Introduction

Imagine a football kicked into the air. Its path isn’t a straight line but a curve - specifically, a parabola. This path, intriguingly, can be described by a quadratic equation. The height of the football at any given time depends on the initial velocity, the angle of the kick, gravity's pull, and, importantly, is represented by an equation of the form:

\[ h(t) = -gt^2 + vt + h_0 \]

Here, \( h(t) \) describes the height at time \( t \), \( g \) is the acceleration due to gravity, \( v \) is the initial velocity, and \( h_0 \) is the initial height. This is a quintessential quadratic equation, where the variable \( t \) (time) is squared.

This example transcends a mere mathematical exercise; it models real-life phenomena like sports, engineering, and even space travel. As we delve into this chapter, we will uncover the principles of quadratic equations, exploring their standard forms, methods of solving them (such as factoring, using the quadratic formula, or completing the square), and their diverse applications.