Chapter 1: Quadratics

A quadratic function takes the form y equals ax squared plus bx plus c where a is nonzero, and its graph is a parabola characterized by a vertex representing either a maximum or minimum point and a vertical line of symmetry passing through the vertex. The direction and width of the parabola depend on the coefficient a: when a is positive the parabola opens upward with a minimum point, and when a is negative it opens downward with a maximum point. The chapter then presents three major algebraic methods for solving quadratic equations. Factorization uses the zero product property to find roots by expressing the quadratic as a product of linear factors. Completing the square transforms the equation into vertex form, which simultaneously reveals the roots and identifies the vertex coordinates. The quadratic formula, derived from completing the square, provides a direct computational approach for any quadratic equation. A critical concept is the discriminant, the expression b squared minus four ac that appears under the radical in the quadratic formula, which determines whether solutions are real or complex and how many distinct solutions exist. When the discriminant is positive there are two distinct real roots, when zero there is one repeated real root, and when negative there are no real roots. The chapter extends these ideas to systems combining linear and quadratic equations by using substitution methods and analyzing intersection points through the discriminant of the resulting quadratic. Solving quadratic inequalities involves rearranging to one side and sketching the parabola to identify intervals where the expression is positive or negative. Advanced applications include recognizing equations that are quadratic in form, such as biquadratic equations or exponential equations, and using substitution to reduce them to standard quadratics. Finally, the vertex form representation f of x equals a times the quantity x minus h squared plus k directly reveals the vertex at coordinates h comma k and the axis of symmetry at x equals h, providing efficient sketching and analysis techniques.