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[...]... somewhere other than on the number line shown in Fig 1-4 -8 -6 Fig 1-4 -4 -2 0 2 4 6 8 The real numbers can be depicted graphically as points on a straight line Later in this chapter you will be introduced to imaginary numbers and complex numbers, which are, in a certain theoretical sense, “nonreal.” For now, however, let’s get back to the task at hand: first-order equations in one variable SOME EXAMPLES... preceding standard form is a one-variable first-order equation Alternative forms are PART 0 10 A Review of Mathematics cx ϭ d x ϭ m/n where c, d, m, and n are constants and n ≠ 0 Here are some examples of single-variable first-order equations: 4x Ϫ 8 ϭ 0 Ϫx ϭ 22 3ex ϭ c x ϭ /c In these equations, , e, and c are known as physical constants, representing the circumference-to-diameter ratio of a circle,... 1 mile One-Variable Second-Order Equations A one-variable second-order equation, also called a second-order equation in one variable or, more often, a quadratic equation, can be written in the following standard form: ax2 ϩ bx ϩ c ϭ 0 where a, b, and c are constants, and x is the variable (The constant c here does not stand for the speed of light.) Equations of this type can have two real-number solutions,... will evolve from arcane to mundane CHAPTER 1 Equations, Formulas, and Vectors 9 One-Variable First-Order Equations In algebra, it is customary to classify equations according to the highest exponent, that is, the highest power to which the variables are raised A one-variable first-order equation, also called a first-order equation in one variable, can be written in the following standard form: ax ϩ... text of Part Zero and take the chapter-ending quizzes If you get less than three-quarters of the answers correct in the quizzes and the section-ending test, find a good desk and study Part Zero It will be a drill, but it will get you “in shape” and make the rest of the book easy In order to learn physics, you must have some mathematical skill Math is the language of physics If I were to tell you otherwise,... want to examine the preceding solution and simplify it Remember that it contains CHAPTER 1 Equations, Formulas, and Vectors Imaginary number line Real number line j4 j2 -4 2 -2 4 - j2 -j4 Zero point (common to both number lines) Fig 1-5 The complex numbers can be depicted graphically as points on a plane, defined by two number lines at right angles (Ϫ8)1/2 An engineer or physicist would write this as... variable, on the left-hand side of an equals sign and some mathematical expression on the right-hand side When denoting a formula, it is important that every constant and variable be defined so that the reader knows what the formula is used for and what all the quantities represent One of the simplest and most well-known formulas is the formula for finding the area of a rectangle (Fig 1-1 ) Let b represent... brute force is the method of choice We’ll define cubic equations, quartic equations, quintic equations, and nth-order equations here but leave the solution processes to the more advanced puremathematics textbooks THE CUBIC A cubic equation, also called a one-variable third-order equation or a third-order equation in one variable, can be written in the following standard form: ax3 ϩ bx2 ϩ cx ϩ d ϭ 0 ... A Review of Mathematics A b Fig 1-1 A rectangle with base length b, height h, and area A A similar formula lets us calculate the area of a triangle (Fig 1-2 ) Let b represent the length (in meters) of the base of a triangle, and let h represent the height (in meters) measured perpendicular to the base Then the area A (in square meters) of the triangle is h A b Fig 1-2 A triangle with base length b,... mentioned that a quadratic equation may have only one real-number solution Here is an example of the factored form of such an equation: (x Ϫ 7) (x Ϫ 7) ϭ 0 Mathematicians might say something to the effect that, theoretically, this equation has two real-number solutions, and they are both 7 However, the physicist is content to say that the only real-number solution is 7 THE QUADRATIC FORMULA Look again .