THE QUADRATIC FORMULA

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1 66 (9-1) Chapter 9 Quadratic Equations and Quadratic Functions the members will sell x tickets. So the total revenue for the tickets is given by R x ( x). a) What is the revenue if the tickets are sold at $8 each? b) For what ticket price is the revenue $30,000? c) Use the accompanying graph to estimate the ticket price that will produce the maximum revenue. a) $7,00 b) $10 and $15 c) $1.50 GETTING MORE INVOLVED 75. Exploration. a) Find the product [x (5 3)][x (5 3)]. b) Use completing the square to solve the quadratic equation formed by setting the answer to part (a) equal to zero. c) Write a quadratic equation (in the form ax bx c 0) that has solutions 3 and 3. d) Explain how to find a quadratic equation in the form ax bx c 0 for any two given solutions. a) x 10x b) 5 3 c) x 3x THE QUADRATIC FORMULA In this section Developing the Quadratic Formula The Discriminant Which Method to Use In Section 9. you learned that every quadratic equation can be solved by completing the square. In this section we use completing the square to get a formula, the quadratic formula, for solving any quadratic equation. Developing the Quadratic Formula To develop a formula for solving any quadratic equation, we start with the general quadratic equation ax bx c 0 and solve it by completing the square. Assume that a is positive for now, and divide each side by a: ax bx c 0 a a Divide by a to get 1 for the coefficient of x. x b a x c 0 Simplify. a x b a x c Isolate the x - and x-terms. a Now complete the square on the left. One-half of b a is b b, and a a. ba x b a x b b c b a Add to each side. a a a x b b a ac a a x b b ac a a b b x a ac Square root property a Factor on the left-hand side, and get a common denominator on the right-hand side. b b ac x Because a 0, a a a. a x b b ac Combine the two expressions. a

2 9.3 The Quadratic Formula (9-15) 67 study tip Try changing subjects or tasks every hour when you study. The brain does not easily assimilate the same material hour after hour. You will learn and remember more from working on it one hour per day than seven hours on Saturday. E X A M P L E 1 We assumed that a was positive so that a a would be correct. If a is negative, then a a. Either way, the result is the same. It is called the quadratic formula. The formula gives x in terms of the coefficients a, b, and c. The quadratic formula is generally used instead of completing the square to solve a quadratic equation that cannot be factored. The Quadratic Formula The solutions to ax bx c 0, where a 0, are given by x b b ac. a Equations with rational solutions Use the quadratic formula to solve each equation. a) x x 3 0 b) x 9 1x a) To use the formula, we first identify a, b, and c. For the equation 1x x 3 0, a b c a 1, b, and c 3. Now use these values in the quadratic formula: b b x ac a x (1)( 3) (1) (1)(3) 1 16 x 16 x x or x x 1 or x 3 Check these answers in the original equation. The solutions are 3 and 1. b) Write the equation in the form ax bx c 0 to identify a, b, and c: x 9 1x x 1x 9 0 Now a, b 1, and c 9. Use these values in the formula: (1) (1) ()(9) x () x Check. The only solution to the equation is 3.

3 68 (9-16) Chapter 9 Quadratic Equations and Quadratic Functions The equations in Example 1 could have been solved by factoring. (Try it.) The quadratic equation in the next example has an irrational solution and cannot be solved by factoring. E X A M P L E calculator close-up Check irrational solutions using the answer key as shown here. An equation with an irrational solution Solve 3x 6x 1 0. For this equation, a 3, b 6, and c 1: (6) (6) (3)(1) x (3) (3 6) (3) The two solutions are the irrational numbers 3 6 and Numerator and denominator have as a common factor. We have seen quadratic equations such as x 9 that do not have any real number solutions. In general, you can conclude that a quadratic equation has no real number solutions if you get a square root of a negative number in the quadratic formula. E X A M P L E 3 A quadratic equation with no real number solutions Solve 5x x 1 0. For this equation we have a 5, b 1, and c 1: 1 (1) (5)(1) x b 1, b 1 (5) x The equation has no real solutions because 19 is not real. The Discriminant A quadratic equation can have two real solutions, one real solution, or no real number solutions, depending on the value of b ac. If b ac is positive, as in Example 1(a) and Example, we get two solutions. If b ac is 0, we get only one solution, as in Example 1(b). If b ac is negative, there are no real number solutions, as in Example 3. Table 9.1 summarizes these facts.

4 9.3 The Quadratic Formula (9-17) 69 Number of real solutions Value of b ac to ax bx c 0 Positive Zero 1 Negative 0 TABLE 9.1 The quantity b ac is called the discriminant because its value determines the number of real solutions to the quadratic equation. E X A M P L E The number of real solutions Find the value of the discriminant, and determine the number of real solutions to each equation. a) 3x 5x 1 0 b) x 6x 9 0 c) x 1 x a) For the equation 3x 5x 1 0 we have a 3, b 5, and c 1. Now find the value of the discriminant: b ac (5) (3)(1) Because the discriminant is positive, there are two real solutions to this quadratic equation. b) For the equation x 6x 9 0, we have a 1, b 6, and c 9: b ac (6) (1)(9) Since the discriminant is zero, there is only one real solution to the equation. c) We must first rewrite the equation: x 1 x x x 1 0 Now a, b 1, and c 1. Subtract x from each side. b ac (1) ()(1) Because the discriminant is negative, the equation has no real number solutions. Which Method to Use If the quadratic equation is simple enough, we can solve it by factoring or by the square root property. These methods should be considered first. All quadratic equations can be solved by the quadratic formula. Remember that the quadratic formula is just a shortcut to completing the square and is usually easier to use. However, you should learn completing the square because it is used elsewhere in algebra. The available methods are summarized as follows.

5 70 (9-18) Chapter 9 Quadratic Equations and Quadratic Functions helpful hint If our only intent is to get the answer, then we would probably use a calculator that is programmed with the quadratic formula. However, by learning different methods we gain insight into the problem and get valuable practice with algebra. So be sure you learn all of the methods. Solving the Quadratic Equation ax bx c 0 Method Comments Examples Square root Use when b 0. If x 7, then x 7. property If (x ) 9, then x 3 Factoring Use when the polynomial x 5x 6 0 can be factored. (x )(x 3) 0 Quadratic Use when the first two x x 6 0 formula methods do not apply. 1 (6) x 1 Completing Use only when this x x 9 0 the square method is specified. x x 9 (x ) 13 WARM-UPS True or false? Explain your answer. 1. Completing the square is used to develop the quadratic formula. True. For the equation x x 1 0, we have a 1, b x, and c 1. False 3. For the equation x 3 5x, we have a 1, b 3, and c 5. False. The quadratic formula can be expressed as x b b ac. False a 5. The quadratic equation x 6x 0 has two real solutions. True 6. All quadratic equations have two distinct real solutions. False 7. Some quadratic equations cannot be solved by the quadratic formula. False 8. We could solve x 6x 0 by factoring, completing the square, or the quadratic formula. True 9. The equation x x is equivalent to x 1 1. True 10. The only solution to x 6x 9 0 is 3. True 9. 3 EXERCISES Reading and Writing After reading this section, write out the answers to these questions. Use complete sentences. 1. What method presented here can be used to solve any quadratic equation? The quadratic formula solves any quadratic equation.. What is the quadratic formula? The quadratic formula is x b b ac. a 3. What is the quadratic formula used for? The quadratic formula is used to solve ax bx c 0.. What is the discriminant? The discriminant is b ac. 5. How can you determine whether there are no real solutions to a quadratic equation? If b ac 0 then there are no real solutions. 6. What methods have we studied for solving quadratic equations? We have solved quadratic equations by factoring, the square root property, completing the square, and the quadratic formula.

6 9.3 The Quadratic Formula (9-19) 71 Solve by the quadratic formula. See Examples x x x 3x , 3 6, 3 9. x 10x x 1x x x x x 15 0, 3 3, x x x 8x 3 0 3, 1 3, y 6y y 6y 0 3 3, , t t w w,, 19. x x x 3x 9 0 No real solution No real solution 1. 8x x. 9y 3y 6y 0, 1 1, w z 0 15, 15 7, h 7h z 6z , , 1 33 Find the value of the discriminant, and state how many real solutions there are to each quadratic equation. See Example. 7. x x x 6x 1 0 0, one 0, one 9. 6x 7x x 5x 7 0 7, none 59, none 31. 5t t w 6w , two 76, two 33. x 1x x 1x 0 0, one 0, one 35. x x y y 0 15, none 7, none 37. x 5 3x 38. 3x x 59, none 5, two Use the method of your choice to solve each equation. 39. x 3 x 1 0. x 7 x, 1 1, 1. (x 1) (x ) 5. x (x 3) 9 0, 3, x x 1 x 1 5 x , 3 5, 5. x 6x x 5x 3 0, 1, 3 7. x 9x 0 8. x 9 0 0, 9 3, 3 9. (x 5) (3x 1) 0 8, 51. x(x 3) 3(x ) No real solution 5. (x 1)(x ) (x ) 3, 5 x x , 3 x 5. x , x x 55. x 3x 0 0, x 5 5, 5 Use a calculator to find the approximate solutions to each quadratic equation. Round answers to two decimal places. 57. x 3x x x , , x x x.3x , , x 3.x x 3.1x , 0.91 Use a calculator to solve each problem. 63. Phasing out freon-1. The emission of CFC-1 (or freon-1) in the U.S. can be modeled by the function Emission (thousands of metric tons) y y 0.87x 1.5x 77.5, Emission of CFC Years after 1980 FIGURE FOR EXERCISE 63 x 1 3

7 7 (9-0) Chapter 9 Quadratic Equations and Quadratic Functions where x is the number of years since 1980 and y is the amount of emission in thousands of metric tons (Energy Information Administration, a) In what years was the emission of CFC-1 gas 106 thousand metric tons? 1983, 1991 b) In what year will the emission of CFC-1 gas be zero? Lottery tickets. The formula R 00x 5000x was used in Exercise 7 of Section 9. to predict the revenue when lottery tickets are sold for x dollars each. For what ticket price is the revenue $5,000? $6.91 and $18.09 In this section Geometric Applications Work Problems Vertical Motion 9. APPLICATIONS OF QUADRATIC EQUATIONS In this section we will solve problems that involve quadratic equations. Geometric Applications Quadratic equations can be used to solve problems involving area. E X A M P L E 1 x + ft x ft FIGURE 9. Dimensions of a rectangle The length of a rectangular flower bed is feet longer than the width. If the area is 6 square feet, then what are the exact length and width? Also find the approximate dimensions of the rectangle to the nearest tenth of a foot. Let x represent the width, and x represent the length as shown in Fig. 9.. Write an equation using the formula for the area of a rectangle, A LW: x(x ) 6 The area is 6 square feet. x x 6 0 We use the quadratic formula to solve the equation: x (1)( 6) 8 (1) 7 (1 7) 1 7 Because 1 7 is negative, it cannot be the width of a rectangle. If x 1 7, then x So the exact width is 1 7 feet, and the exact length is 1 7 feet. We can check that these dimensions give an area of 6 square feet as follows: LW (1 7)(1 7) Use a calculator to find the approximate dimensions of 1.6 and 3.6 feet. Work Problems The work problems in this section are similar to the work problems that you solved in Chapter 5. However, you will need the quadratic formula to solve the work problems presented in this section.