Quadratic Equation Solver
Enter coefficients a, b, c for the equation ax² + bx + c = 0. Our calculator finds all roots, shows step-by-step solutions (Quadratic Formula, Completing the Square, Factoring when possible), and plots the parabola.
Parabola Visualization
Educational Notes
- Quadratic Equation: Any equation of the form ax² + bx + c = 0, where a ≠ 0.
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The Discriminant: D = b² - 4ac. It determines the type and number of real roots:
- D > 0: Two distinct real roots
- D = 0: One real repeated root
- D < 0: Two complex (nonreal) roots
- Graph Shape: The graph is a parabola. If a > 0, it opens upwards; if a < 0, it opens downwards.
- Factoring: If the equation can be factored over real numbers, the roots are rational and factoring will show them directly.
- Completing the Square: Rewriting the equation and isolating x by forming a perfect square trinomial.
- Quadratic Formula: Universal formula to solve any quadratic: x = (-b ± √(b²-4ac)) / (2a)
Quadratic Equation Solver – Instantly Find Roots, Steps & Graphs
Struggling with algebra? Whether you’re a high school student or a data science pro, solving quadratic equations is a must. With our Quadratic Equation Solver, you can easily find the roots of any equation in seconds. Just enter your equation—and get real, complex, or repeated roots with complete step-by-step solutions.
🧠 What Is a Quadratic Equation?
A quadratic equation is any equation in the form:
ax² + bx + c = 0
Where:
a,b, andcare constantsxis the variablea ≠ 0(if a = 0, it becomes linear)
For example:2x² + 5x - 3 = 0
These equations always graph as parabolas and can have:
- Two distinct real roots
- One real root (double root)
- Two complex roots
🧮 How the Quadratic Equation Solver Works
Simply enter the values of a, b, and c. Then, the calculator:
- Applies the quadratic formula:
x = [-b ± √(b² – 4ac)] / 2a - Determines the type of root:
- ✅ Real and unequal if discriminant > 0
- ✅ Real and equal if discriminant = 0
- ✅ Complex if discriminant < 0
- Shows the full steps for how it got there
- Optionally graphs the parabola for visual aid
📊 Example Problem
Solve: x² – 4x + 3 = 0
- a = 1, b = -4, c = 3
- Discriminant = (-4)² – 4(1)(3) = 16 – 12 = 4
- Roots:
x = [4 ± √4] / 2
x = (4 ± 2)/2 → x = 1 and x = 3
✅ You now know your roots and how to find them!
⚙️ Features of the Solver
- 🧮 Handles real and complex roots
- 🔄 Supports positive or negative coefficients
- ✍️ Shows detailed steps with explanations
- 📈 Displays graph of the equation
- 🧠 Ideal for learning, not just solving
- 📱 Fully mobile-compatible
🎓 Who Should Use This Calculator?
- 👨🎓 High school students learning algebra
- 👩🏫 Teachers preparing math worksheets
- 📐 Engineers and architects
- 💻 Developers needing curve fitting
- 📊 Anyone solving equations with powers of 2
🔥 Features Included
- Quadratic equation solver
- Solve ax² + bx + c = 0
- Free online quadratic calculator
- Find roots of a quadratic equation
- Step-by-step algebra solver
- Complex root calculator
- Quadratic graphing tool
- Math equation solver for students
🧭 Related Tools on Desmos Calculators
- Graphing Calculator – plot quadratic functions easily
- Discriminant Calculator – understand the nature of roots
- Polynomial Calculator – solve higher-degree equations
🔗 External References
- Khan Academy – Quadratic Equations
- Wolfram Alpha – Solve Quadratic
- Purple Math – How to Solve Quadratics
✅ Final Thoughts
The Quadratic Equation Solver turns a tricky math problem into a smooth, visual solution. Whether you’re solving for real-world physics or finishing your math homework, this tool saves time and boosts understanding. And because it explains each step, it helps you learn—not just guess.