# The Math Dude's Quick and Dirty Guide to Algebra

**by**Marshall, Jason

### 9780312569563

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## Summary

## Author Biography

When not writing and hosting the *Math Dude’s Quick and Dirty Tips to Make Math Easi*er podcast, Jason Marshall works as a staff research scientist at the California Institute of Technology (Caltech) studying the infrared light emitted by starburst galaxies and quasars. Before that, he was a postdoctoral scholar at NASA's Jet Propulsion Laboratory (JPL). Jason obtained a PhD from Cornell University, where he worked with the team of astronomers that built the IRS (nothing to do with taxes) instrument for the *Spitzer Space Telescope *and helped teach many physics and astronomy classes. In addition to these astronomical pursuits, Jason has many earthly interests: traveling the world, tinkering with technology, watching and playing soccer, and spending time with his wife, Shannon, fixing up their small but increasingly comfortable Los Angeles area home.

## Table of Contents

Introduction | p. 1 |

What Can This Book Do for You? | p. 1 |

How Should You Use This Book? | p. 7 |

Prologue: Why Math Isn't An Awful Nerd | p. 9 |

Basic Number Properties | p. 10 |

Basic Arithmetic | p. 11 |

Let the Game Begin! | p. 13 |

Looking for Patterns in Numbers | p. 17 |

Exponents and Perfect Squares | p. 19 |

A Surprising Sequence of Numbers | p. 20 |

Wrap-up | p. 26 |

Final Exam | p. 27 |

What Is Algebra, Really? | |

Taking Algebra To The Streets | p. 33 |

The Secret Algebra You've Already Been Doing | p. 34 |

Using Variables | p. 38 |

Writing Equations | p. 40 |

What's the Point of Algebra? | p. 42 |

Square Roots | p. 46 |

The Pythagorean Theorem | p. 49 |

Why Algebra Matters in the Real World | p. 56 |

Algebra in Your Backyard | p. 62 |

Algebra Tutorial: How to Make a Graph | p. 63 |

Challenge Problems | p. 71 |

Final Exam | p. 72 |

Algebra Basics | p. 76 |

What Are Variables? | p. 77 |

How Do Variables Work? | p. 82 |

What Are Algebraic Expressions? | p. 86 |

Algebra Tutorial: The Order of Operations | p. 88 |

Algebra Tutorial: Practice Problem | p. 92 |

Intro to Equations | p. 95 |

Halftime Recap | p. 99 |

How You Should Think About Equations | p. 106 |

Algebra Tutorial: How to Solve an Equation | p. 107 |

Algebra Tutorial: How to Solve an Algebra Problem | p. 123 |

Wrap-up | p. 129 |

Final Exam | p. 130 |

Understanding Algebra Better | |

Walk The Number Line | p. 135 |

A Brief History of Numbers | p. 136 |

Algebra and Decimal Numbers | p. 141 |

Algebra and the Number Line | p. 143 |

Absolute Values | p. 144 |

Number Boot Camp | p. 155 |

Linear Equations | p. 164 |

Algebra Tutorial: How to Solve Single Variable Linear Equations | p. 167 |

Linear Equations with Absolute Values | p. 174 |

Algebra Tutorial: How to Solve Absolute Value Equations | p. 176 |

Linear Inequalities | p. 179 |

Algebra Tutorial: How to Solve Linear Inequalities | p. 181 |

Wrap-up | p. 193 |

Final Exam | p. 194 |

Arithmetic 2.0: Math With Variables, Exponents, And Roots | p. 197 |

Arithmetic 1.0: Preparing for Algebra | p. 198 |

Doing Arithmetic with "Real" Numbers | p. 201 |

Arithmetic 2.0 (Here Comes the Algebra) | p. 214 |

Algebra Tutorial: How to Simplify Expressions | p. 220 |

Math Properties | p. 223 |

Algebra Tutorial: How to Combine Like Terms | p. 238 |

Exponentiation | p. 240 |

Roots | p. 248 |

Exponentiation and Roots Combined | p. 254 |

Irrational Exponents | p. 256 |

Can You Remind Me of the Point of Algebra? | p. 259 |

Final Exam | p. 260 |

Solving Algebra Problems | |

Polynomials, Functions, And Beyond | p. 265 |

What Are Polynomials? | p. 266 |

Evaluating Polynomials | p. 274 |

Functions | p. 279 |

Visualizing Polynomial Functions | p. 287 |

How to Solve Problems with Polynomials | p. 296 |

Algebra Tutorial: How to Find and Write the Equations of Lines | p. 300 |

Equations of Horizontal and Vertical Lines | p. 309 |

Systems of Equations | p. 320 |

Algebra Tutorial: How to Solve a System of Equations | p. 321 |

Systems of Inequalities | p. 328 |

Algebra Tutorial: How to Solve a System of Inequalities | p. 329 |

Challenge Problem #1 | p. 333 |

Final Exam | p. 335 |

The Root Of The Problem | p. 339 |

Challenge Problem #2 | p. 339 |

Roots of Polynomials | p. 344 |

Factoring Polynomials | p. 360 |

Algebra Tutorial: How to Factor Polynomials | p. 372 |

Solving Quadratic Equations | p. 383 |

Challenge Problem #3 | p. 401 |

Game Over? | p. 409 |

Final Exam | p. 410 |

The Math Dude's Solutions | p. 413 |

Acknowledgments | p. 475 |

Index | p. 477 |

Table of Contents provided by Ingram. All Rights Reserved. |

## Excerpts

**Math Dude's Quick and Dirty Guide to Algebra**

*** PART I***

**WHAT IS ALGEBRA, REALLY?**

*Though this be madness, yet there is method in't.*

**CHAPTER 1**

**Taking Algebra to the Streets**

**THE SECRET ALGEBRA YOU'VE ALREADY BEEN DOING**

**STEP 1**

**STEP 2**

**STEP 3**

*"Wow, thanks! All of that effort just to solve one little problem that I, uh, already knew how to solve anyway. Great job, Math Dude."*

**INEQUALITIES: <, >, =, AND =**

*not*equal. For example, here's how the less than ("<") and greater than (">") symbols work:

**POP QUIZ:**

**WHICH IS BIGGER? SMALLER? EQUAL?**

**1.**The size of the moon is ____ the size of a mountain

**2.**The numbers 5, 8, 13, and 21 are ____ the number 5.

**3.**100 ____ 102

**4.**The size of the planet Saturn is ____ the size of the planet Jupiter.

**5.**The numbers 5, 8, 13, and 21 are ____ the number 21.

**USING VARIABLES**

*m*. That's right, we're going to use a symbol to represent a number. In algebra (and all of math), we call symbols like

*m*, variables (we'll have a lot more to say about variables in the next chapter). As we've said, we don't actually care what

*m*stands for yet--it could be ten cents, ten dollars, or ten thousand dollars. It doesn't matter. We just need to know that there is

*some*amount of money in your wallet. And once we know that, we call it

*m*so that we can talk about it and, as you'll see, use it to solve problems.

*c*. And let's also add a little number to each

*c*to show which item we're talking about on our list. In other words,

*c*1 represents the cost of the first item on the list (maybe it's pizza dough),

*c*2 represents the cost of the second item on the list (perhaps cheese), and so on.

**WRITING EQUATIONS**

*c*1 +

*c*2 +

*c*3 + ... >

*m*

*c*values), then if that number is greater than the amount of money in your wallet (which we've called

*m*), the answer is "Yes, you need to go to the bank."

**ALGEBRA MAKES THINGS EASIER!**

*c*variables for the additional costs.

*your*life easier--but rest assured that it'll start making more and more sense as we see more and more examples of algebra in action throughout this book.

**POP QUIZ:**

**CAN YOU USE ALGEBRA TO MAKE A BUDGET?**

**Bonus question:**Once you answer this quiz question, go back and think about why we used ">" and not "=" in our earlier grocery shopping problem.

**WHAT'S THE POINT OF ALGEBRA?**

*and*fun: cars and computers, for example. All of which might leave you wondering if these two views ever run into each other? Or do they just quietly live out their separate lives?

*big*pyramid--about 750 feet long on each side, and 450 feet high! Not having much of a social life, Knot Dude spent a lot of time in the evenings listening to his father complain about the difficulty of his job. In particular, one problem was proving quite vexing: how to line up the four walls of the foundation of the pyramid so they all would meet to form a giant square. After all, without a perfectly square base, the pyramid wouldn't be a pyramid--it'd just be a mess.

*ALGEBRA DECODER!*

*** RIGHT ANGLE**

*Extra details about the origin and meaning of some math terms--aka "jargon"--will be given in "Algebra Decoder!" sections like this.*

**right angle**? Well, the quick and dirty explanation isfound by looking at the similarity between the phrase "right angle" and the word "rectangle" (think rect-angle). They almost look and sound the same, right? And this makes perfect sense because all four corners of a rectangle form right angles! On a related note, the symbol used to indicate that an angle is a right angle looks kind of like an "L" (which you can see on the pyramid foundation drawing). Again, this symbol makes sense since it looks like one corner of a rectangle.

**POP QUIZ:**

**HOW MANY RIGHT ANGLES CAN YOU FIND?**

*inside*of each shape. In other words, we're not talking about the total number of corners in each shape! This distinction really only matters for one of these shapes--which is it?

*very*special triangle. In fact, Knot Dude found that as long as the rope was pulled nice and taut, two sides of this triangle will always come together to form a perfect right angle.

**SQUARE ROOTS**

*or*-7. Aren't both of these correct solutions? Well, not exactly. Here's what I mean: In algebra, you'll sometimes see problems that look like

*x*2 = 49

*x*== 7 or

*x*= -= - 7 (you should try plugging both these values of

*x*into the equation tocheck that they both work). But saying that this equation has two possible numbers that make it true isn't the same thing as saying thatitself is equal to either 7 or -7. The bottom line is thatis just a number ... and it's a positive number!

**HOW TO CALCULATE SQUARE ROOTS**

**WATCH OUT!**

**The Peril of Square Roots**

*There are some things in life ... and math ... that you really want to avoid. And those are exactly the kinds of things you'll find in Watch Out! sections.*

*any*number is always positive. And, if you think about it, you'll see that this means that a negative number does not have a square root--since there's no number that you can square to equal it (at least none that you know of at this point). So, consider yourself warned--you can't take the square root of a negative number!

**POP QUIZ:**

**CALCULATING SQUARE ROOTS**

*approximately*... but not

*precisely*... equal to the thing on the right. In other words, p˜3.14 says that the number p (pi!) is approximately equal to 3.14, but not exactly. Does that help you figure out which one-quarter of the problems you should answer exactly?

**THE PYTHAGOREAN THEOREM**

*The Wizard of Oz*knew about the Pythagorean theorem! Remember? When he receives his diploma from the Wizard, he declares: "The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side. Oh, joy ... rapture! I've got a brain!" Well, the only problem is ... the Scarecrow got it wrong! So, what does the Pythagorean theorem actually say?

*any*right triangle have a very special relationship to the length of the long side of the triangle (which is called its hypotenuse). Specifically, if you square the lengthsof the two legs and then add the resulting numbers together, that number will always equal the square of the length of the hypotenuse. Okay, words are nice, but this is an algebra book, so let's express the Pythagorean theorem algebraically. Here it is:

*a*2 +

*b*2 =

*c*2

*a, b,*and

*c*mean? Well,

*a*and

*b*represent the length of each leg of the triangle, and

*c*represents the length of the hypotenuse. Think about it for a minute, and you'll see that this formula "says" the exact same thing as that other more wordy description--but it's a lot more concise.

*a*and

*b*), and you can tell me how long the hypotenuse must be (in other words,

*c*). Just to make sure I'm not crazy, let's check and see if this works for Knot Dude's rope. Remember, Knot Dude's rope triangle had

*a*= 3 and

*b*= 4. That gives us

*a*2 = 9 and

*b*2 = 16, which means that

*a*2 +

*b*2 = 9 + 16 = 25

*c*2? Well, in Knot Dude's triangle

*c*= 5, which means that

*c*2 = 25. And, of course, that's exactly the same answer that we got for

*a*2 +

*b*2. So, who would've guessed it: the Pythagorean theorem actually works! (For the record, I might have guessed it.)

**ALGEBRA DECODER!***** WHAT IS A "PYTHAGOREAN"?**

*really*long time ago ... about 2,500 years ago, in fact! Given this length of time, it shouldn't be too surprising that many of the details of Pythagoras' life are a little fuzzy. But we do know that he was a philosopher and a mathematician, and that he started a kind of religious movement known as "Pythagoreanism," where, among other things, everybody was a vegetarian and lived in awe of the awesomeness of math. So, now you know!

**OVERACHIEVER**

**BADGE**

**THE WONDERFUL WORLD OF "PYTHAGOREAN TRIPLES"**

*If you're an overachiever, you're going to love these sections. While you don't have to know all this stuff, you'll definitely be glad that you do!*

*a*2 +

*b*2 =

*c*2? Maybe there just aren't any others. Well, actually, there are others--plenty of them. In fact, there are an infinite number of them! And I can say this without even having to plug in a single number to check. How can I be so sure? Take a look at this:

*not*the only one of these special right triangles with whole number sides.

*a*= 6,

*b*= 8, and

*c*= 10. So,

*a*2 +

*b*2 = 62 + 82 = 36 + 64 = 100. And

*c*2 = 102 = 100. Thank goodness,

*a*2 +

*b*2 =

*c*2. It works! If you think about it, you'll see that instead of doubling the sides of the triangle, we could have multiplied them all by 3, 4, or any other whole number too. Each of these scaled-up triangles would have satisfied the Pythagorean theorem, and they all would have had whole number sides.

**WHAT IS PYTHAGORAS' MESSAGE?**

*(name of fruit or vegetable)*. Your mission today, if you choose to accept it, is to figure out the __________

*(silly adjective)*secret message that Pythagoras left embedded in his name for new math secret agents to __________

*(unnecessary adverb)*discover. If you succeed, you will graduate from secret agent math academy and become a full-blown secret agent. __________

*(overly enthusiastic exclamation)*!

*(judgmental adjective)*discoveries: the Pythagorean theorem.

*a*2 +

*b*2 =

*c*2. In the problems below, I give the lengths of the two legs (

*a*and

*b*) and the hypotenuse (

*c*) of six right triangles that satisfy the Pythagorean theorem. Well, almost. I don't actually give you the lengths of all three sides of each triangle--I only tell you the lengths of two of the three sides. Your job is to figure out the length of the third side, and then to match that to the corresponding letter from the name "Pythagoras" above.

*a*= 12 and

*c*= 20 are given. Your task is, therefore, to find the value of

*b*. If you do the calculation, or just notice that this must be a scaled-up version of the 3 - 4 - 5 triangle (all the sides are multiplied by 4), you'll find that

*b*= 16. What letter does that correspond to in the name "Pythagoras" above? The very first letter: "P". So you then write that in the space to the right of the problem below. Got it? Good. Remember to look for problems that are just scaled-up versions of the 3 - 4 - 5 triangle, and to look for problems that are repeats of ones you've already solved but that ask you to find a different one of the three sides. (Hint: number 4 below looks awfully similar to number 1, doesn't it?) Now have at it!

*good*advice--perhaps Pythagoras was getting a little eccentric in his later years. Also, just in case you'rewondering what I mean that Pythagoras left this message for you "in his name," the letters in the words "SPY," "OATH," and "ARG(H)" (that last "H" was a bit of a cheat) can be rearranged to spell "Pythagoras." That's right--it's what's known as an anagram!

*(name of fruit or vegetable)*.

**WHY ALGEBRA MATTERS IN THE REAL WORLD**

*"No, let me think about it."*

*"Yeah, actually--I played around with something a while ago that behaves exactly like that. Have a seat, let me show you how it works ..."*

**HELP, MY DECK IS CROOKED!**

*not*a square! So what went wrong? Well, clearly the folks building your deck hadn't heard the story of Knot Dude, Papa Knot, and the Great Pyramid of Giza. Otherwise, they would've used a piece of rope with knots in it to help build a deck with square corners. After seeing the deck, you're so angry that you decide to tear it out and rebuild it yourself--properly this time! But while you're fumbling around trying to make a triangle like Knot Dude's, you come to the realization that rope is pretty floppy and isn't a very practical tool. So you stop and think for a while ... and you come up with something ingenious: a way to use your rope to build a square deck that doesn't require tying a single knot!

**HOW TO MAKE A SQUARE - METHOD ONE**

**HOW TO MAKE A SQUARE - METHOD TWO**

**1.**The lengths of all four sides must be equal.

**2.**The lengths of the two diagonals stretching from opposite corners of the square must also be equal.

**ALGEBRA IN YOUR BACKYARD**

*a*and

*b,*are both 15 feet, and we can find the length of the diagonal stretching from one corner of your deck to the other using the good old Pythagorean theorem. How? Well, first we know that

*a*2 +

*b*2 = 152 + 152 = 225 + 225 = 450.

*a*2 +

*b*2 =

*c*2, this means that

*c*

*2*= 450.

*c*. Which, after running the numbers through a calculator, gives us a number that's approximately equal to 21.21.

*must*have a corner-to-corner length of about 21.21 feet. And that means that instead of using a rope to check that the two diagonals of your deck are the same length, we could instead

**ALGEBRA TUTORIAL***

**HOW TO MAKE A GRAPH**

*Throughout the book, you'll find sections like this containing "algebra tutorials." Each of these sections gives an in-depth look at a particular type of algebra problem and includes a step-by-step guide showing you how to solve them. Some of the steps are left for you todo, but if you get stuck you can find solutions at the end of the book in Math Dude's Solutions.*

**STEP 1: DRAW CARTESIAN COORDINATES**

*x*-axis (sometimes called the abscissa) runs horizontally and the

*y*-axis (sometimes called the ordinate) runs vertically. The little marks along each axis label the

*x*and

*y*values at those locations. Both axes have a value of 0 at the origin--the place where the two axes meet. So that's our playing field ... now let's start doing something on it.

**STEP 2: CREATE LIST OF ORDERED PAIRS**

*c,*of the triangle formed by half a deck:

*c,*let's start by switching around the left and right sides of the Pythagorean theorem and write it like

*c*2 =

*a*2 +

*b*2. Then, let's take the square root of both sides to get

*"b"*in our drawing of the deck. Instead, both sides are labeled

*"a"*since this deck is shaped like a square--which means that both sides have the same length. So let's use the fact that

*a*=

*b*for this deck to rewrite our equation like this

*c*= v2 *

*a*

*any*square deck will be ... we simply plug in values for

*a*, and in return we get values for

*c*. For example, if we do this for

*a*= 1, we get

*c*= v2 If we do it for

*a*= 2, we get

*c*= v2 * 2--which is more often written 2v2 (we can omit the "*" since it's clear we're talking about multiplying 2 by v2). And if we do this for a bunch of values of

*a*, we end up creating a list like this one:

**STEP 3: PLOT POINTS ON THE PLANE**

*x*-

*y*plane (which is just another name for the Cartesian coordinate system we set up) is defined by a single pair of numbers. So by plotting several points, we can begin to investigate how these pairs of numbers are related to each other. How does this all work? Well, before we tackle our deck problem and the ordered pairs we just came up with, let's take a look at this graph:

*x*and

*y*axes. Ordered pairs are typically written inside parentheses with the

*x*coordinate given first. For example, the location of the origin is specified by the ordered pair (0, 0), the location of the point labeled "A" at

*x*= 1 and

*y*= 2 is specified by the ordered pair (1, 2), and the location of the point labeled "B" at

*x*= 3 and

*y*= 5 is ... well ... I'll leave it to you to finish up writing this ordered pair. And while you're at it, go ahead and fill in the ordered pair for point C too.

*x*-axis we have an "

*a*-axis," and instead of a

*y*-axis, we have a "

*c*-axis." Why? Because we don't have (

*x, y*) ordered pairs, we have (

*a*,

*c*) ordered pairs. In other words, since the table we made earlier contains

*a*and

*c*values, the ordered pairs we get from it are (

*a*,

*c*) ordered pairs. So far I've plotted the ordered pairs (0, 0) and (30, 30v2 ). I'll leave it up to you to plot the rest of the points from the table we made earlier. And while you're at it, go ahead and add the two points for

*a*= 40 and

*a*= 50 that aren't in the table. Feel free to use a calculator to help you figure out the approximate values of all those numbers multiplied by v2. Once you're done with all that, you'll be ready to move from plotting points to plotting curves.

**STEP 4: CONNECT THE DOTS TO DRAW A CURVE**

*a, c*) ordered pairs on the plot? Okay, there should be at least six points on your graph showing the locations of the ordered pairs corresponding to deck sizes of

*a*= 0, 10, 20, 30, 40, and 50 feet (you might have some additional points too). While plotting these points, you should start to see a pretty clear trend developing that tells you about how these points are related to one another. In this case, you should be able to see that the points appear to be ascending in a straight line from the lower left to the upper right of the plot. So, let's go ahead and draw a line that connects the dots and shows the overall trend. We'll learn more about this in chapter 3, which will help you understand why the points in this problem end up in a straight line.

*a*= 24 feet on the "

*a*-axis," then moving vertically up from this point until you hit the diagonal line that we drew, and finally moving horizontally to the left until you hit the "

*c*-axis." The value of

*c*at that point is the diagonal size of thedeck. With just a quick glance at the graph we can see that this value must be a bit more than 30 feet (the actual number is 24 v2 ... which is close to 34 feet). So, in the future, when you need to make a graph, just remember to follow the four steps we used in this tutorial:

**How to Make a Graph**

**1.**Draw Cartesian coordinates.

**2.**Create a list of ordered pairs.

**3.**Plot ordered pairs on the x-y plane.

**4.**Connect the dots to draw a curve.

**POP QUIZ:**

**CONNECT THE DOTS**

*n*,

*n*2). Don't worry,

*n*here isn't anything crazy--it's just representing allthe various whole numbers. So the first ordered pair for

*n*= 1 is (1, 1), the second ordered pair for

*n*= 2 is (2, 4), and so on. After you come up with a list of ordered pairs (going up to

*n*= 10 should do the trick), plot the points on the axes, and then draw a curve that goes through the points and shows the overall trend.

**Bonus question:**What does this curve you've drawn mean? In other words, all of the perfect squares are sitting nicely on the points you plotted, so what do the parts of the curve that are located between the points represent?

**CHALLENGE PROBLEMS**

**1.**Pick a whole number greater than 2. Now multiply it by 2, and then subtract it from the square of your originalnumber. Then add 1 to this result. After that, take the square root of this number, and then add 1 to the result. What's the answer? After all that work, did you get back exactly the same number you started with? How can that be? Magic, right? No, it's algebra.

**2.**You are told that the product of two consecutive whole numbers is 1056. Can you figure out what those two numbers are? If you think about it, this seems like a nearly impossible problem to solve without simply trying to multiply a bunch of numbers together until you get the right pair. But how can we do it smarter and faster?

**3.**You drop a rock into a well and hear a splash two seconds later. Can you figure out how far it is from the top of the well to the surface of the water? There's a lot going on here. We have to worry about the physics of gravity, the motion of the rock, the speed of sound, and how this all comes together mathematically. Do you think we'll be able to do it?

**FINAL EXAM***

*** ALGEBRA IN THE EVERYDAY WORLD**

*** INEQUALITIES**

*** SQUARE ROOTS**

*** THE PYTHAGOREAN THEOREM**

*b*, and how far you need to position the bottom of the ladder from the house to keep it from falling over,

*a*. Given all that, can you figure out the answers to the following two questions?

*** GRAPHING POINTS AND LINES**

*y*1 =

*x*and

*y*2 = 2 *

*x*on the same set of axes. In other words, first choose a bunch of

*x*values (thewhole numbers from 1 through 10 will do), plug these

*x*values into these two relationships to find ordered pairs for (

*x*,

*y*1) and (

*x*,

*y*2), plot these points (use different symbols for each relationship), and then draw lines connecting the dots. What effect does the 2 in

*y*2 have on the result?