Quadratic trinomial (other)

A quadratic trinomial is a polynomial of the form $ax^2 + bx + c$, where $a \neq 0$. Beyond basic factorization and solving, quadratic trinomials can be analyzed in other ways: completing the square, finding the vertex, and understanding their graphs (parabolas).

Important properties

• The graph of $ax^2 + bx + c$ is a parabola.

• Completing the square rewrites $ax^2 + bx + c$ as $a(x + \frac{b}{2a})^2 + (c - \frac{b^2}{4a})$.

• The vertex of the parabola is at $x = -\frac{b}{2a}$.

• The axis of symmetry is the vertical line $x = -\frac{b}{2a}$.

• The direction of opening depends on the sign of $a$ (upwards if $a > 0$, downwards if $a < 0$).