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:

\[ x^2 + 4x + 4 \]

\[ 2x^2 + 3x  1 \]

\[ 3x^2  5x + 2 \]

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

\[ 5x^2  10 \]