Solving Equations

Prof. Keely's Math Online Lecture Notes
Clark College, Vancouver WA
Copyright © 2000 Sally J. Keely. All Rights Reserved.

ELEMENTARY & INTERMEDIATE ALGEBRA

These brief notes are intended to guide you through the textbook and/or other course readings/materials. As you read the textbook pay particular attention to the "topics of importance" and be sure you know how to accomplish each. The "supplemental sites" may provide additional resources on the internet that supplement the topics. Note: This material is extensively elaborated upon in my optional e-book GOLDen Mathematics: Intermediate Algebra. This section of material only is downloadable for nominal fee at www.lulu.com/content/378812. See "Tell me more about Keely's GM book".

Quadratic Equations

GOLDen Mathematics - Intermediate Algebra: Section 9.1
Supplemental Sites: MathOL Links - Alg 9.1

Topics of Importance
Solve Quadratic Equations algebraically by the following methods:
Factoring, Square Root Method, Completing the Square, Quadratic Formula
Solve quadratic equations graphically (calc: ROOT or ZERO)
Determine the type of solutions by graphing or using the discriminant
Introduction to solving equations that are quadratic in form
Applications of quadratic equations
Modeling data with a quadratic function

Comments and Cautions
Quadratic equations (i.e. equations of the form ax²±bx±c=0) can be solved by several methods. The first and foremost is by factoring. But what about when the quadratic is not factorable (i.e. prime)? Luckily, we can resort to three other methods. These methods will open up a world of real-life applications where the equations are usually not "nice" and factorable. The root method which is very useful if the quadratic is in a particular form. Completing the square works has some pretty cool history, but is not used very much any more other than to derive the quadratic formula. The quadratic formula is a particularly useful method since it can be used to solve any quadratic equation, factorable or not. However, the QF is a bit tedious and lends itself to several common errors which we will discuss on the boards.

Below is a summary of the four different ways of solving quadratic equations algebraically by:
* factoring, then using the zero-product rule, i.e. the form ( )( )=0
* the root method, but this only works when the quadratic is in the form ( )²=#
* completing the square, but this only works when the x² term's coefficient is 1
* the quadratic formula - very useful, be sure to memorize it!
Be sure to study all of these methods carefully and know when to use which one!

Real solutions to quadratic equations can be found by graphing. Please read my online Calculator Guide: x-Intercept Points for keystrokes and an example. A quadratic equation always has two answers. But, are they always different from one another? When are they Real vs. Complex? Answers to these questions can be found by graphing or algebraically using the discriminant D = b²- 4ac. Be sure to study the connection between the discriminant, the graph, and the roots of the equation.

Text Notes (These notes refer to Introductory & Intermediate Algebra for CS 3rd ed by Blitzer sections 6.6 + 11.1-11.2.)

In week 2 (math 095) or week 7 (math 091):

	ch 6.6 is an extremely important section. It covers solving algebraically by factoring as well as solving graphically by finding the roots of the equation. This material will be used throughout the remainder of the course so be sure to get it down pat! There are also some important application problems to study. (Yeah, word problems!)
	ch 6.6 example 7 Read through the modeling motion example but pay particular attention to the use of the zero product rule to solve the equation. We will discuss the height function further much later in the course.
	ch 6.6 example 8 This and similar geometry applications are important to be able to work. Fair game for the quiz!

In week 7 (math 095) or week 5 (math 093):

	ch 11.1 is jam packed with material! You are expected to already know how to solve a polynomial equation algebraically by factoring, i.e. solving via the form ( )( )=0, and also graphically, i.e. graphing the polynomial and using ZERO to find the x-intercept values. Review chapter 6.6 and the Calculator Guide: x-Intercept Points as necessary.
	ch 11.1 Next tackle the "square root method". Be careful that you end up with two answers, i.e. don't forget to include the plus-or-minus sign at the point that you apply the square root function to both sides of the equation. Also, watch for imaginary numbers!
	ch 11.1 Next study the "completing the square method". This method only works when the coefficient of the x-squared term is 1. Example 7 shows what to do if the coefficient isn't 1.
	ch 11.1 includes several applications. Cover the compound interest formula and its applications. Cover the Pythagorean Theorem. See Lecture Notes: Applications of Quadratic, Rational, and Radical Equations for lecture notes on this topic.
	ch 11.1 pg 745-747 You can SKIP all the "distance formula" and "midpoint formula" problems. This material is covered in a pre-calculus class.
	ch 11.2 covers the last of the four algebraic methods of solving quadratic equations, using the "quadratic formula". It is imperative that you MEMORIZE the quadratic formula prior to taking a 100-level math class!
	ch 11.2 pg 753 shows the derivation of the quadratic formula. It is not important to understand this process, but it helps some people to know from where the formula comes -- it's not out of the blue! This page shows how you can complete the square on the general quadratic equation to solve it for x and the solution is the quadratic formula. Thus the quadratic formula is really just a general form of the solutions to a quadratic equation.
	ch 11.2 pg 755 Take note of the caution. Not doing so will cause you to make frequent errors while simplifying the answers from the quadratic formula.
	ch 11.2 pg 757 Be sure you understand the connection between the graph of the quadratic function and the number of real solutions of the associated quadratic equation.
	ch 11.2 pg 757-758 discusses the "discriminant" D=b²-4ac which is the radicand in the quadratic formula. Be sure you understand the connection 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.
	ch 11.2 pg 759 has a useful chart comparing the four methods of solving a quadratic equation and some tips for choosing the best method.
	ch 11.2 bottom of pg 759 and example 5 works the "backwards" process of starting with given solutions and finding a quadratic equation that would have those solutions.
	Whew! Lots to talk about on the boards this week ;-}

Rational Equations

GOLDen Mathematics - Intermediate Algebra: Section 9.2
Supplemental Sites: MathOL Links - Alg 9.2

Topics of Importance
Solve rational equations that are proportions by the cross products method
Solve rational equations that are not proportions by the LCD method
Applications of rational equations

Comments and Cautions
An important concept here is that we are working with equations not expressions -- so we can actually solve for x. And since there is an equals sign we are able to multiply both sides by something to eliminate the denominators. This certainly wasn't true with expressions. When working with a rational expression all you could do would be to, for example, add them together by building up the denominators to be the same -- you could not multiply through by the LCD and eliminate it. This is a vital difference to keep in mind as you work through this section!

Text Notes (These notes refer to Introductory & Intermediate Algebra for CS 3rd ed by Blitzer sections 7.6.)

	ch 7.6 When solving a rational equation, concentrate mostly on the algebraic approach. Solving graphically is covered more in a pre-calculus class. You may opt to use the graph to check your answers if you would like (see my Calculator Guide: Intersection Points for detailed steps).
	ch 7.6 pg 511 has a vitally important "study tip" box that compares simplifying rational expressions to solving rational equations. I can't stress how important it is to differentiate expressions from equations and to know what to do with each!
	ch 7.6 When finding the domain restriction, concentrate on the algebraic approach (finding the x-values that make the denominators in the original equation zero) not the graphical approach (which represents domain restrictions of rational expressions as vertical asymptote lines). Always make sure that you do not include any domain restrictions in your final list of solutions. This text calls this step "checking the proposed solutions".
	ch 7.6 "Solving a Formula for a Variable" starting on the bottom of page 512 is covered in a later date's lecture notes.

Radical Equations

GOLDen Mathematics - Intermediate Algebra: Section 9.3
Supplemental Sites: MathOL Links - Alg 9.3

Topics of Importance
Solve radical equations algebraically that contain:
a single radical (any index), or fractional exponents, or two square roots
Recognize extraneous solutions
Check number of solutions and solve radical equations graphically

Comments and Cautions
Today we will transition from radical expressions to radical equations. Thus we will 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 etc as needed) 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 graphing each side of the original equation as a separate function and then finding the intersection point (using INTERSECTION or ISECT). You may want to review my Calculator Guide: Intersection Points for detailed steps.

Text Notes (These notes refer to Introductory & Intermediate Algebra for CS 3rd ed by Blitzer sections 10.6.)

	ch 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.
	ch 10.6 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.
	ch 10.6 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!

Formulas - Intermediate

GOLDen Mathematics - Intermediate Algebra: Section 9.4
Supplemental Sites: MathOL Links - Alg 9.4

Topics of Importance
Evaluate formulas given specific input values
Solve formulas for a specific variable

Comments and Cautions
In order to effectively solve application problems, we first need to be able to work with formulas. Typical formulas you have seen before include geometry formulas like area or perimeter, or banking formulas like simple interest. Today we will concentrate on evaluating formulas given specific input values and rearranging formulas by solving for a specific variable. The formulas in this section may include rational quadratic or rational expressions. It is a sequel to the "Formulas - Elementary" material in elementary algebra, so it may be worth reviewing that material first.

Text Notes (These notes refer to Introductory & Intermediate Algebra for CS 3rd ed by Blitzer section 7.6, cont'd.)

	These notes refer only to the part of ch 7.6 called "Solving a Formula for a Variable" starting at the bottom of page 512 and continuing through the end of the section.
	ch 7.6 In this section the text provides only one example of solving a formula for a specific variable. They expect you to recall, and I would strongly recommend that you review, section 2.4 (elementary formulas) before covering ch 7.6 (intermediate formulas).
	ch 7.6 You don't need to know what the formulas are used for or memorize any of them, just how to evaluate them when given specific input values and how to rearrange them by solving for (a.k.a. "isolating") a specific variable.

Applications of Quadratic, Rational, and Radical Equations

GOLDen Mathematics - Intermediate Algebra: Section 9.5
Supplemental Sites: MathOL Links - Alg 9.5

Topics of Importance
Solving intermediate word problems of the following types:
Work problems
Motion problems
Geometry problems & the Pythagorean Theorem
Applications of quadratic, rational, and radical equations
Modeling data with quadratic, rational, and radical functions

Comments and Cautions
Today we are going to apply our knowledge of equations to word problems. We will start with "work problems" where people or machines are working and "motion problems" where people or machines are moving (often cars, trains, planes, etc.). The chart method is useful to organize the information for these. Once you get the equation it will likely be rational in nature and you can use the skills gained in a previous lesson to solve.

Next we will study "geometry problems" and the "Pythagorean Theorem". Drawing pictures and labeling them carefully will help you to translate these problems into an equation to solve.

Word problems is often a sticking point for students, but if you keep an optimistic attitude, take one step at a time, and read the problem carefully, you can do it! If you need additional assistance, I recommend a little book: How to Solve Word Problems in Algebra by Mildred Johnson. There are others available that are good too, but this one is easy to follow and full of examples like those we cover in this class.

Text Notes (These notes refer to Introductory & Intermediate Algebra for CS 3rd ed by Blitzer section 7.7 + part of 11.1.)

In week 4 (math 095) or week 10 (math 091):

	ch 7.7 covers several types of word problems. The most important are the "work" and "motion" types. You should also cover the "proportion" types which are solved by the "cross-products method".
	ch 7.7 pg 522-524 You can SKIP all the "similar triangles" problems. This material is covered in a pre-calculus class.
	ch 7.8 SKIP