Solving a Quadratic in Standard Form by Factoring

Factoring only works against zero. The zero product rule says that if two things multiply to give zero, at least one of them has to be zero — there's no equivalent rule for a product equaling 28 or 5 or any other number. That's why the first move, always, is herding every term onto one side so the equation reads ax² + bx + c = 0. Skip this step and even a perfectly factored expression tells you nothing about the solutions.

Once the equation equals zero, factor the quadratic into two binomials, (x - p)(x - q) = 0. Each binomial becomes its own mini-equation: x - p = 0 and x - q = 0, giving solutions x = p and x = q. This connects directly to the graph: p and q are the x-intercepts of the parabola y = ax² + bx + c, the points where the curve crosses the x-axis. It also runs in reverse — if you're told the zeros of a quadratic, you can rebuild the factored form and expand it back to standard form.

This skill sits right next to the quadratic formula and completing the square as the three standard ways to solve ax² + bx + c = 0. Factoring is fastest when the numbers cooperate; when they don't, the quadratic formula always works. Recognizing which tool fits a given equation — and never dividing by a variable to 'simplify' — is what separates a clean solve from a lost solution.