A58 APPENDIX Review 145. Find an equation that has no solution and give it to a fellow student to solve. Ask the fellow student to write a critique of your equation. 146. Describe three ways you might solve a quadratic equation. State your preferred method; explain why you chose it. 147. Explain the benefits of evaluating the discriminant of a quadratic equation before attempting to solve it. 148. Find three quadratic equations: one having two distinct solutions, one having no real solution, and one having exactly one real solution. 149. The word quadratic seems to imply four ( quad ), yet a quadratic equation is an equation that involves a polynomial of degree 2. Investigate the origin of the term quadratic as it is used in the expression quadratic equation . Write a brief essay on your findings. 150. The equation = − x 2 has no real solution. Why? ‘Are You Prepared?’ Answers 1. ( )( )( ) − + − x x x 2 2 1 2. ( )( ) − + x x 2 3 1 3. { } − 5 3 , 3 4. True A.7 Complex Numbers; Quadratic Equations in the Complex Number System* OBJECTIVES 1 Add, Subtract, Multiply, and Divide Complex Numbers (p. A59) 2 Solve Quadratic Equations in the Complex Number System (p. A63) Complex Numbers One property of a real number is that its square is nonnegative. For example, there is no real number x for which = − x 1 2 To remedy this situation, we introduce a new number called the imaginary unit . DEFINITION The Imaginary Unit The imaginary unit, which we denote by i , is the number whose square is −1. That is, = − i 1 2 This should not surprise you. If our universe were to consist only of integers, there would be no number x for which = x2 1. This was remedied by introducing numbers such as 1 2 and 2 3 , the rational numbers . If our universe were to consist only of rational numbers, there would be no number whose square equals 2. That is, there would be no x for which = x 2. 2 To remedy this, we introduced numbers such as 2 and 5, 3 the irrational numbers . Recall that the real numbers consist of the rational numbers and the irrational numbers. Now, if our universe were to consist only of real numbers, then there would be no number x whose square is −1. To remedy this, we introduce the number i , whose square is −1. In the progression outlined, each time we encountered a situation that was unsuitable, a new number system was introduced to remedy the situation. The number system that results from introducing the number i is called the complex number system . *This section may be omitted without any loss of continuity.
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