Mastering Quadratic Equations and Inequalities

Factoring Quadratic EquationsPage 1

Quadratic equations follow the standard form ax^2 + bx + c = 0. When a=1, factoring involves finding two numbers that multiply to c and add up to b. The signage of factors depends on the signs within the equation (e.g., if both constants are positive, factors are positive; if the middle term is negative, factors may be negative). For equations where a > 1, the coefficient of x^2 must also be accounted for during the grouping process.

Understanding Quadratic InequalitiesPage 1

Quadratic inequalities use symbols like <, >, ≤, or ≥, representing regions on a parabola. They are solved by setting the equation to zero, factoring, identifying critical points, and using a sign test across resulting intervals to determine which regions satisfy the original inequality.

Step-by-Step Solving MethodologyPage 1

To solve a quadratic inequality: 1) Rewrite in standard form; 2) Factor the expression; 3) Find critical points by setting factors to zero; 4) Define intervals based on critical points; 5) Perform a sign test on each interval using a test point; 6) Express the valid regions using interval or set notation. Inclusive inequalities (≤, ≥) include critical points in the solution set, while strict inequalities (<, >) do not.