerous advantages:

🎯 Accurate Solutions Every Time

Eliminate calculation errors and understand the complete solution process. Perfect for learning, homework verification, and professional applications.

  • Save Time: Instant solutions instead of manual calculations.
  • Avoid Errors: Accurate results every time.
  • Learn Better: See step-by-step solutions.
  • Visual Understanding: Graph helps conceptualize solutions.
  • Verify Work: Check your manual calculations.
  • Handle Complex Numbers: Solves all types of quadratics.
  • Always Available: Access from any device.

🎓 Understanding Quadratic Equations

Standard Form

ax² + bx + c = 0

Where a ≠ 0, and a, b, c are real numbers (coefficients).

The Quadratic Formula

x = [-b ± √(b² - 4ac)] / (2a)

This formula always works for any quadratic equation!

The Discriminant

Δ = b² - 4ac

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

The Vertex

Vertex: (-b/(2a), f(-b/(2a)))

The vertex is the minimum (a > 0) or maximum (a < 0) point of the parabola.

Methods to Solve Quadratic Equations

  • Quadratic Formula: Works for all quadratics
  • Factoring: When equation factors nicely
  • Completing the Square: Useful for deriving vertex form
  • Graphing: Visual method to find approximate roots
📐 Real-World Example: Projectile Motion

Problem: A ball is thrown with equation h(t) = -5t² + 20t + 2. When does it hit the ground?

Solution: Set h(t) = 0. Solve -5t² + 20t + 2 = 0. Roots: t ≈ 4.1s (positive root is the answer)

❓ Frequently Asked Questions (FAQ)

What is a quadratic equation?
A quadratic equation is a second-degree polynomial equation in the form ax² + bx + c = 0, where a ≠ 0. The highest power of x is 2.
How do I know if my equation is quadratic?
Check if the highest power of the variable is 2 (x²). The equation must have an x² term with a non-zero coefficient.
Can a quadratic equation have no real solutions?
Yes! When the discriminant (b² - 4ac) is negative, the equation has two complex (imaginary) solutions instead of real solutions.
What does the discriminant tell me?
The discriminant (b² - 4ac) determines the nature of roots: positive = two real roots, zero = one repeated root, negative = two complex roots.
How do I verify my solution?
Substitute each root back into the original equation. If both sides equal zero, your solution is correct!
What if coefficient a is 0?
If a = 0, it's not a quadratic equation—it's linear (bx + c = 0). The coefficient of x² must be non-zero for a quadratic.
What is the vertex of a parabola?
The vertex is the turning point (maximum or minimum) of the parabola. It's located at x = -b/(2a), and represents the peak or valley of the curve.
Will this work on my phone?
Yes! The quadratic equation solver is fully responsive and works perfectly on smartphones and tablets. Solve equations anywhere.
Can I use this for physics problems?
Absolutely! Quadratic equations appear in projectile motion, acceleration, optics, and many physics applications. Our solver handles all these cases.
How accurate is this solver?
Our solver uses the standard quadratic formula and is mathematically precise. Results are accurate for academic, professional, and personal use.

🚀 Getting Started

Ready to solve quadratic equations? Scroll back up and enter your coefficients! Input values for a, b, and c, click solve, and see complete solutions with steps, discriminant, vertex, and graph. Try the example buttons to learn different types of quadratic equations!

🎯 Quick Start Tips

  • Enter coefficients a, b, and c from your equation ax² + bx + c = 0
  • Coefficient a cannot be zero (must be quadratic)
  • Try example buttons to see different solution types
  • Check the discriminant to understand root nature
  • Verify solutions by substituting back into original equation

📊 Conclusion

A quadratic equation solver is an indispensable tool for students, teachers, engineers, and anyone working with second-degree polynomials. Our free online solver provides instant, accurate solutions with detailed steps and visual graphs.

Whether you're solving algebra homework, physics problems, or engineering calculations, our quadratic equation solver gives you the complete solution with step-by-step explanations, helping you understand not just the answer, but the process.

Start solving quadratic equations now and master algebra with confidence!