Quadratic Equation Solver | Solve ax² + bx + c = 0

📖 How to Use This Quadratic Equation Solver

Enter coefficients - Input values for a, b, and c in the quadratic equation ax² + bx + c = 0.

Click "Solve Quadratic Equation" - Get instant results including roots, discriminant, and vertex.

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 (referring to the x² term).

Quadratic equations appear frequently in physics, engineering, economics, and many other fields. They describe parabolic curves (U-shaped graphs) when plotted.

🧮 Quadratic Formula

x = [-b ± √(b² - 4ac)] / 2a

Where:
• a, b, c are coefficients (a ≠ 0)
• The discriminant Δ = b² - 4ac determines the nature of roots

📊 Understanding the Discriminant

Δ > 0 (Positive): Two distinct real roots
Δ = 0 (Zero): One real root (double root)
Δ < 0 (Negative): Two complex conjugate roots

📍 Vertex of a Parabola

The vertex is the turning point of the parabola. For a quadratic equation ax² + bx + c, the vertex coordinates are:

Vertex x-coordinate = -b / 2a
Vertex y-coordinate = f(-b/2a) = a(-b/2a)² + b(-b/2a) + c

💡 Applications of Quadratic Equations

Physics: Projectile motion, free fall, and optics
Engineering: Bridge design, signal processing
Economics: Profit maximization, cost analysis
Biology: Population growth models
Computer Graphics: Parabolic curves and animations

❓ Frequently Asked Questions (FAQ)

What if a = 0?

If a = 0, the equation becomes linear (bx + c = 0). The solver will handle this case and provide the linear solution instead.

What does it mean if the discriminant is negative?

A negative discriminant means the quadratic equation has no real roots. Instead, it has two complex (imaginary) roots. The calculator displays them in the form a ± bi.

What is the vertex of a parabola?

The vertex is the highest or lowest point of a parabola. It tells you the maximum or minimum value of the quadratic function. Our calculator shows both the x and y coordinates of the vertex.

Can the quadratic formula be used for any quadratic equation?

Yes, the quadratic formula works for all quadratic equations, whether the roots are real, rational, irrational, or complex. It's the most reliable method.

What is the difference between factoring and using the quadratic formula?

Factoring works only for quadratics with integer roots. The quadratic formula works for every quadratic equation, regardless of the nature of the roots.