Solve a Real-World Problem by FactoringYEARBOOK DESIGN A sponsor for the schoolyearbook has asked that the length andwidth of a photo in their ad be increased bythe same amount in order to double the areaof the photo. If the original photo is 12centimeters wide by 8 centimeters long,what should be the new dimensions ofthe enlarged photo?ExplorePlanx12 cmBegin by making a diagram like the one shown above, labelingthe appropriate dimensions.Let x = the amount added to each dimension of the photo.The new length times the new width equals twice the old area.x + 12 · x + 8 = 2(8)(12)8 cmxSolve (x + 12)(x + 8) = 2(8)(12) Write the equation.x 2 + 20x + 96 = 192 Multiply.x 2 + 20x - 96 = 0 Rewrite the equation so that one side equals 0.(x + 24)(x - 4) = 0 Factor.x + 24 = 0 or x - 4 = 0 Zero Product Propertyx = -24 x = 4 Solve each equation.CheckThe solution set is {-24, 4}. In the context of the situation, only4 is a valid solution because dimensions cannot be negative.Thus, the new dimensions of the photo should be 4 + 12 or16 centimeters, and 4 + 8 or 12 centimeters.5. GEOMETRY The height of a parallelogram is 18 centimeters less than itsbase. If the parallelogram has an area of 175 square centimeters, what isits height?Personal Tutor at ca.algebra1.comExamples 1–3(pp. 435–436)Example 4(p. 436)Example 5(p. 437)Factor each trinomial.1. x 2 + 11x + 24 2. n 2 - 3n + 23. w 2 + 13w - 48 4. p 2 - 2p - 355. y 2 + y - 20 6. 72 + 27a + a 2Solve each equation. Check the solutions.7. n 2 + 7n + 6 = 0 8. a 2 + 5a - 36 = 09. y 2 + 9 = 10y 10. d 2 - 3d = 7011. NUMBER THEORY Find two consecutive integers x and x + 1 with aproduct of 156.Lesson 8-3 Factoring Trinomials: x 2 + bx + c 437

HELPHOMEWORKFor SeeExercises Examples12–23 1–324–31 432, 33 5Factor each trinomial.12. x 2 + 12x + 27 13. c 2 + 12c + 3514. y 2 + 13y + 30 15. d 2 - 7d + 1016. p 2 - 17p + 72 17. g 2 - 19g + 6018. x 2 + 6x - 7 19. n 2 + 3n - 5420. y 2 - y - 42 21. z 2 - 18z - 4022. -72 + 6w + w 2 23. -30 + 13x + x 2Solve each equation. Check the solutions.24. b 2 + 20b + 36 = 0 25. y 2 + 4y - 12 = 026. d 2 + 2d - 8 = 0 27. m 2 - 19m + 48 = 028. z 2 = 18 - 7z 29. h 2 + 15 = -16h30. 24 + k 2 = 10k 31. c 2 - 50 = -23c32. GEOMETRY The triangle has an area of 40 squarecentimeters. Find the height h of the triangle.33. SUPREME COURT When the justices of the Supreme Courtassemble each day, each justice shakes hands with eachof the other justices. The total number of handshakes hpossible for n people is given by h = _ n 2 - n. Write and2solve an equation to determine the number of justices onthe Supreme Court.h cm(2h 6) cmReal-World LinkThe “Conferencehandshake” has been atradition since the late19th century. Each day,there is a total of 36handshakes by thejustices.Source: supremecourtus.govPRACTICEEXTRASee pages 733, 751.Self-Check Quiz atca.algebra1.comH.O.T. ProblemsRUGBY For Exercises 34 and 35, use the following information.The length of a Rugby League field is 52 meters longer than its width w.34. Write an expression for the area of the field.35. The area of a Rugby League field is 8160 square meters. Find the dimensionsof the field.GEOMETRY Find an expression for the perimeter of a rectangle with thegiven area.36. area = x 2 + 24x - 81 37. area = x 2 + 13x - 90SWIMMING For Exercises 38–40, use the following information.The length of a rectangular swimming pool is 20 feet greater than its width. Thearea of the pool is 525 square feet.38. Define a variable and write an equation for the area of the pool.39. Solve the equation.40. Interpret the solutions. Do they both make sense in the context of theproblem? Explain.41. REASONING Explain why, when factoring x 2 + 6x + 9, it is not necessary tocheck the sum of the factor pairs -1 and -9 or -3 and -3.42. OPEN ENDED Give an example of an equation that can be solved using thefactoring techniques presented in this lesson. Then solve your equation.438 Chapter 8 Factoring