Quadratic inequality and its solution

An inequality with a quadratic function on the left and zero on the right is called a quadratic inequality: ax² + bx + c > 0 (or <, ≥, ≤), where a ≠ 0. A solution is any value of x that turns it into a true numerical inequality; to solve it means to find all solutions or to show there are none. If the roots of ax² + bx + c = 0 are x₁ < x₂, the trinomial factors as a(x − x₁)(x − x₂). A product of two factors is positive when they have the same sign and negative when their signs differ, so the inequality leads to two systems of first-degree inequalities. If a < 0 we first multiply both sides by −1 and reverse the inequality sign. The result: (x − x₁)(x − x₂) > 0 gives x < x₁ or x > x₂, while (x − x₁)(x − x₂) < 0 gives x₁ < x < x₂.