Quadratic Equation Calculator
Solve any quadratic equation in the form ax² + bx + c = 0. Get real or complex roots, discriminant analysis, vertex, and step-by-step solutions instantly.
Enter Your Quadratic Equation
Solution Results
Enter coefficients a, b, c to solve your quadratic equation.
What is a Quadratic Equation?
A quadratic equation is a second-degree polynomial equation in a single variable x, with the general form: ax² + bx + c = 0, where a, b, and c are coefficients and a ≠ 0.
The "quadratic" part of the name comes from "quadratus," which is Latin for "square." This refers to the x² term, which is the highest power in the equation. Quadratic equations are fundamental in algebra and appear in various real-world applications including physics, engineering, economics, and computer graphics.
Key Components of a Quadratic Equation:
Coefficient a (x² term)
Determines the parabola's direction: positive (opens upward) or negative (opens downward)
Coefficient b (x term)
Affects the position of the axis of symmetry and vertex
Coefficient c (constant)
Determines the y-intercept of the parabola
How to Use This Quadratic Calculator
- Enter Coefficients: Input values for a, b, and c from your equation ax² + bx + c = 0
- Click Solve: Press the "Solve Quadratic Equation" button to calculate
- Review Results: View the roots, discriminant value, vertex coordinates, and complete solution steps
- Try Examples: Use the example buttons to see different types of quadratic equations
The Quadratic Formula Explained
This is the quadratic formula - it solves ANY quadratic equation. The ± symbol means there are usually two solutions.
Why Use This Quadratic Calculator?
100% Accurate Solutions
Get mathematically precise roots for any quadratic equation
Instant Results
Solve complex equations in seconds instead of minutes
Learning Tool
Step-by-step solutions help understand the process
Complete Analysis
Get discriminant, vertex, graph properties, and more
Mobile Friendly
Use on any device - phone, tablet, or computer
Free & No Login
Completely free tool, no registration required
Understanding the Discriminant
The discriminant (Δ) determines the nature of roots
Who Should Use This Quadratic Calculator?
Students & Teachers
- Homework help and verification
- Test preparation and practice
- Understanding solution steps
Professionals & Engineers
- Quick calculations for projects
- Physics and engineering problems
- Data analysis and modeling
Real-World Applications of Quadratic Equations
1. Physics & Motion
Projectile motion, free fall, and acceleration problems often involve quadratic equations to calculate time, distance, or velocity.
2. Business & Economics
Profit maximization, cost minimization, and revenue optimization problems frequently use quadratic models.
3. Engineering & Architecture
Structural design, parabolic shapes (bridges, satellite dishes), and electrical circuit analysis.
4. Computer Graphics
Curve rendering, animation paths, and collision detection in games and simulations.
Frequently Asked Questions (FAQ)
What if 'a' equals zero in my equation?
If a = 0, the equation becomes linear (bx + c = 0), not quadratic. Our calculator will show an error because quadratic equations require a ≠ 0. For linear equations, use a linear equation solver instead.
What are complex roots and when do they occur?
Complex roots occur when the discriminant (Δ = b² - 4ac) is negative. They involve the imaginary unit i (where i² = -1). Complex roots always come in conjugate pairs: a ± bi. They indicate the parabola doesn't intersect the x-axis.
How accurate are the calculator results?
Our calculator provides 100% mathematically accurate results using the quadratic formula. For real roots, we show up to 6 decimal places. For complex roots, both real and imaginary parts are calculated precisely.
Can I use fractions or decimals?
Yes! You can enter coefficients as decimals (e.g., 2.5) or whole numbers. The calculator handles both. For exact fractional results, you may want to simplify the results manually or use a fractions calculator.
What's the difference between roots, zeros, and x-intercepts?
These terms are often used interchangeably: Roots are solutions to the equation. Zeros are x-values where the function equals zero. X-intercepts are points where the graph crosses the x-axis (only for real roots).
How do I find the vertex of the parabola?
The vertex (h, k) is calculated as: h = -b/2a and k = f(h) = a(h)² + b(h) + c. Our calculator automatically computes and displays the vertex coordinates in the results.
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