LESSON OBJECTIVES &KDSWHU

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1 LESSON OBJECTIVES &KDSWHU 4.1 Solving Quadratic Equations by Graphing Write quadratic equations in standard form. Solve quadratic equations by graphing. 4.2 Solving Quadratic Equations Using Square Roots Solve quadratic equations of the form ax2 c = 0 using square roots. 4.3 Solving Quadratic Equations by Completing the Square Solve quadratic equations by completing the square. 4.4 Solving Quadratic Equations by Factoring Solve quadratic equations by factoring. &RQWHQWV 6ROYLQJ 4XDGUDWLF (TXDWLRQV E\ *UDSKLQJ 6ROYLQJ 4XDGUDWLF (TXDWLRQV 8VLQJ 6TXDUH 5RRWV 6ROYLQJ 4XDGUDWLF (TXDWLRQV E\ &RPSOHWLQJ WKH 6TXDUH 6ROYLQJ 4XDGUDWLF (TXDWLRQV E\ )DFWRULQJ 6ROYLQJ 4XDGUDWLF (TXDWLRQV 8VLQJ WKH 4XDGUDWLF )RUPXOD 4.5 Solving Quadratic Equations Using the Quadratic Formula Find the number of solutions to a quadratic equation. Solve quadratic equations by using the quadratic formula. 4.6 Solving Quadratic Equations That Have Complex Solutions Solve quadratic equations that have complex number solutions. Interpret complex number solutions. 152 Chapter 4 Quadratic Equations 6ROYLQJ 4XDGUDWLF (TXDWLRQV 7KDW +DYH &RPSOH[ 6ROXWLRQV 0DWK /DEV 7RVV LQ WKH &DQ 0HDVXULQJ DQ 8PEUHOOD 0DWK $SSOLFDWLRQV &KDSWHU 5HYLHZ &KDSWHU $VVHVVPHQW &KDSWHU 4XDGUDWLF (TXDWLRQV Mental Math Skills Review Factor ± ±3 1. x2 + 2x 24 (x 4)(x + 6) 3. ( 2) ± x2 10x + 21 (x 7)(x 3) ( 12)2 ± x2 + 10x + 3 (2x + 1)(4x + 3) 4. 9x2 + 12x 45 (3x 5)(3x + 9)

2 Look to Your Future Evidence of quadratic relations are present in the tasks performed by skilled professionals and in activities people choose to do in their leisure time. Areas where professionals use quadratic relationships include the fields of architecture and engineering, finance, physics, and marketing. Leisure activities where quadratic relationships play a role could involve a bouncing ball. Ask students if they see themselves using quadratic relationships more as a skilled professional or in leisure activities. PLANNING THE CHAPTER Math Labs, pp Data Sheet (Lab Data Sheets) Math Applications, pp Chapter Review, pp Chapter Test, p. 196 Software Generated Assessment Standardized Test Practice, p. 197 Grid Response Form (CRB) Chapter Resource Book (CRB) Reteaching, pp. 145, 149, 153, 159, 163, 167 Extra Practice, pp. 147, 151, 155, 161, 165, 169 Enrichment, pp. 157, 171 Standardized Test Response Form, pp. 173, 174 Standardized Test Answers, p. 175 Classroom/Journal Topics What s Ahead? In this chapter, students will learn multiple methods for solving quadratic equations. These methods include graphing, algebraic manipulation, and using the quadratic formula. Students will also be introduced to complex numbers. Chapter 4 Quadratic Equations 153

3 LESSON PLANNING Vocabulary quadratic equation parabola standard form minimum maximum vertex axis of symmetry roots zeros Extra Resources Reteaching 4.1 Extra Practice 4.1 Assignment In-class practice 1 5 Homework 6 35 Math Applications Exercises 1, 8, and 16 from pages START UP A projectile is any object thrown or otherwise projected into the air at an angle. The object has both horizontal and vertical motion. While the horizontal motion is constant, the vertical motion is affected by gravity. The object s trajectory is the parabolic path of the object as it falls to the ground. 154 Chapter 4 Quadratic Equations

4 x = b 2 a x = b 2 a INSTRUCTION When discussing the line of symmetry, remind students that vertical lines have an equation of the form x = a and horizontal lines have an equation of the form y = a, where a is a constant. ACTIVE LEARNING Students should verify the solutions of x = 5 and x = 1 by plugging them into the equation y = x 2 4x 5. Emphasize that students should get a value of zero, thus making the solutions zeros of the quadratic equation. Instruction Emphasize to students that a parabola has either a minimum or a maximum value, but not both. Ask students to describe a real world situation which has either a minimum or a maximum value, but not both. (sample answer: the number of points scored on a video game has a minimum of 0, but does not have a maximum) R.E.A.C.T. Strategy Transferring Have students use a spreadsheet to determine the zeros for the function y = 3x 2 4x + 7. Students will use the Intermediate Value Theorem to determine where they need to focus on the table of values. An excerpt of a possible spreadsheet is shown. 4.1 Solving Quadratic Equations by Graphing 155

5 INSTRUCTION As students make the table of values, point out the symmetry in the y-values. Help students connect the symmetry in the table of values to the axis of symmetry for the graph of the parabola. You can expand on this lesson by asking students to make a conjecture as to how many solutions a cubic or quartic equation might have. Have them verify their conjectures using a graphing calculator. (cubic: 3 solutions; quartic: 4 solutions) Answer to Ongoing Assessment b = 64 = 64 = 2 2a 2( 16) 32 Reteaching 4.1 (CRB) R.E.A.C.T. Strategy Cooperating Ask students to work together to write a new quadratic function for the fire hose. Advise students that they should only make changes to the linear term of the function, since the quadratic term represents the acceleration due to gravity and the constant term represents the height of the ladder. The new function should result in the water reaching a maximum height of 224 and the first drop of water reaching the ground between 6 and 7 seconds. Students should use a graphing calculator to continue to revise their function until the given conditions are met. h(t) = 16t t Chapter 4 Quadratic Equations

6 WRAP UP To ensure mastery of objectives, students should be able to: Put a given quadratic equation into standard form. Graph a quadratic equation in standard form. Use the graph of a quadratic equation to determine the solutions of a quadratic equation. Assignment In-class practice 1 5 Homework 6 35 Math Applications Exercises 1, 8, and 16 from pages Extra Practice 4.1 (CRB) 2 3, 6 Think and Discuss Answers 1. Graph the quadratic equation and find the x-values where the graph crosses the horizontal axis. 2. yes; if its graph does not cross the x-axis. 3. Answers will vary. Sample answer: the flight of an arrow 4. yes; if the vertex of the graph is on the x-axis. 5. If the coefficient of the quadratic term is positive, the graph opens upwards. If it is negative, the graph opens downward. 4.1 Solving Quadratic Equations by Graphing 157

7 Practice and Problem Solving Additional Answers 30. a Mixed Review Additional Answers Answers to Math Applications Math Applications for this chapter are on pages The notes and solutions shown below accompany the suggested applications to assign with this lesson. 1. a. The coefficient of the equation that models the distance an object has traveled after being dropped is 2 1 times the gravitational force, 32 feet per second. b. h(t) = = 16t2 Find the value of h when t = 4: 256. Because the pebble traveled down, the distance is negative, but the height of the cliff must be positive. The cliff is 256 feet high. 158 Chapter 4 Quadratic Equations 8. a. Find the maximum height on the graph: 450 m. b. Find the x-intercept of the graph: about 9 s 16. a. h(t) = 16t b. Find the x-intercept of the graph: about 6.5 and 6.5. Because time must be positive, it took about 6.5 s for the raft to reach the water. c. Find the value of y when t = 5: 300 ft.

8 x c LESSON PLANNING Extra Resources Reteaching 4.2 Extra Practice 4.2 Assignment In-class practice 1 5 Homework 6 36 Math Applications Exercise 14 from pages START UP Review the concept of using inverses to solve equations by asking students to identify the inverse operation used to solve these example equations: 1. x + 4 = 16 (subtraction) 2. 4x = 16 (division) 3. x 2 = 16 (square root) =± Diversity in the Classroom Visual Learners After solving the equations on this page, ask students to describe the graphs of y = 5x and y = 4x 2 16 based on their knowledge of x-intercepts and the value of the lead coefficient. Call attention to the fact that y = 4x cannot cross the x-axis because it has no real solutions. The graph of y = 5x will open upward, will have a vertex below the x-axis, and will cross the x-axis at ±5. The graph of y = 4x 2 16 will open downward, will have a vertex below the x-axis, and will not cross the x axis. 4.2 Solving Quadratic Equations Using Square Roots 159

9 INSTRUCTION Example 2 Remind students that the acceleration of a freefalling object is not affected by the mass/weight of the object. A pebble will take the same amount of time to hit the water as a 150- pound cliff diver. Students often forget to give both solutions when using the square root property to solve quadratic equations. Remind students that writing x =± 2 is a short cut way 3 of showing that there are actually two solutions to the equation: x = 2 3 and x = =± 2 3 =±2 3 Problem Solving Understand the Problem The height of the cliff is given. You need to find the amount of time it takes a rock to fall 100 feet. Develop a Plan Use the formula from Example 2 and a height of 100 ft. h = 16t 2 + s Carry Out the Plan 16t = 0 16t 2 = 100 t 2 = 6.25 t = 2.5 s Check the Results Check: 16(2.5) = 0 R.E.A.C.T. Strategy Relating Have students access an Internet site which provides animated simulations of free-falling objects. These simulations will help students understand that the speed of the falling object does not remain constant, and thus graphs of the height functions of free-falling objects are parabolas rather than straight lines. 160 Chapter 4 Quadratic Equations

10 WRAP UP To ensure mastery of objectives, students should be able to: Solve quadratic equations of the form ax 2 c = 0 using square root. Assignment In class practice 1 5 Homework 6 36 Math Applications Exercise 14 from pages Reteaching 4.2 (CRB) Think and Discuss Answers 1. c = 0 2. c < 0 3. c > 0 4. Divide both sides by 3. Then take the square root of each side. 5. ax 2 = c 4.2 Solving Quadratic Equations Using Square Roots 161

11 Mixed Review Additional Answers paperback books and 1 hardcover Answers to Math Applications Math Applications for this chapter are on pages The notes and solutions shown below accompany the suggested applications to assign with this lesson. 14. a. 2s 2 = 450 s 2 = 225 s = ±15 Length must be positive, so the dimensions of the bedroom are 15 ft 15 ft. b. P = 4s = 4 15 = 60 ft Extra Practice 4.2 (CRB) s t 64h k Chapter 4 Quadratic Equations

12 Diversity in the Classroom Kinesthetic Learners Before completing the activity with the Algebra Tiles, solve the following equations with the students. Use the lack of solution for the last equation as motivation for completing the Algebra Tiles activity. 1. x 2 = 16 ±4 2. (x + 2) 2 = 16 6, 2 3. x 2 + 4x + 4 = 16 6, 2 4. x x + 36 = 16 10, 2 5. x x + 2 = 16 no solution using the square root process LESSON PLANNING Vocabulary completing the square perfect square trinomial Extra Resources Reteaching 4.3 Extra Practice 4.3 Enrichment 4.3 Assignment In-class practice 1 4 Homework 5 33 Math Applications Exercises 3 and 6 from pages START UP Have students begin the lesson by multiplying the polynomial expressions (x + 2) 2 and (x + 9) 2. Have students describe the pattern of perfect square binomials in their own words. (x 2 + 4x + 4; x ; sample description of pattern: The quadratic term is the square of the first term of the binomial, the linear term is 2x (product of terms of the binomial), and the constant term is the square of the second term of the binomial.) 4.3 Solving Quadratic Equations by Completing the Square 163

13 ACTIVE LEARNING Have students use Algebra Tiles to complete the square for x x +. Present students with x x +. Have students explain why Algebra Tiles would not be useful in solving this problem. (11x cannot be cut in half and represented using Algebra tiles.) Instruction Tell students that completing the square is a useful tool both for solving quadratic equations and for finding the vertex of a parabola. Point out that the formula x = b was derived by 2a completing the square. ( ) Diversity in the Classroom Visual Learners Write the following steps on the board and ask students to identify the errors in the solution process. x 2 + 4x + 2 = 18 x 2 + 4x = 16 2 x + 4x = 16 x + 2 = ±4 x = ±4 2 x = 6, 2 The 2 x + 4x x + 2. The square must be completed prior to taking the square root of both sides. 164 Chapter 4 Quadratic Equations

14 ( ) ( ) = = 2 = = 4± 19 INSTRUCTION Help students connect with the 2 notation b ( 2) by having them repeat the translation take half of the b value and then square it. Students often simplify expressions such as x = 4 ± 5 incorrectly. Often, a student will find the answer of x = 1 and will then write the solution set as x = ±1. Encourage students to rewrite the expression x = 4 ± 5 as: x = or x = 4 5. Doing so will help them obtain both of the correct solutions to the solution set. Diversity in the Classroom Visual Learners Help students visualize the equation using a balance scale. Draw a balance scale on the board and write x 2 + 4x on one side of the scale and 5 on the other side of the scale. Describe how adding 4 to the left side of the scale will put the scale out of balance. Describe how then adding 4 to the right side of the scale will bring it back into balance again. 4.3 Solving Quadratic Equations by Completing the Square 165

15 INSTRUCTION Students will often ask if the term that completes the square will always be a positive number. Remind students of the examples in the start-up notes for the class. The constant term obtained when the expression (x + c) 2 is multiplied will be positive both when c is a positive number and when c is a negative number. Reteaching 4.3 (CRB) ( ) ( ) = = 3 = Think and Discuss Answers 1. To complete the square means finding what needs to be added to the constant term to make a perfect square trinomial. 2. Use a large square to represent x 2 and 8 rectangles, 4 horizontal and 4 vertical, to represent 8x. Then add 16 small unit tile squares to complete the square. 3. 1; if the leading coefficient, a, is not 1, divide both sides of the equation by a. Then complete the square. 4. Divide both sides by 2. Then add 4 to both sides to form the perfect square trinomial (x + 2) 2 on the left hand side. Then take the square root of both sides and solve for x. 166 Chapter 4 Quadratic Equations

16 1 4 WRAP UP To ensure mastery of objectives, students should be able to: Use the strategy of completing the square to solve a quadratic equation. Assignment In-class practice 1 4 Homework 5 33 Math Applications Exercises 3 and 6 from pages Extra Practice 4.3 (CRB) Enrichment 4.3 (CRB) 4.3 Solving Quadratic Equations by Completing the Square 167

17 Practice and Problem Solving Additional Answers 30. a. (50, 3,250); this point represents the maximum daily sales (3,250) when the price is $50. Mixed Review Additional Answers 33. a. s = 6A 6 Answers to Math Applications Math Applications for this chapter are on pages The notes and solutions shown below accompany the suggested applications to assign with this lesson. 3. a. 0 = x 2 630x 0 = x(x 630) 0 = x or 0 = x 630 x = 0 ft or x = 630 ft The height of the arch is 630 ft. b. 220 = x2 + 4x 34,650 = x 2 630x 34, ,225 = x 2 630x + 99,225 64,575 = (x 315) 2 ± = x = x 315 or = x 315 x 569 ft or x 61 ft c. y = (200)2 + 4(200) 546 ft a. 2 = t 2 6t = t 2 6t = t 2 6t = t 2 6t = (t 3) 2 ±2 = t 3 2 = t 3 or 2 = t 3 t = 1 s or t = 5 s b. 2 = t 2 6t = t 2 6t = t 2 6t = t 2 6t = (t 3) 2 0 = t 3 or 0 = t 3 t = 3 s c. I = (7) 2 6(7) + 7 = 14 amps 168 Chapter 4 Quadratic Equations

18 R.E.A.C.T. Strategy Experiencing Many students find factoring both tiresome and difficult. A fun way to motivate students to practice factoring is to play Factoring Bingo. Each square of the BINGO card contains a monomial or binomial factor. As students factor polynomials read aloud by the teacher, they can mark the squares that are on their BINGO boards. LESSON PLANNING Vocabulary factoring Zero Product Property perfect square trinomial difference of two squares Extra Resources Reteaching 4.4 Extra Practice 4.4 Assignment In-class practice 1 3 Homework 4 30 Math Applications Exercises 4 and 11 from pages START UP Review multiplying binomials with the FOIL method using the following examples. Emphasize how the signs of the binomials affect the resulting trinomial. 1. (x + 7)(2x + 5) (2x x + 35) 2. (x 7)(2x 5) (2x 2 19x + 35) 3. (x + 7)(2x 5) (2x 2 + 9x 35) 4. (x 7)(2x + 5) (2x 2 9x 35) Remind students that factoring is like Jeopardy. Given the polynomial answer, the original polynomial factors must be determined. 4.4 Solving Quadratic Equations by Factoring 169

19 INSTRUCTION Encourage students to always attempt to factor out the greatest common factor before attempting to use the guess and check method. Use the following example to illustrate why this is important: 6x 2 12x 48 = 6(x 2 2x 8) = 6(x 4)(x + 2) 6x 2 has more than one set of factors to consider, whereas x 2 only one set of factor to consider. Students often attempt to solve an equation such as the following using a nonzero product property. (x + 6)(x + 4) = 8 x + 6 = 8 or x + 4 = 8 x = 2 or x = 4 Remind students that this equation must be set equal to zero before solving and using the Zero Product Property R.E.A.C.T. Strategy Cooperating Have students work in pairs. Each student should write a trinomial of the form ax 2 + bx + c and give it to his or her partner to be factored. Many of the trinomials that the students write for each other will not be factorable. As students discover this difficulty, ask students to come up with a strategy for always writing trinomials that can be factored. Write two binomials and multiply them. 170 Chapter 4 Quadratic Equations

20 1, INSTRUCTION Remind students that the quadratic equations being solved by factoring could also be solved by using the Square Root Property or a combination of completing the square and the Square Root Property. It is often helpful to show students both strategies for the same equation. Students will often try to apply the pattern for the difference of two squares to a problem such as x = 0. Help students to understand that this does not work by using the FOIL method. Also, remind students that solving this equation using the Square Root Property would result in having no real solutions. Common Student Misconceptions Students might think that equations that are not factorable do not have any solutions. Show that an equation that cannot be factored (such as x 2 4x 6 = 0) does not have rational solutions, but that it does have solutions. Show them that the solutions do exist by graphing the parabola and pointing out that it does indeed cross the x-axis. 4.4 Solving Quadratic Equations by Factoring 171

21 Answer to Critical Thinking A b a + b ab (x a)(x b) x 2 3x x 2 + 7x x 2 8x x 2 12x x 2 + 4x 21 The sum of the numbers (a + b) is the opposite of the coefficient of the x-term, and the product (ab) is the constant term. WRAP UP To ensure mastery of objectives, students should be able to: Use factoring to solve a quadratic equation. Recognize special case quadratic equations. Write a quadratic equation or parabola given the solutions of the equation. Assignment In-class practice 1 3 Homework 4 30 Math Applications Exercises 4 and 11 from pages Think and Discuss Answers 1. If the product of two factors of a quadratic equation is equal to 0, then either factor can be equal to 0. Set each factor equal to 0 and solve for the variable. 2. no; the left side of the equation does not factor. 3. Since 1 and 4 are roots, you know that (x 1)(x 4) = 0 or x 2 5x + 4 = Chapter 4 Quadratic Equations

22 Reteaching 4.4 (CRB) 1 3, 3 3 2, 4 3 Extra Practice 4.4 (CRB) Common Student Misconceptions Students often express the total width of the photograph and mat as 5 + x and the total length of the photograph and mat as 7 + x. Remind students that both sides of the mat must be added to the dimensions of the photograph resulting in a width of: 5 + x + x = 5 + 2x and a length of 7 + x + x = 7 + 2x. 4.4 Solving Quadratic Equations by Factoring 173

23 Answers to Math Applications Math Applications for this chapter are on pages The notes and solutions shown below accompany the suggested applications to assign with this lesson. 4. a. Let x represent the first integer and x + 2 represent the second integer. x 2 + 2x 1,763 = 0 b. (x 41)(x + 43) = 0 x 41 = 0 or x + 43 = 0 x = 41 or x = 43 Because 43 is not a positive integer, so the first integer is 41. The second integer is x + 2 = = a. h(t) = 16t t b. t = b 2 a = 128 2( 16) = 4 s c. 0 = 16t t 0 = 16t(t 8) 16t = 0 or t 8 = 0 t = 0 or t = 8 When t = 0, the firework has not left the ground, so the shell will hit the ground when t = 8 s. 174 Chapter 4 Quadratic Equations

24 x = b ± 2 b 4ac 2a LESSON PLANNING Vocabulary quadratic formula Extra Resources Reteaching 4.5 Extra Practice 4.5 Assignment In-class practice 1 5 Homework 6 36 Math Applications Exercises 5, 9, 13, 15, and 17 from pages START UP Tell students that the quadratic formula was derived by completing the square of the equation ax 2 + bx + c = 0. Show students the following partial derivation. 2 ax + bx + c = 0 2 ax + bx = c 2 x + b a x = c a 2 x b a x b c b 2a a 2a ( ) = + ( ) R.E.A.C.T. Strategy Experiencing Encourage students to use a graphing calculator to check their answers when using the quadratic formula. Remind students that the solutions to the quadratic equation will be x-intercepts of the graph of the parabola. 4.5 Solving Quadratic Equations Using the Quadratic Formula 175

25 INSTRUCTION Have students compute the discriminant first and then use the rest of the formula if necessary. Point out to students that if the discriminant turns out to be a perfect square then the quadratic is factorable. Make sure that students have put the equation into standard form so that they accurately identify a, b, and c prior to using the quadratic formula. Ask students to expand the chart on the bottom of the page by adding columns for sketch of the equation and an example equation. Have students use examples worked in previous sections to help fill out their tables. Reteaching 4.5 (CRB) 2 x = b± b 4ac 2a 2 ( 19) ± ( 19) 4( 16)( 4) x = 2( 16) ( 19) ± x = 32 ( 19) ± 105 x = 32 ( 19) ( 19) 105 x = or x = x 027. or x Enriching the Lesson Ask students to solve the equation 2x 3 18x + 10 = 0 through a combination of factoring and the quadratic formula. (x = 0, 0.6, 8.4). After students have determined the solutions to the equation, ask students to sketch a possible graph for the equation y = 2x 3 18x Have students compare the sketch to the actual graph using a graphing calculator. 176 Chapter 4 Quadratic Equations

26 INSTRUCTION Using an example such as x 2 + 6x + 8 = 0, have students solve the equation by completing the square, by factoring, by graphing, and by using the quadratic formula. Ask students to rank the methods in their order of personal preference. WRAP UP To ensure mastery of objectives, students should be able to: Use the discriminant to determine the number of real solutions for a given quadratic equation. Use the quadratic formula to solve a quadratic equation. Assignment In-class practice 1 5 Homework 6 36 Math Applications Exercises 5, 9, 13, 15, and 17 from pages Think and Discuss Answers 1. It is useful to solve using the quadratic formula when the equation does not factor. 2. Subtract 8 from each side to write the equation as 3x 2 + 6x 8 = 0. Then substitute 3 for a, 6 for b, and 8 for c into the quadratic formula and simplify. 3. 0, 1, or ; 2 real roots 5. Answers will vary. 4.5 Solving Quadratic Equations Using the Quadratic Formula 177

27 Extra Practice 4.5 (CRB) Answers to Math Applications Math Applications for this chapter are on pages The notes and solutions shown below accompany the suggested applications to assign with this lesson. 5. a. f(t) = 16t t + 3 b. f(0.5) = 16(0.5) (0.5) + 3 = 39 ft c. t = b 2 a = 80 2( 16) = 2.5 s f(2.5) = 16(2.5) (2.5) + 3 = 103 ft d. f(2.25) = 16(2.25) (2.25) + 3 = 102 ft Yes, the ball will clear a 20 foot tall fence. 9. a. p = b 2 a = 400 2( 20) = $10 b. R(p) = 20(10) (10) + 20,000 = $22,000 c. 0 = 20p p + 20,000 p = ± ( 20)( 20, 000) 2( 20) p or p The price of a shirt must be positive, so the company will start to lose money when the price of a shirt is $ d. $ = $8 R(p) = 20(8) (8) + 20,000 = $21, Chapter 4 Quadratic Equations

28 x y = y 1 4 = x a. Draw a diagram. Let x represent the width of the walkway. 3 y 1 =± x 9 (40 2x)(25 2x) = , x + 4x 2 = 544 4x 2 130x = 0 2 x = 130 ± ( )( 456) 24 ( ) x = 28.5 or x = 4 Because 28.5 is an extraneous solution, the width is 4 ft. b. 40 2(4) = 32 ft; 25 2(4) = 17 ft Answers to Math Applications Math Applications for this chapter are on pages The notes and solutions shown below accompany the suggested applications to assign with this lesson. 13. a. 93 = (12 + 2x)(16 + 2x) = 4x x 0 = 4x x 93 x = ± ( )( 93) 24 ( ) x = 15.5 or x = 1.5 Because length must be positive, x = 1.5 m. b (1.5) = 19 m 15. a. x = b 2 a = (. ) 20, since x = 0 represents 1985, x = 20 represents b. y = (10) (10) = million tons = 95,404,000 tons c. 1990: y = (5) (5) = million tons 2001: y = (16) (16) = million tons There were more emissions in Solving Quadratic Equations Using the Quadratic Formula 179

29 LESSON PLANNING Vocabulary imaginary number complex number Extra Resources Reteaching 4.6 Extra Practice 4.6 Enrichment 4.6 Assignment In-class practice 1 5 Homework 6 35 Math Applications Exercises 2, 7, 12, 18, and 19 from pages START UP To help students accept the notion of imaginary numbers, tell them that ancient mathematicians once believed that all numbers were both positive and rational. As mathematical applications surfaced which required a number to be either irrational or negative, mathematicians revised their beliefs and the number systems were updated to include numbers such as 5 and 2. Have students solve the equations x = 0 using the square root method, x 2 + 6x + 8 = 0 by completing the square, and 3x 2 + x + 8 = 0 by using the quadratic formula. Use a graphing calculator to verify that each of these equations have no real solutions. Indicate to students that by the end of class, they will be able to solve each of these equations. ( x =± 36 ; x = 3± 8; does not cross the x-axis) 1= 1 2 b± b 4ac x = 2a 2 ( 112) ± ( 112) 416 ( )( 220) x = 216 ( ) 112 ± 12, , 080 x = Chapter 4 Quadratic Equations

30 112 ± 1, 536 x = 32 x = 112 ± 16i 6 32 x = i 6 or x = i x i or x i 1 3 ( ) ± ( ) ()( 50 ) 21 () 10 ± INSTRUCTION Help students understand that imaginary and complex numbers cannot be graphed on the Cartesian coordinate system since number lines only consist of real numbers. Emphasize that this is why quadratic equations which have no real solution do not cross the x-axis. Answer to Critical Thinking When the vertex of graph of a quadratic equation is on the x axis, then there is only one solution. This is the only time there is one solution to a quadratic equation. INSTRUCTION Review the process of examining the discriminant to determine the number of solutions of a quadratic equation. Discuss the situation in which the equation has no solutions and emphasize that because the situation arises frequently, it is necessary to find ways to express the imaginary solutions. 4.6 Solving Quadratic Equations Using Any Method 181

31 Reteaching 4.6 (CRB) R.E.A.C.T. Strategy Applying Have students research the Internet and make a list of three real world applications which involve the use of imaginary and/or complex numbers. Allow class time for students to briefly describe each application. 182 Chapter 4 Quadratic Equations

32 WRAP UP To ensure mastery of objectives, students should be able to: Solve quadratic equations which have non-real solutions. Interpret the meaning of complex solutions that occur in quadratic equation application problems. Assignment In-class practice 1 5 Homework 6 35 Math Applications Exercises 2, 7, 12, 18, and 19 from pages Extra Practice 4.6 (CRB) Enrichment 4.6 (CRB) 2± Think and Discuss Answers 1. The graph does not cross the x-axis. 2. Answers may vary. Sample answer: the sum or a real number and an imaginary number; 2 + 3i They are all equal to The solutions are complex. 4.6 Solving Quadratic Equations Using Any Method 183

33 Practice and Problem Solving Additional Answers 26. no; the solutions to the equation P = 30 are both complex numbers, and the number of items sold must be real. 29. a. Both solutions to the equation r(x) = c(x) are complex, so there are no break even points. 30. a. b. x = 2, 10; the revenue generated equals the cost of manufacturing, when the company sells 2 units or 10 units. b. no; the graph of the quadratic equation modeling the rocket s height never reaches 550. c. t = 4 ± 1.84i; both solutions are complex numbers. Since amounts of time cannot be imaginary, the rocket never reaches a height of 550 feet. d. t = 4; the rocket reaches a height of 496 feet 4 seconds after it is launched Answers to Math Applications Math Applications for this chapter are on pages Complete notes and detailed solutions for the suggested applications to assign with this lesson can be found on pages Chapter 4 Quadratic Equations

34 1 2 MATH LAB Activity 1 PREPARE Ask students to think about a free throw in a basketball game. Discuss the path the ball follows and be certain that students know the path is parabolic shaped. Students can practice throwing paper wads in a trash can to see the parabolic-shaped curve. TEACH Students should complete the lab in pairs. Roles for groups of 2 students: 1. recorder and timer 2. free throw shooter Students will both have to play each role. Verify that the recorder measures the initial height of the ball s release accurately. FOLLOW-UP Compare the data from all the class members to make generalizations about the height of the free throw and time it was in the air. You can have the students keep record of the hits and misses. You may be able to find a relationship between the height of the curve and the success of the shot. This is a good opportunity for students to use a graphing calculator to find an equation for a curve of best fit. Math Labs 185

35 MATH LAB Activity 2 PREPARE Use an open umbrella to demonstrate that a cross section of the domed part is a parabola. Discuss how the handle serves as the line of symmetry and an axis of rotation about which to spin the parabola. TEACH Point out that some umbrellas have a part of the handle that protrudes about the domed part. Subtract the length of this protrusion from the height when you measure from the ground to the tip of the dome. Students will need to help each other during the measuring steps of this project. One student will need to hold the umbrella perpendicular to the ground while the other student takes the measurement. Ask students to compare different types of umbrellas being measured. Some will be wider, while some will have a greater height than others. FOLLOW-UP Have students present their umbrellas and equations to the class. Suggest that the student compare these to determine how the values of a and b affect the shape of the umbrella. 1 2 Math Lab Solutions and Notes ; 0.4; your throw 0.6 s; your partner s throw 0.4 s; divide the total time the paper is in the air by 2 7. Graphs will vary, but should be a parabola with a maximum point. 186 Chapter 4 Quadratic Equations

36 y x 2 40 Math Lab Solutions and Notes 6. x y No; Sample answer: The shape of the umbrella is that of a parabolic curve rotated around its axis of symmetry, in this case, the handle of the umbrella. Since any cross-section of the shape of the umbrella is the same parabolic curve, it does not matter which cross-section is chosen. Math Labs 187

37 Math Applications Solutions and Notes 1. a. The coefficient of the equation that models the distance an object has traveled after being dropped is 2 1 times the gravitational force, 32 feet per second. b. h(t) = = 16t2 Find the value of h when t = 4: 256. Because the pebble traveled down, the distance is negative, but the height of the cliff must be positive. The cliff is 256 feet high. 2. a. 13 = 16t t = 16t t 9 t = ± ( 16)( 9) 2( 16) b. t = b 2 a = 26 2( 16) = s c. h(t) = 16t t a. 0 = x 2 630x 0 = x(x 630) 0 = x or 0 = x 630 x = 0 ft or x = 630 ft The height of the arch is 630 ft. b. 220 = x2 + 4x ,650 = x 2 630x 34, ,225 = x 2 630x + 99,225 64,575 = (x 315) 2 ± = x = x 315 or = x 315 x 569 ft or x 61 ft c. y = (200)2 + 4(200) 546 ft 188 Chapter 6 Quadratic Equations

38 relationship between the number of people and the ticket prices: (15, 200) and (16, 190). Find the slope of the linear relationship: = 10, and y-intercept: 200 = 10(15) + b, therefore b = 350. Write an equation in slope intercept form: y = 10x + 350, where y is the number of people and x is the ticket price. Write an equation for the revenue. R(x) = number of people number of tickets R(x) = x( 10x + 350) = 10x x x = b 2 a = 350 2( 10) = $17.50 b. R(17.5) = 10(17.5) (17.5) = $3, Math Applications Solutions and Notes 4. a. Let x represent the first integer and x + 2 represent the second integer. x 2 + 2x 1,763 = 0 b. (x 41)(x + 43) = 0 x 41 = 0 or x + 43 = 0 x = 41 or x = 43 Because 43 is not a positive integer, so the first integer is 41. The second integer is x + 2 = = a. f(t) = 16t t + 3 b. f(0.5) = 16(0.5) (0.5) + 3 = 39 ft c. t = b 2 a = 80 2( 16) = 2.5 s f(2.5) = 16(2.5) (2.5) + 3 = 103 ft d. f(2.25) = 16(2.25) (2.25) + 3 = 102 ft Yes, the ball will clear a 20 foot tall fence. 6. a. 2 = t 2 6t = t 2 6t = t 2 6t = t 2 6t = (t 3) 2 ±2 = t 3 2 = t 3 or 2 = t 3 t = 1 s or t = 5 s b. 2 = t 2 6t = t 2 6t = t 2 6t = t 2 6t = (t 3) 2 0 = t 3 or 0 = t 3 t = 3 s c. I = (7) 2 6(7) + 7 = 14 amps 7. a. To write an equation for the revenue, find an expression for the Math Applications 189

39 Math Applications Solutions and Notes 7. c. y = 10(17.5) = 175 people d. Find an expression for the relationship between the number of people and the ticket prices: (15, 200) and (17, 190). Find the slope of the linear relationship: = 5, and y-intercept: 200 = 5(15) + b, therefore b = 275. Write an equation in slope intercept form: y = 5x + 275, where y is the number of people and x is the ticket price. Write an equation for the revenue. R(x) = x( 5x + 275) = 5x x x = b 2 a = 275 2( 5) = $27.50 R(27.5) = 5(27.5) (27.5) = $3, a. Find the maximum height on the graph: 450 m. b. Find the x-intercept of the graph: about 9 s. 9. a. p = b 2 a = 400 2( 20) = $10 b. R(p) = 20(10) (10) + 20,000 = $22,000 c. 0 = 20p p + 20,000 p = ± ( 20)( 20, 000) 2( 20) p or p The price of a shirt must be positive, so the company will start to lose money when the price of a shirt is $ d. $ = $8 R(p) = 20(8) (8) + 20,000 = $21, a. x = b 2 a = ( 003. ) (. ) = 1,000 seats b. C = (1,000) (1,000) + 35 = $20 c. C = (2,000) (2,000) + 35 = $35 d. Answers may vary. Sample answers: storage costs, machine wear, and/or labor costs 190 Chapter 6 Quadratic Equations

40 Math Applications Solutions and Notes 11. a. h(t) = 16t t has not left the ground, so the shell will hit the ground when t = 8 s. b. t = b 2 a = 128 2( 16) = 4 s c. 0 = 16t t 0 = 16t(t 8) 16t = 0 or t 8 = 0 t = 0 or t = 8 When t = 0, the firework 13. a. 93 = (12 + 2x)(16 + 2x) = 4x x 0 = 4x x 93 x = 2 56 ± ( )( 93) 24 ( ) x = 15.5 or x = 1.5 Because length must be positive, x = 1.5 m. b (1.5) = 19 m Math Applications 191

41 Math Applications Solutions and Notes 14. a. 2s 2 = 450 s 2 = 225 s = ±15 Length must be positive, so the dimensions of the bedroom are 15 ft 15 ft. b. P = 4s = 4 15 = 60 ft 15. a. x = b 2 a = (. ) 20, since x = 0 represents 1985, x = 20 represents b. y = (10) (10) = million tons = 95,404,000 tons c. 1990: y = (5) (5) = million tons 2001: y = (16) (16) = million tons There were more emissions in a. h(t) = 16t b. Find the x-intercept of the graph: about 6.5 and 6.5. Because time must be positive, it took about 6.5 s for the raft to reach the water. c. Find the value of y when t = 5: 300 ft. 17. a. Draw a diagram. Let x represent the width of the walkway. (40 2x)(25 2x) = , x + 4x 2 = 544 4x 2 130x = 0 2 x = 130 ± ( )( 456) 24 ( ) x = 28.5 or x = 4 Because 28.5 is an extraneous solution, the width is 4 ft. b. 40 2(4) = 32 ft; 25 2(4) = 17 ft 192 Chapter 6 Quadratic Equations

42 2 80 Math Applications Solutions and Notes 18. a. b. 4(x 8) 2 = 48 4(x 2 16x + 64) = 48 4x 2 64x = 48 4x 2 64x = 0 x = ( ) ± ( ) ( )( 208 ) 24 ( ) x 4.54 or x If x = 4.54, then = Length must be positive, so x = x 8 = = 3.46 The dimensions of the boxes are 3.46 in in. 4 in. 19. a. d = s2 80 b. d = 40 2 = 20 km 80 c = s2 80 3,136 = s 2 ±56 = s Because speed must be positive, the speed of the automobile was 56 km/h. d. 150 = s ,000 = s 2 ±109.5 s Because speed must be positive, the maximum speed of an automobile should be km/h. Math Applications 193

43 Vocabulary Review axis of symmetry (4.1) completing the square (4.3) difference of two squares (4.4) factoring (4.3) maximum (4.1) minimum (4.1) parabola (4.1) perfect square trinomial (4.3, 6.4) quadratic equation (4.1) quadratic formula (4.5) roots (4.1) standard form (4.1) vertex (4.1) zeros (4.1) Zero Product Property (4.4) Chapter Review Additional Answers = 2 = ( ) = ( 4) Chapter 6 Quadratic Equations

44 Chapter Review Additional Answers 12. The daily profit will never be as much as $35,000 for any price. The equation P(x) = 35,000 does not have any real number solutions. 2 x = b± b 4ac 2a 2 6± 6 41 ()( 22) x = 21 () x = 6± x = 6± ± 4( 13) x = 2 x = 3± i Chapter Review 195

45 Chapter Test Additional Answers = 7± cm 196 Chapter 6 Quadratic Equations

46 Standardized Test Practice Additional Answers Open Ended Response 8. no; 5i + 5i = 0, and 0 is not a purely imaginary number. Extended Response 9. a. x b., x 150, 000 c. x + 5 d. 15; $10,000 Standardized Test Response Form (CRB) Chapter Assessments 197