What is a quadratic trinomial?

Definition

A quadratic trinomial is an expression of the form:

f(x) = ax² + bx + c

where a, b and c are real numbers and a is not equal to 0 (otherwise it is no longer a quadratic trinomial, but a linear expression or a constant).

• a is the leading coefficient (coefficient of x²)
• b is the coefficient of x
• c is the constant term

Examples

Expression	a	b	c
f(x) = 2x² - 3x + 1	2	-3	1
f(x) = -x² + 4	-1	0	4
f(x) = x² - 5x	1	-5	0

Common pitfall

Do not confuse ‘trinomial’ (3 terms: ax^2, bx, c) with the idea that there must absolutely be three non-zero terms. Even if b = 0 or c = 0, the expression remains a second-degree trinomial as long as a is not equal to 0.

Why study its sign?

Knowing the sign of f(x) for different values of x enables you to solve inequalities of the form ax^2 + bx + c >= 0, which are very common in physics (trajectories), economics (costs, profits) and geometry.

This lesson teaches you how to determine, for any value of x, whether f(x) is positive, negative or zero, without having to plot the corresponding graph.