Vieta’s theorem. Factoring a quadratic trinomial

An equation x² + px + q = 0 with leading coefficient 1 is called a reduced quadratic equation; any quadratic equation can be brought to this form by dividing by a. Vieta’s theorem: if x₁ and x₂ are the roots of this equation, then x₁ + x₂ = -p and x₁ · x₂ = q. The converse: if x₁ + x₂ = -p and x₁x₂ = q, then x₁ and x₂ are the roots of x² + px + q = 0; hence roots can be found by guessing, and an equation can be built from given roots. A quadratic trinomial ax² + bx + c with roots x₁ and x₂ factors as a(x - x₁)(x - x₂).