Quadratic Formula Calculator

Solve algebraic equations, quadratic formulas, systems of equations, and polynomial problems. Master algebra for education and practical applications.

Quadratic Formula Calculator

Solve quadratic equations using the quadratic formula with step-by-step solutions and vertex analysis

Quadratic Equation

- 5x + 6 = 0

Coefficient of x² (cannot be 0)

Coefficient of x

Constant term

📐 Common Examples

🧮 Quadratic Formula

x = (-b ± √(b² - 4ac)) / 2a
Discriminant: Δ = b² - 4ac
• Δ > 0: Two real solutions
• Δ = 0: One solution (repeated root)
• Δ < 0: Complex solutions

Solutions

Solution 1

x₁ = 3

Solution 2

x₂ = 2

Discriminant

1

Two real

Vertex

(2.5, -0.25)

Minimum

Axis of Symmetry

x = 2.5

Y-Intercept

6

When x = 0

Factored Form

1(x - 3.000)(x - 2.000)

Step-by-Step Solution

1Given equation: x² - 5x + 6 = 0
2Coefficients: a = 1, b = -5, c = 6
3Discriminant: b² - 4ac = -5² - 4(1)(6) = 1
4Since discriminant > 0, there are two distinct real solutions
5Using quadratic formula: x = (-b ± √(b² - 4ac)) / 2a
6x = (5 ± √1) / 2
7x₁ = (5 + 1.000) / 2 = 3.000
8x₂ = (5 - 1.000) / 2 = 2.000

Parabola Properties

Opens: upward (a > 0)
Vertex: Minimum point at (2.5, -0.25)
Axis of symmetry: x = 2.5
Domain: All real numbers
Range: y ≥ -0.25

1Mastering Quadratic Formula Calculations

Solve quadratic equations using the quadratic formula with step-by-step solutions. Whether you're a student, educator, or professional, understanding the mathematical principles behind quadratic formula will enhance your problem-solving abilities. This guide breaks down complex concepts into understandable components and provides practical applications for real-world scenarios.

Mathematical Foundations

The underlying mathematics involves fundamental principles that have wide-ranging applications. Understanding these concepts helps you: • Verify calculator results manually • Apply knowledge to related problems • Develop mathematical intuition • Solve more complex variations • Teach others effectively

Step-by-Step Problem Solving

Approach problems systematically: 1. Identify what you're solving for 2. Gather all necessary information 3. Choose the appropriate formula 4. Perform calculations carefully 5. Verify results make sense 6. Consider alternative methods


2Real-World Applications

Mathematical concepts aren't just theoretical - they have practical applications in many fields.

Professional Applications

Professionals in various fields use these calculations: • Engineers for design and analysis • Scientists for research and experiments • Financial analysts for modeling • Architects for planning and construction • Data scientists for analysis and predictions

Educational Value

For students and educators: • Builds foundational math skills • Develops logical thinking • Prepares for advanced topics • Provides exam preparation • Enables peer tutoring and collaboration

This comprehensive guide is regularly updated to ensure accuracy. Last reviewed: 7/25/2025

Frequently Asked Questions

How do I solve quadratic equations?

Use factoring, completing the square, or the quadratic formula: x = (-b ± √(b²-4ac)) ÷ 2a. Choose the method based on the equation form.

What are the fundamental algebra rules?

Balance equations (same operation on both sides), combine like terms, distribute multiplication, and follow order of operations (PEMDAS).

When do I use algebra in real life?

Algebra helps with budgeting, calculating discounts, determining rates and ratios, solving engineering problems, and analyzing data trends.