B Apply

37. Think About a Plan Suppose you want to put a frame around the painting shown below. The frame will be the same width around the entire painting. You have 276  in. 2 276 . in. squared  of framing material. How wide should the frame be?

    

    • What does 276  in. 2 276 . in. squared  represent in this situation?
    • How can you write the dimensions of the frame using two binomials?
38. The period of a pendulum is the time the pendulum takes to swing back and forth. The function L = 0.81 t 2 l equals , 0.81 , t squared  relates the length L in feet of a pendulum to the time t in seconds that it takes to swing back and forth. A convention center has a pendulum that is 90 feet long. Find the period.
39. Landscaping Suppose you have an outdoor pool measuring 25 ft by 10 ft. You want to add a cement walkway around the pool. If the walkway will be 1 ft thick and you have 304  ft 3 304 , ft cubed  of cement, how wide should the walkway be?

    

40. Error Analysis A classmate solves the quadratic equation as shown. Find and correct the error. What are the correct solutions?
41. Open-Ended Write an equation with the given solutions.
    1. 3 and 5
    2. − 3 negative 3  and 2
    3. − 1 negative 1  and − 6 negative 6

Solve each equation by factoring, using tables, or by graphing. If necessary, round your answer to the nearest hundredth.

42. x 2 + 2 x = 6 − 6 x x squared , plus 2 x equals 6 minus 6 x
43. 6 x 2 + 13 x + 6 = 0 6 x squared , plus 13 x plus 6 equals 0
44. 2 x 2 + x − 28 = 0 2 x squared , plus x minus 28 equals 0
45. 2 x 2 + 8 x = 5 x + 20 2 x squared , plus 8 x equals 5 x plus 20
46. 3 x 2 + 7 x = 9 3 x squared , plus 7 x equals 9
47. 2 x 2 − 6 x = 8 2 x squared , minus 6 x equals 8
48. ( x + 3 ) 2 = 9 open x plus 3 , close squared , equals 9
49. x 2 + 4 x = 0 x squared , plus 4 x equals 0
50. x 2 = 8 x − 7 x squared , equals 8 x minus 7
51. x 2 − 3 x = 6 x squared , minus 3 x equals 6
52. 4 x 2 + 5 x = 4 4 x squared , plus 5 x equals 4
53. 7 x − 3 x 2 = − 10 7 x minus , 3 x squared , equals negative 10

Reasoning The graphs of each pair of functions intersect. Find their points of intersection without using a calculator. (Hint: Solve as a system using substitution.)

54. y = x 2 y = − 1 2 x 2 + 3 2 x + 3 table with 2 rows and 1 column , row1 column 1 , y equals , x squared , row2 column 1 , y equals negative , 1 half , x squared , plus , 3 halves , x plus 3 , end table
55. y = x 2 − 2 y = 3 x 2 − 4 x − 2 table with 2 rows and 1 column , row1 column 1 , y equals , x squared , minus 2 , row2 column 1 , y equals , 3 x squared , minus 4 x minus 2 , end table
56. y = − x 2 + x + 4 y = 2 x 2 − 6 table with 2 rows and 1 column , row1 column 1 , y equals negative , x squared , plus x plus 4 , row2 column 1 , y equals , 2 x squared , minus 6 , end table

C Challenge

57. The equation x 2 − 10 x + 24 = 0 x squared , minus 10 x plus 24 equals 0  can be written in factored form as ( x − 4 ) ( x − 6 ) = 0 . open x minus 4 close open x minus 6 close equals 0 .  How can you use this fact to find the vertex of the graph of y = x 2 − 10 x + 24 ? y equals , x squared , minus 10 x plus 24 question mark
58. 1. Let a > 0 . eh greater than 0 .  Use algebraic or arithmetic ideas to explain why the lowest point on the graph of y = a ( x − h ) 2 + k y equals eh . open x minus h close squared . plus k  must occur when x = h.
    2. Suppose that the function in part ( a )  is  y = a ( x − h ) 3 + k . open eh close , is , y equals eh . open x minus h close cubed . plus k .  Is your reasoning still valid? Explain.

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