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Algebra I / Elem. Algebra

The Basics

1.1 Intro to Algebra
1.2 Real Number System and Calculators

Linear Equations and Inequalities

Functions and Graphs I

3.1^s Introduction to Graphing
3.2 Solving Equations Graphically
3.3 Functions I - Introduction
3.4 Functions II - Domain, Range, & Operations

Lines and thier Graphs

4.1 Intercept Points
4.2^s Slopes
4.3 Equations of Lines
4.4 Graphing Linear Inequalities

Algebra II / E&I Algebra

Linear Systems

5.1 2x2 Systems of Equations I - Graphically
5.2 2x2 Systems of Equations II - Substitution Method
5.3 2x2 Systems of Equations III - Elimination Method
5.4^s 3x3 Systems of Equations
5.5 Applications
5.6 Systems of Linear Inequalities

Exponents & Polynomials

Intermediate Algebra starts here!

Factoring

Rational Expressions

Rational Equations and Applications

Algebra III / Inter. Algebra

Radical Expressions

10.1 Roots & Radicals
10.2 Fractional Exponents
10.3 Operations I - Simplify & Multiply
10.4 Operations II - Add, Subtract, & Divide
10.5 Operations III - Distribute, Foil, & Factor
10.6^s Complex Numbers

Nonlinear Equations and Applications

11.1 Radical Equations
11.2 Quadratic Equations I - Root Method
11.3^s Quadratic Equations II - Complete the Square
11.4^s Quadratic Equations III - Quadratic Formula
11.5^s Applications - Geometry & Pythagorean Theorem

Functions and Graphs II

12.1 Parabolas I - Algebraic Approach
12.2 Pareabolas II - Graphical Approach
12.3^s Functions III - Composite & Inverse

Exponential and Logarithmic Functions

13.1^s Exponential Functions & Graphs
13.2 Logarithmic Functions & Graphs
13.3 Exponential & Logarithmic Equations

Liberal Arts Maths

Under Construction

College Algebra

Equations and Inequalitites

Functions and Graphs

Polynomial and Rational Functions

3.1 Polynomial Functions I - Introduction, Zeros
3.2 Polynomial Functions II - Division, Remainder Thm, Factor Thm
3.3 Polynomial Functions III - Complex Zeros, Fundamental Thm of Algebra, Descartes Rule
3.4 Rational Functions & Graphs
3.5 Partial Fractions

Exponential and Logarithmic Functions

Systems and Matrices

Conic Sections

Sequences and Series

7.1^s Introduction to Sequences and Series
7.2 Arithmetic Sequences and Series
7.3 Geometric Sequences and Series

Trigonometry

The Six Trigonometric Functions

Right Triangle Trigonometry

Circular Functions

Graphs of Trigonometric Functions

4.1^s Basic Graphs
4.2^s Amplitude & Period
4.3^s Vertical Translations & Phase Shift
4.4 General Equation & Graph to Equation
4.5^s Combination Graphs & Harmonic Motion
4.6^s Inverse Trigonometric Functions

Trigonometric Identities

5.1 Proving Trig Identities
5.2 Sum/Difference Identities
5.3 Double-Angle Identities
5.4^s Half-Angle Identities
5.5 IDs Involving Inverse Trig Functions

Trigonometric Equations

Oblique Triangles and the Laws

Vectors

Complex, Parametric, and Polar Forms

9.1 Trigonometric Form of Complex Numbers
9.2 Products and Quotients in Trig Form
9.3 Powers and Roots in Trig Form
9.4 Parametric Equations and Graphs
9.5^s Polar Equations and Graphs

Calculus I

Limits and Continuity

Derivatives

2.1 Definition of Derivative
2.2 Derivatives of Basic Functions
2.3 Motion and the Derivative
2.4 Product Rule, Quotient Rules, & Deriv. of Trig Functions
2.5 Chain Rule
2.6 Implicit Differentiation
2.7 Derivatives of Exponential and Logarithmic Functions
2.8 Logarithmic Differentiation
2.9^s Derivatives of Inverse Functions
2.10^s Derivatives of Inverse Trig Functions
2.11 Related Rate Applications

Analysis of Curves

3.1 Extreme Values
3.2 Mean Value Theorem for Derivatives
3.3 First Derivative Test & Increasing-Decreasing Functions
3.4 Second Derivative Test & Concavity
3.5^s Curve Sketching
3.6 Fun with Reform Calculus
3.7 Optimization Applications

Antiderivatives

Calculus II

Transcendental Functions

5.0^s Review of Transcendental Functions and thier Derivatives
5.1 Integrals Resulting in Logarithmic Functions
5.2 Integrals Resulting in Inverse Trig Functions
5.3 Hyperbolic Trigonometric Functions
5.4 Numeric and Electronic Integration

Geometry

Physics

Integration Techniques

Calculus of Infinity

Parametric, Polar, and Conic Curves

Calculus III

Under Construction

S = contains supplemental resources

Course: Algebra III / Intermediate Algebra
Topic: Nonlinear Equations and Applications
Subtopic: Radical Equations

Overview

Today we will transition from radical expressions to radical equations. We will now be able to solve equations that contain a radical (or two!) for x. The process involves eliminating the radical and solving the resulting equation. You eliminate the radical by isolating it and squaring (or cubing, fourthing, etc.) both sides of the equation.

CAUTION: Be sure to square (or cube, etc.) the entire side of the equation not individual terms. I.e., squaring both sides of √(x)=√(y)+2 does not make x=y+4. Actually it makes x=(√(y)+2)2=(√(y)+2)(√(y)+2) which must then by FOILed.

CAUTION: You must always check your answers when you even-power both sides due to the potential for extraneous solutions. Check by plugging each answer back into the original equation. Then simplify each side to see if they match (and don't power each side to eliminate the radical as that is where the potential error can enter in the first place!).

CAUTION: Recall that an even root can never be equal to a negative number, so something like √(x-5)=-3 is automatically no solution. Don't waste your time trying to solve it.

Besides solving radical equations algebraically you should be able to solve them graphically by electronically graphing each side of the original equation as a separate function and then finding the intersection point (using the INTERSECTION feature of your grapher). If you are using a TI graphing calculator you may want to review my Calculator Guide: Intersection Points for detailed steps.

Objectives

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

11.1.1 Solve radical equations algebraically that contain:
a single radical (any index), or fractional exponents, or two square roots
11.1.2 Recognize extraneous solutions to radical equations
11.1.3 Determine the number of solutions a radical equation will have by graphing the equation
11.1.4 Solve a radical equation by graphing electronically

Terminology

Define: radical equation, extraneous solution

Text Notes
Text: Intro & Inter Algebra for CS 3ed by Blitzer, sect. 10.6

Practice solving radical equations both algebraically and graphically -- it is important to be able to solve both ways! The "using technology" box on pg 712 discusses solving a radical equation graphically by finding the x-intercept values.
Pay attention to the bold-faced statement toward the bottom of page 709 regarding the need to check answers. Not doing so may lead to "extraneous solutions" discussed further at the end of example 2.
pg 711 Pay attention to the caution/"study tip" -- not doing so will lead to a very common procedural error and lots of wrong answers!

Prof. Keely's Lesson Notes for Mathematics Online