What is a Quadratic Equation?

A quadratic equation is a second-degree polynomial equation in a single variable, typically written in the standard form:

ax² + bx + c = 0

where a, b, and c are constants, and a ≠ 0 (if a = 0, the equation becomes linear rather than quadratic).

The Quadratic Formula

The quadratic formula is a powerful tool for solving any quadratic equation. It is derived from the process of completing the square and is expressed as:

x = (-b ± √(b² - 4ac)) / (2a)

This formula provides the solutions (roots) to the quadratic equation, which represent the points where the parabola intersects the x-axis.

Discriminant Analysis

The expression inside the square root in the quadratic formula, b² - 4ac, is called the discriminant. It determines the nature of the roots:

• Positive discriminant (b² - 4ac > 0): Two distinct real roots
• Zero discriminant (b² - 4ac = 0): One real root (a repeated root)
• Negative discriminant (b² - 4ac < 0): Two complex conjugate roots

Applications of Quadratic Equations

Quadratic equations appear in various fields including physics, engineering, economics, and mathematics. They are particularly useful for:

• Modeling projectile motion in physics
• Calculating maximum and minimum values in optimization problems
• Designing curved structures in architecture and engineering
• Analyzing profit and loss scenarios in business