1.57 Quadratic Factoring Solution Definitions

Quadratic Factoring Solution Definitions establishes the vocabulary for solving quadratic equations by rewriting them as a product of factors set equal to zero, describing which quadratic equations are eligible for this method, the specific equation form the method produces along the way, and the special circumstance in which the method yields only a single repeated value rather than two distinct solutions.

Factorable Quadratic Equation

A factorable quadratic equation is a quadratic equation, generally written as ax² + bx + c = 0, whose left-hand expression can be rewritten as a product of two binomial factors with integer or rational coefficients. Not every quadratic equation is factorable in this sense; when integer or rational factors cannot be found, solving by factoring is not a viable method, and an alternative solving technique must be used instead.

Factored Quadratic Equation

A factored quadratic equation is the equivalent form of a factorable quadratic equation obtained once its expression has been rewritten as a product of two binomial factors set equal to zero, such as (x − 2)(x − 3) = 0. Once a quadratic equation has been rewritten into this factored form, the zero-product principle applies directly: since the product of the two factors equals zero, at least one of the factors itself must equal zero, allowing each factor to be set equal to zero and solved individually as a simple one-step equation.

Repeated Quadratic Root

A repeated quadratic root is a single solution value produced by a factored quadratic equation in which both binomial factors are identical, such as (x − 4)(x − 4) = 0, resulting in only one distinct solution, x = 4, rather than two separate solutions. A repeated quadratic root arises specifically from a perfect square trinomial factored form, and geometrically it corresponds to a parabola whose vertex touches the horizontal axis at exactly one point rather than crossing it at two separate points.

Together, these definitions describe the factoring method for solving quadratic equations: identifying whether a quadratic equation is factorable, rewriting it into a factored quadratic equation to which the zero-product principle applies, and recognizing the special case of a repeated quadratic root when both binomial factors produced by the factoring process happen to be identical.