Elementary Algebra
Introduction to Algebra Linear Equations and Inequalities Functions and Graphs I Lines and thier Graphs Linear Systems Exponents & Polynomials
Intermediate Algebra
Factoring Rational Expressions Rational Equations and Applications Radical Expressions Nonlinear Equations and Applications Functions and Graphs II Exponential and Logarithmic Functions
Precalculus I / College Algebra
Equations and Inequalitites Functions and Graphs Polynomial and Rational Functions Exponential and Logarithmic Functions Systems and Matrices Geometry Basics Conic Sections Sequences and Series
Precalculus II / Trigonometry
The Six Trigonometric Functions Right Triangle Trigonometry Circular Functions Graphs of Trigonometric Functions Trigonometric Identities Trigonometric Equations Oblique Triangles and the Laws Vectors Complex, Parametric, and Polar Forms
Calculus I
Limits and Continuity Derivatives Analysis of Curves Antiderivatives
Calculus II
Transcendental Functions
Geometry Physics Integration Techniques Calculus of Infinity Parametric, Polar, and Conic Curves
Calculus III

Course: Intermediate Algebra
Topic: Nonlinear Equations and Applications

Overview

This lesson covers the last of four methods for algebraically solving a quadratic equation. From the quadratic equation ax2+bx+c=0, identify the a,b,c, and plug into the quadratic formula , and simplify to give the exact x-solutions. The quadratic formula is a particularly useful method since it can be used to solve any quadratic equation, factorable or not. However, it is a bit tedious and lends itself to several common errors worth discussing.

The four different ways of solving a quadratic equation algebraically:

• Factoring into the form ( )( )=0 and using the zero-product rule, but only works if the quadratic is factorable
• Root method, but this only works when the quadratic is in the form (expression)2=#
• Completing the square, but this only works when the x2 term's coefficient is 1
• Quadratic formula is particularly versatile, memorize it!

Study all of these methods carefully and know when to use which one!

Real solutions to a quadratic equation can also be found by graphing electronically (the x-intercept values are the Real solutions). Non-Real solutions can only be found algebraically.

A quadratic equation always has two answers. But, are they always different from one another? When are they Real vs. non-Real? The discriminant (radicand of the quadratic formula) can be used to answer these questions without having to fully solve or graph the quadratic equation.

Objectives

By the end of this topic you should know and be prepared to be tested on:

• 11.4.1 Recognize that all quadratic equations can be solved by the quadratic formula method
• 11.4.3 Find the discriminant of a quadratic equation
• 11.4.4 Determine the type of solutions to a quadratic equation by graphing or using the discriminant
• 11.4.5 Understand the connections between the discriminant being zero/positive/negative, the effect on the graph of the quadratic function, the number of x-intercept points on the graph of the quadratic function, and the number of real/non-real solutions of the associated quadratic equation
• 11.4.6 Given two numbers, form a quadratic equation to which they would be solutions (i.e., working "backwards" from the solutions to the equation)

Terminology