Concept preview

What you will be able to do

1. 01Read the roots (x-intercepts) of a parabola straight from factored form $y = a(x - r)(x - s)$.
2. 02Apply the zero-product rule to solve a factored quadratic.
3. 03Factor trinomials with leading coefficient $1$ using the sum-and-product method.
4. 04Factor trinomials with a leading coefficient $\ne 1$, differences of squares, and perfect-square trinomials.
5. 05Always remove a greatest common factor first, and factor completely.

Vertex form told you where a parabola turns; factored form tells you where it crosses the $x$-axis. Written as $y = a(x - r)(x - s)$, the two numbers $r$ and $s$ are the roots of the quadratic — the same objects also called the zeros or the $x$-intercepts. The beauty of this form is that the roots are handed to you with no calculation at all: $y = (x - 2)(x - 5)$ must cross the axis at $x = 2$ and $x = 5$, because those are the only inputs that make the output zero. Reading a graph's crossings straight from an equation is the payoff that makes factoring worth learning.