Introduction

Algebra Before Symbols

Algebra lets us solve problems that would be difficult or impossible using arithmetic alone, and it underpins nearly every advanced branch of mathematics, science, engineering and industry that follows. It was not always written the way you see it today, though. The 9th-century Persian scholar al-Khwarizmi — working at the House of Wisdom in Baghdad — was among the first to systematically solve linear and quadratic equations, but he did it entirely in words, with no symbols at all (the word algorithm is derived from his name). It took centuries for mathematicians to develop the compact symbolic notation — $x$, $=$, $+$, exponents — that turned al-Khwarizmi's word-problems into the equations you manipulate today. That shift let mathematicians reason about entire types of problems at once instead of solving one instance at a time, and the quadratic expression is the first, and most important, type this chapter builds that reasoning around.

What This Chapter Covers

A quadratic expression in one variable is anything of the form $ax^2+bx+c$ with $a\neq 0$. This chapter studies such expressions and the equations built from them in real depth: solving quadratic equations (including when the roots are complex, using the previous two chapters' machinery), tracking how the sign of a quadratic expression changes as $x$ varies, finding the maximum or minimum value it can take, and finally solving quadratic inequations — inequalities built from a quadratic expression rather than an equation.