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Understanding Quadratic Equations

A quadratic equation is a second-degree polynomial equation in a single variable x, typically written in the form ax² + bx + c = 0, where a, b, and c are coefficients (real or complex numbers) and 'a' is not equal to zero. Solving the equation means finding the values of x (called roots or solutions) that satisfy the equation. A quadratic equation always has two roots, which may be real and distinct, real and equal, or complex conjugates.

Methods for solving quadratic equations date back to ancient Babylonian, Egyptian, Greek, Chinese, and Indian mathematicians. While early methods were often geometric or algorithmic, the general algebraic solution, known as the quadratic formula, was consolidated over centuries. The formula x = [-b ± √(b² - 4ac)] / 2a provides the roots in terms of the coefficients a, b, and c. The term inside the square root, Δ = b² - 4ac, is called the discriminant, and its sign determines the nature of the roots.

Quadratic equations appear frequently in various fields, including physics (e.g., projectile motion), engineering, economics, and computer graphics. They model parabolic curves and situations where quantities depend on the square of a variable. This calculator uses the quadratic formula to find the roots of a given equation, providing solutions even when they are complex numbers.