📖 How to Use This Quadratic Equation Solver
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Enter coefficients - Input values for a, b, and c in the quadratic equation ax² + bx + c = 0.
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Click "Solve Quadratic Equation" - Get instant results including roots, discriminant, and vertex.
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View step-by-step solution - See the complete calculation process using the quadratic formula.
📐 What is a Quadratic Equation?
A quadratic equation is a second-degree polynomial equation in a single variable x, with the form ax² + bx + c = 0, where a ≠ 0. The name "quadratic" comes from "quadratus," the Latin word for square.
Quadratic equations appear frequently in physics, engineering, economics, and many other fields. They describe parabolic curves when plotted.
🧮 Quadratic Formula
x = [-b ± √(b² - 4ac)] / 2a
Where: a ≠ 0 and Δ = b² - 4ac is the discriminant
📊 Understanding the Discriminant
- Δ > 0: Two distinct real roots
- Δ = 0: One real root (double root)
- Δ < 0: Two complex conjugate roots
📍 Vertex of a Parabola
The vertex is the turning point of the parabola. For ax² + bx + c:
Vertex x = -b / 2a
Vertex y = f(-b/2a)
💡 Applications of Quadratic Equations
- Physics: Projectile motion, free fall, optics
- Engineering: Bridge design, signal processing
- Economics: Profit maximization, cost analysis
- Biology: Population growth models
- Computer Graphics: Parabolic curves
❓ Frequently Asked Questions
What if a = 0?
If a = 0, the equation becomes linear (bx + c = 0). The solver handles this and provides the linear solution.
What does a negative discriminant mean?
A negative discriminant means no real roots. The equation has two complex (imaginary) roots displayed as a ± bi.
What is the vertex of a parabola?
The vertex is the highest or lowest point of the parabola. Our calculator shows both x and y coordinates.
Does the quadratic formula work for all quadratics?
Yes, the quadratic formula works for all quadratic equations, regardless of the nature of the roots.
What is the difference between factoring and the quadratic formula?
Factoring works only for integer roots. The quadratic formula works for every quadratic equation.