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10 10.1 Factoring and Completing the Square 10.2 The Quadratic Formula 10.3 Graphing Parabolas 10.4 More on Quadratic Equations 10.5 Quadratic and Rational Inequalities Chapter Q uadratic Equations and Inequalities Is it possible to measure beauty? For thousands of years artists and philosophers have been challenged to answer this question. The seventeenth-century philoso- pher John Locke said, Beauty consists of a certain composition of color and figure causing delight in the beholder. Over the centuries many architects, sculptors, and painters have searched for beauty in their work by exploring numerical pat- terns in various art forms. Today many artists and architects still use the concepts of beauty given to us by the ancient Greeks. One principle, called the Golden Rectangle, concerns the most pleasing proportions of a rec- tangle. The Golden Rectangle appears in nature as well as in many cultures. Examples of it can be seen in Leonardo da Vincis Proportions of the Human Figure as well as in Indonesian temples and Chinese pagodas. Perhaps one of the best-known examples of the Golden Rectangle is in the faade and floor plan of the Parthenon, built in Athens in the fifth century B . C . In Exercise 91 of Section 10.4 we will see that the principle of the Golden Rectangle is based on a proportion that we can solve using the quadratic formula. L W W W W L 2 W dug22241_ch10a.qxd 11/10/2004 18:29 Page 617 618 Chapter 10 Quadratic Equations and Inequalities 10-2 In this Section Review of Factoring Review of the Even-Root Property Completing the Square Miscellaneous Equations Imaginary Solutions 10.1 Factoring and Completing the Square Factoring and the even-root property were used to solve quadratic equations in Chapters 5, 6, and 9. In this section we first review those methods. Then you will learn the method of completing the square, which can be used to solve any quadratic equation. Review of Factoring A quadratic equation has the form ax 2 bx c 0, where a , b , and c are real num- bers with a 0. In Section 5.6 we solved quadratic equations by factoring and then applying the zero factor property. Of course we can only use the factoring method when we can factor the quadratic poly- nomial. To solve a quadratic equation by factoring we use the following strategy. Zero Factor Property The equation ab 0 is equivalent to the compound equation a o r b 0. Strategy for Solving Quadratic Equations by Factoring 1. Write the equation with 0 on one side. 2. Factor the other side. 3. Use the zero factor property to set each factor equal to zero. 4. Solve the simpler equations. 5. Check the answers in the original equation. E X A M P L E 1 Solving a quadratic equation by factoring Solve 3 x 2 4 x 15 by factoring.... View Full Document

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