What Is a Quadratic Expression?

A quadratic expression in one variable $x$ is anything of the form

The condition $a \neq 0$ is what makes it genuinely quadratic — if $a = 0$ the $x^2$ term vanishes and we are left with a linear (or constant) expression instead. Here $a$ is the coefficient of $x^2$, $b$ is the coefficient of $x$, and $c$ is the constant term; $ax^2$ is called the quadratic term and $bx$ the linear term.

The same idea extends naturally to more variables. A quadratic expression in two variables $x$ and $y$ collects every term of total degree at most 2 in $x, y$:

which is the general second-degree expression you will meet again when you study pairs of straight lines and conics — every term is either degree 2 ($x^2$, $xy$, $y^2$) or lower. This chapter, however, is entirely devoted to the one-variable case, because that is where equations, sign behaviour, extreme values, and inequations all have a clean, complete theory — the two-variable case is picked up separately once conics are introduced. …