376 CHAPTER 6 Algebra, Graphs, and Functions Pablo and Maya recently installed an in-ground rectangular swimming pool in their backyard. They have enough money budgeted to pay for the installation of a brick border around the pool of uniform width that has an area of 296 square feet. They would like to know the width of this border. This problem can be modeled using a quadratic equation. In this section, we will learn two methods to solve quadratic equations. Solving Quadratic Equations by Using Factoring and by Using the Quadratic Formula SECTION 6.9 LEARNING GOALS Upon completion of this section, you will be able to: 7 Multiply binomials. 7 Factor trinomials of the form + + x bx c. 2 7 Factor trinomials of the form + + ≠ ax bx c a , 1. 2 7 Solve quadratic equations by factoring. 7 Solve quadratic equations by using the quadratic formula. 7 Use quadratic equations to model application problems. Why This Is Important Quadratic equations play an important role in modeling many applications from the fields of engineering, physics, chemistry, statistics, business, building construction, and scientific research. As our society becomes more technologically advanced, many occupations rely on basic mathematical understanding, which includes knowledge of quadratic equations. of beef and 150 lb of pork available, how many packs of all-beef and how many packs of regular hot dogs should the manufacturer make to maximize the profit? What is the profit? * 56. MODELING—Car Seats and Strollers A company makes car seats and strollers. Each car seat and stroller passes through three processes: assembly, safety testing, and packaging. A car seat requires 1 hr in assembly, 2 hr in safety testing, and 1 hr in packaging. A stroller requires 3 hr in assembly, 1 hr in safety testing, and 1 hr in packaging. Employee work schedules allow for 24 hr per day for assembly, 16 hr per day for safety testing, and 10 hr per day for packaging. The profit for each car seat is $25, and the profit for each stroller is $35. How many units of each type should the company make per day to maximize the profit? What is the maximum profit? Three car seats and seven strollers, $320 $ Concept/Writing Exercises 57. a) Do all systems of linear inequalities have solutions? Explain. No b) Write a system of linear inequalities that has no solution. * 58. Can a system of linear inequalities have a solution set consisting of a single point? Explain. Yes. One example is x 0, ≥ y x y 0, 0, 0, ≥ ≤ ≤ which has solution (0, 0). 59. Which of the following inequalities have the same graph? Explain how you determined your answer. a) x y 2 8 − < b) x y 2 8 − + > − c) x y 2 4 16 − < (a), (b), and (d) d) y x2 8 > − Research Activity 60. Operations Research The study of operations research draws on several disciplines, including mathematics, probability theory, statistics, and economics. George B. Dantzig (see the Profile in Mathematics on page 370) was one of the key people in developing operations research. Write a paper on Dantzig and his contributions to operations research and linear programming. *See Instructor Answer Appendix Wavebreakmedia/Shutterstock Tetiana Mandziuk/Shutterstock
RkJQdWJsaXNoZXIy NjM5ODQ=