Solving by factoring

A quadratic equation sets a degree-2 polynomial to zero: x² + 2x − 3 = 0. Unlike the linear equations of module 1, you can't just undo your way to x — the x² and the x can't be merged. The way in is a fact so simple it looks like a trick.

The zero-product rule

If a product of numbers is zero, at least one factor is zero:

$A \cdot B = 0 \quad\Rightarrow\quad A = 0 \ \text{ or } \ B = 0$

No other number does this — a product equal to 12 tells you almost nothing about its factors (1 × 12? 3 × 4? −2 × −6?), but a product equal to zero pins a factor down. This is why we factor: it converts one hard quadratic into two easy linear equations.

The method

Solve x² + 2x − 3 = 0.

Two solutions — quadratics generally have two, one per factor.

The picture

Graph y = (x + 3)(x − 1) and the solutions become visible: they're the points where the parabola crosses the x-axis, because that's where y = 0. Solving a quadratic and finding a parabola's x-intercepts are the same act.

The non-negotiable zero

The rule needs an actual 0 on one side. Faced with x² + 2x = 3, moving the 3 over comes first — factoring x(x + 2) = 3 and trying "x = 3 or x + 2 = 3" produces nonsense, because a product of 3 doesn't constrain its factors. Zero on one side, always, before you factor.

Why this matters

This is the fastest way to solve any quadratic that factors — but not all of them do. The final two lessons build the tools for the rest: completing the square, and the formula it gives birth to.