Definition

A quadratic expression is an expression of the form:

\[ ax^2 + bx + c \]

where:

  • \( a, b, \) and \( c \) are real numbers (i.e., \( a, b, c \in \mathbb{R} \)),
  • \( a \neq 0 \),
  • \( x \) is the variable.

In this expression, \( ax^2 \) is the quadratic term, \( bx \) is the linear term, and \( c \) is the constant term.

Note

The requirement that \( a \neq 0 \) ensures that the expression is indeed quadratic, as the degree of the polynomial is two.

Examples of quadratic expressions:

  1. \[ x^2 + 4x + 4 \]
  2. \[ -2x^2 + 3x - 1 \]
  3. \[ 3x^2 - 5x + 2 \]
  4. \[ \frac{1}{2}x^2 - \frac{3}{4}x + \frac{5}{6} \]
  5. \[ 5x^2 - 10 \]