1010.1Factoring andCompleting the Square10.2The Quadratic Formula10.3Graphing Parabolas10.4More on QuadraticEquations10.5Quadratic and RationalInequalitiesChapter Quadratic Equations and InequalitiesIs it possible to measure beauty? For thousands of years artists and philosophershave been challenged to answer this question. The seventeenth-century philoso-pher John Locke said, “Beauty consists of a certain composition of color and ±gurecausing 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 usby the ancient Greeks. One principle,called the Golden Rectangle, concernsthe most pleasing proportions of a rec-tangle. The Golden Rectangle appearsin nature as well as in many cultures.Examples of it can be seen in Leonardoda Vinci’s Proportions of the HumanFigureas well as in Indonesian templesand Chinese pagodas. Perhaps one of the best-known examples of theGolden Rectangle is in the façade and²oor plan of the Parthenon, built inAthens in the ±fth 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.LWWWWL 2 Wdug22241_ch10a.qxd 11/10/2004 18:29 Page 617
618Chapter 10Quadratic Equations and Inequalities10-2In this Section•Review of Factoringeview of the Even-RootProperty•Completing the Square•Miscellaneous Equations• Imaginary Solutions10.1Factoring and Completing the SquareFactoring and the even-root property were used to solve quadratic equations inChapters 5, 6, and 9. In this section we ﬁrst review those methods. Then you willlearn the method of completing the square, which can be used to solve anyquadratic equation.Review of FactoringA quadratic equation has the form ax2±bx±c²0, where a, b, and care real num-bers with a³0. In Section 5.6 we solved quadratic equations by factoring and thenapplying 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 PropertyThe equation ab²0 is equivalent to the compound equationa²0orb²0.Strategy for Solving Quadratic Equations by Factoring1.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.EXAMPLE 1Solving a quadratic equation by factoringSolve 3x2´4x²15 by factoring.
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