Quadratic expressions

Here's a trap that catches a lot of people: seeing an x² in an expression and immediately calling it quadratic. That's not quite right. What actually defines a quadratic is that x² is the highest power present , nothing bigger, like x³, is allowed.

Quadratic Expression: An algebraic expression in one variable where the greatest exponent is 2, written in the form $a{x}^2+bx+c$ where $a\ne 0$.

Think of it like a speed limit sign , the number on the sign is the maximum, not just any speed you happen to be going. The "2" in a quadratic is the ceiling, not just a passenger.

Some quick examples to build your eye for this. $x^2-5x-14$ is quadratic. $3x^2+2x$ is quadratic (the $c$ term is just zero). $2y-5y^2+7$ is also quadratic , the terms being written in a scrambled order doesn't change anything. Now for the non-quadratics: $x^3-5x-14$ is cubic because the highest power is 3. $8t+12$ is linear , no squared term at all. And $x^2+3y-25$ is not a quadratic in one variable because it mixes two different variables, $x$ and $y$.