Quadratic equations and functions

Introduction

A quadratic equation is an equation that can be written in the form,

$ax^2+bx+c=0$, where $a\ne 0$.

This is the standard form of the quadratic equation.

Sometimes the quadratic equation has no $x$ term ($b=0$) or no constant term ($c=0$).

There are three methods for solving a quadratic equation: (i) factoring and applying the zero product rule, (ii) completing the square, and (iii) using the quadratic formula. Here we will focus on the last two, which can be used to solve any quadratic equation. Note that the first method, factoring, cannot solve every quadratic equation — it works only when the equation is factorable.

Sections in this chapter

Solving a quadratic equation by completing the square

Writing the equation as a perfect square and applying the square root property.

Solving a quadratic equation by quadratic formula

Quadratic functions

Quadratic functions and their graphs: parabolas, vertex, axis of symmetry, standard (vertex) form, shapes, intercepts, the discriminant and the number of x-intercepts, and maximum or minimum.

Deriving and using the quadratic formula, applications, and the discriminant.