The Quadratic Formula, Discriminant Analysis, and Vieta's Relations

Solve multi-step linear equations and inequalities with numerical and literal coefficients, justify each algebraic transformation, and rearrange scientific formulas for specified variables.

Solve quadratic equations using the zero-product property, square root extraction, and completing the square for monic and non-monic quadratics.

Solve any quadratic equation using the quadratic formula, analyze root types and multiplicities using the discriminant, and relate roots to coefficients using Vieta's formulas.

Solve simple rational, radical, and absolute value equations and inequalities in one variable, and identify extraneous solutions introduced by algebraic operations.

Solve systems of linear equations in two and three variables using graphing, substitution, and elimination, and prove equivalence under linear combinations.

Solve systems of linear equations using augmented matrices, row echelon transformations (Gaussian elimination), and matrix inverses (AX = B).

Solve systems involving linear, quadratic, and circular equations in two variables algebraically and graphically, and interpret points of intersection.

Graph linear inequalities in two variables as half-planes, identify polygonal feasible regions for systems of linear inequalities, and interpret constraints and viable solutions in modeling contexts.