CHAPTER 3: Quadratic Functions and Equations; Inequalities
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1 MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 3: Quadratic Functions and Equations; Inequalities 3.1 The Complex Numbers 3.2 Quadratic Equations, Functions, Zeros, and Models 3.3 Analyzing Graphs of Quadratic Functions 3.4 Solving Rational Equations and Radical Equations 3.5 Solving Equations and Inequalities with Absolute Value This program calculates the sum, difference, product, and quotient of two complex numbers. This program will be used in Section 3.2 when graphing quadratic functions. Oct 2 8:13 AM Complex Numbers & TI calculator i 4 = 1 (5+3i)( 6 i) = 27 23i Oct 2 8:01 AM Sep 28 7:02 AM 1
2 2.2 The Complex Numbers Perform computations involving complex numbers. We can define a non real number that is a solution of the equation x = 0. The complex numbers are formed by adding real numbers and multiples of i. Oct 2 3:34 PM The Complex Number System Example Express each number in terms of i. Some functions have zeros that are not real numbers. The complex number system is used to find zeros of functions that are not real numbers. When looking at a graph of a function, if the graph does not cross the x axis, then it has no x intercepts, and thus it has no real number zeros. 2
3 Example (continued) continued Complex Numbers A complex numberis a number of the form a + bi, where a and b are real numbers. The number a is said to be the real part of a + bi and the number bi is said to be the imaginary part of a + bi. Imaginary Number a + bi, a 0, b 0 Addition and Subtraction Pure Imaginary Number a + bi, a = 0, b 0 Complex numbers obey the commutative, associative, and distributive laws. We add or subtract them as we do binomials. We collect the real parts and the imaginary parts of complex numbers just as we collect like terms in binomials. Example Add or subtract and simplify each of the following. a. (8 + 6i) + (3 + 2i) b. (4 + 5i) (6 3i) a. (8 + 6i) + (3 + 2i) = (8 + 3) + (6i + 2i) = 11 + (6 + 2)i = i Multiplication When and are real numbers, This is not true when and are not real numbers. Note: Remember 2 i = 1 b. (4 + 5i) (6 3i) = (4 6) + [5i ( 3i)] = 2 + 8i 3
4 Example Multiply and simplify each of the following. Example (continued) continued Simplifying Powers of i Recall that 1 raised to an even power is 1, and 1 raised to an odd power is 1. Simplifying powers of i can then be done by using the fact that i 2 = 1 and expressing the given power of i in terms of i 2. Note that powers of i cycle through i, 1, i, and 1. Conjugates The conjugate of a complex number a + bi is a bi. The numbers a + bi and a bi are complex conjugates. Examples: 3 + 7i and 3 7i 14 5i and i 8i and 8i The product of a complex number and its conjugate is a real number. 4
5 Multiplying Conjugates Example Multiply each of the following. a. (5 + 7i)(5 7i) b. (8i)( 8i) a. (5 + 7i)(5 7i) = 52 (7i) 2 = 25 49i 2 = 25 49( 1) = = 74 b. (8i)( 8i) = 64i 2 = 64( 1) = 64 Dividing Using Conjugates Example Divide 2 5i by 1 6i. : Write fraction notation. Multiply by 1, using the conjugate of the denominator to form the symbol for 1. This program calculates the sum, difference, product, and quotient of two complex numbers. Express the number in terms of i : (By hand and by TI) 241/4. 241/6. Express the number in terms of i : (By hand and by TI) 241/9. 241/10. Oct 2 2:08 PM 5
6 241/ /19. Oct 2 9:44 AM 242/36. 3i(6 + 4i) 242/42. (3 5i)(8 2i) 241/22. ( 3 4i) (8 i) 242/ /50. (5 + 9i)(5 9i) 242/30. (10 4i) (8 + 2i) 6
7 242/ / /56. (5 4i) 2 242/58. ( 3 + 2i) 2 242/ / /60. (2 4i) 2 242/64. (6 + 5i) 2 242/82. i /86. ( i) 6 242/87. (5i) 4 242/88. (2i) 5 Sep 27 8:34 PM Oct 2 8:37 AM 7
CHAPTER 3: Quadratic Functions and Equations; Inequalities
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