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C-104 Appendix C MINI PROJECT A Geometric Interpretation of Complex Roots As you know, if a quadratic equation f(x) x 2 Bx C 0 has two real roots, there is a geometric interpretation: The roots are the x-intercepts for the graph of f. Less well known is the fact that there is a geometric interpretation when the quadratic equation has nonreal complex roots. If these roots are a bi, where a and b are real numbers, then the coordinates of the vertex of the graph of f are (a, b 2 ). (a) Check that this is true in the case of the quadratic equation f(x) x 2 6x 25 0. (b) Show that the result holds in general. That is, assume that the roots of f(x) x 2 Bx C 0 are a bi, where a and b are real numbers with b 0. Show that the vertex of the graph of f is (a, b 2 ). Hint: Use the theorem on sums and products of the roots of a quadratic equation to express A and B in terms of a and b.
Appendix C C-105 PROJECT Two Methods for Solving Certain Cubic Equations The first person known to have solved cubic equations algebraically was del Ferro but he told nobody of his achievement. On his deathbed, however, del Ferro passed on the secret to his (rather poor) student Fior. Fior began to boast that he was able to solve cubics and a challenge between him and Tartaglia was arranged in 1535.—From the MacTutor History of Mathematics website: http://www-groups.dcs.st-and.ac.uk/ ~history. The site is maintained by the University of St. Andrews, Scotland. Stated in modern language, one of the problems that Fior challenged Tartaglia to solve in 1535 was essentially as follows. A Restatement of One of Fior’s Challenge Problems for del Ferro A tree is 12 units high. At what height should it be cut so that the length of the portion left standing equals the cube root of the length of the portion cut off? 12 12-x x In this project you’ll learn two different methods that can be used to determine the root of the cubic equation that solves Fior’s tree-cutting problem. As is indicated in the figure above, we let x denote the length of the part of the tree left standing. The condition stated in the problem then gives us x 1 3 12 x. After cubing both sides and rearranging, we have the cubic equation to be solved: x 3 x 12 0 (1) Method 1 To find a root of an equation of the form x 3 ax b 0, when a a 0, make the substitution x y After simplifying, this will produce a 3y . sixth-degree equation that is of quadratic type and that therefore can be solved as in Section 2.2.* Apply this strategy to solve equation (1). Hints: At the beginning, you have a 1, and so the substitution to make in equation (1) is 1 x y After simplifying and then solving the resulting equation of quadratic type, you should obtain y 16 236 1 272 13 . Then, choosing the pos- 3y . itive square root for the moment, you’ll obtain the required value for x: x y 1 3y 16 236 1 27 2 13 1 316 236 1 27 2 13 2.14404 using a calculator What value do you obtain for x if you use the negative square root, that is, y (6 ) 13 ? 236 1 27 *Precalculus: A Problems-Oriented Approach, 7th edition
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APPENDIX C Title and Page Number Pr
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Appendix C C-3 MINI PROJECT Discuss
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Appendix C C-5 MINI PROJECT Thinkin
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Appendix C C-7 1. equilateral trian
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Appendix C C-9 MINI PROJECT Flying
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Appendix C C-11 MINI PROJECT An Ine
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Appendix C C-13 (a) The two data po
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Appendix C C-15 and http://www.svob
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Appendix C C-17 For Figure D we can
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Appendix C C-19 MINI PROJECT A Grap
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Appendix C C-21 MINI PROJECT Who Ar
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Appendix C C-23 MINI PROJECT 1 How
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Appendix C C-25 either the quadrati
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Appendix C C-27 PROJECT The Least-S
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Appendix C C-29 Then, for any line
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Appendix C C-31 PROJECT Finding Som
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Appendix C C-33 3. When a person co
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Appendix C C-35 2. Use part (b) of
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Appendix C C-37 PROJECT Coffee Temp
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Appendix C C-39 MINI PROJECT More C
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Appendix C C-41 The calculations co
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Appendix C C-43 PROJECT Transits of
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Appendix C C-45 E A ¨ V ¨ Bª d S
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Appendix C C-47 PROJECT A Linear Ap
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Appendix C C-49 Continuing with the
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Appendix C C-51 PROJECT Constructin
Page 53 and 54: Appendix C C-53 1 y Figure A A grap
Page 55 and 56: Appendix C C-55 PROJECT Fourier Ser
Page 57 and 58: Appendix C C-57 sine and cosine fun
Page 59 and 60: Appendix C C-59 PROJECT The Motion
Page 61 and 62: Appendix C C-61 PROJECT The Design
Page 63 and 64: Appendix C C-63 denote the angles o
Page 65 and 66: Appendix C C-65 inverse trigonometr
Page 67 and 68: Appendix C C-67 PROJECT Inverse Sec
Page 69 and 70: Appendix C C-69 Alternatively, usin
Page 71 and 72: Appendix C C-71 PROJECT Snell’s L
Page 73 and 74: Appendix C C-73 vertically downward
Page 75 and 76: 0 0 0 Appendix C C-75 PROJECT Vecto
Page 77 and 78: Appendix C C-77 Exercise 8 We outli
Page 79 and 80: Appendix C C-79 y y=2x+5 Îy=2 A (1
Page 81 and 82: 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Appendi
Page 83 and 84: Appendix C C-83 PROJECT Parameteriz
Page 85 and 86: Appendix C C-85 It is worth noting
Page 87 and 88: Appendix C C-87 6. Show that the li
Page 89 and 90: Appendix C C-89 2. An altitude of a
Page 91 and 92: Appendix C C-91 PROJECT The Leontie
Page 93 and 94: Appendix C C-93 n x units of that
Page 95 and 96: Appendix C C-95 him/herself through
Page 97 and 98: Appendix C C-97 PROJECT The Leontie
Page 99 and 100: Appendix C C-99 As in the solution
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Page 103: Appendix C C-103 PROJECT Using Hype
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Page 109 and 110: Appendix C C-109 MINI PROJECT An Un
Page 111 and 112: Appendix C C-111 PROJECT More on Su
Page 113 and 114: Appendix C C-113 SOLUTION 25 25 a (
Page 115: Appendix C C-115 4. Simplify the fo
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