Solve quadratic inequalities

Learning Objectives Rearrange any quadratic inequality into the standard form ax^2 + bx + c > 0. Find the critical points of a quadratic inequality by solving the related quadratic equation through factoring or the quadratic formula. Construct a sign chart or number line to test the intervals defined by the critical points. Solve quadratic inequalities by interpreting the graph of the corresponding parabola. Express the solution set of a quadratic inequality using correct interval notation, including the use of parentheses and square brackets. Identify and solve special cases, such as when the quadratic has no real roots or one repeated root. Apply the process of solving quadratic inequalities to problems in calculus, such as finding where a function is increasing or decr...

TermDefinitionExample Quadratic InequalityAn inequality that can be written in one of the standard forms: ax^2 + bx + c > 0, ax^2 + bx + c < 0, ax^2 + bx + c ≥ 0, or ax^2 + bx + c ≤ 0, where a, b, and c are real numbers and a ≠ 0.x^2 - 5x + 4 ≤ 0 is a quadratic inequality. Related Quadratic EquationThe equation formed by replacing the inequality symbol with an equals sign. The solutions to this equation are the critical points.For the inequality x^2 - 5x + 4 ≤ 0, the related equation is x^2 - 5x + 4 = 0. Critical PointsThe real roots (or zeros) of the related quadratic equation. These are the x-values where the expression equals zero, and they divide the number line into intervals.The critical points of x^2 - 5x + 4 = 0 are x = 1 and x = 4. Sign Chart (or Number Line Analysis)A visu...

Standard Form of a Quadratic Inequality ax^2 + bx + c > 0 (or <, ≥, ≤) Before solving, always manipulate the inequality so that one side is zero. This allows you to find the critical points where the expression changes sign. Quadratic Formula x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} Use this formula to find the critical points (roots) of the related equation ax^2 + bx + c = 0, especially when the expression is not easily factorable. Discriminant Analysis \Delta = b^2 - 4ac The discriminant tells you about the nature of the critical points. If Δ < 0, there are no real roots, meaning the parabola never crosses the x-axis and the expression is always positive or always negative.