Solve each equation by completing the square. Round to the nearest tenth if necessary.

Transcription

1 Find the value of c that makes each trinomial a perfect square. 1. x 18x + c In this trinomial, b = 18. So, c must be 81 to make the trinomial a perfect square.. x + x + c In this trinomial, b =. So, c must be 11 to make the trinomial a perfect square. 3. x + 9x + c In this trinomial, b = 9. So, c must be to make the trinomial a perfect square. 4. x 7x + c In this trinomial, b = 7. esolutions Poweredto bymake Cognero So, cmanual must -be the trinomial a perfect square. Solve each equation by completing the square. Round to the nearest tenth if necessary. Page 1

2 9-4 Solving Quadratic by Completing Square So, c must be toequations make the trinomial a perfectthe square. 4. x 7x + c In this trinomial, b = 7. So, c must be to make the trinomial a perfect square. Solve each equation by completing the square. Round to the nearest tenth if necessary. 5. x + 4x = 6 The solutions are about 5. and x 8x = 9 The solutions are about 1.4 and x + 9x 1 = 0 esolutions Manual - Powered by Cognero Page

3 The solutions are about 1.4 and x + 9x 1 = 0 The solutions are about.4 and x + 10x + = 4 Page 3

4 The solutions are about.4 and x + 10x + = 4 The solutions are about 1.4 and CCSS MODELING Collin is building a deck on the back of his family s house. He has enough lumber for the deck to be 144 square feet. The length should be 10 feet more than its width. What should the dimensions of the deck be? Let x = the width of the deck and x + 10 = the length of the deck. Page 4 The dimensions of the deck can not be negative. So, x = or 8. The width of the deck is 8 feet and the length of the deck is or 18 feet.

5 The solutions are about 1.4 and CCSS MODELING Collin is building a deck on the back of his family s house. He has enough lumber for the deck to be 144 square feet. The length should be 10 feet more than its width. What should the dimensions of the deck be? Let x = the width of the deck and x + 10 = the length of the deck. The dimensions of the deck can not be negative. So, x = or 8. The width of the deck is 8 feet and the length of the deck is or 18 feet. Find the value of c that makes each trinomial a perfect square. 10. x + 6x + c In this trinomial, b = 6. So, c must be 169 to make the trinomial a perfect square. 11. x 4x + c In this trinomial, b = 4. So, c must be 144 to make the trinomial a perfect square. 1. x 19x + c Page 5

6 9-4 Solving Quadratic by Completing Square So, c must be 169 toequations make the trinomial a perfectthe square. 11. x 4x + c In this trinomial, b = 4. So, c must be 144 to make the trinomial a perfect square. 1. x 19x + c In this trinomial, b = 19. So, c must be to make the trinomial a perfect square. 13. x + 17x + c In this trinomial, b = 17. So, c must be to make the trinomial a perfect square. 14. x + 5x + c In this trinomial, b = 5. So, c must be to make the trinomial a perfect square. 15. x 13x + c Page 6

7 9-4 Solving Quadratic Equations Completing thesquare. Square So, c must be to make the by trinomial a perfect 14. x + 5x + c In this trinomial, b = 5. So, c must be to make the trinomial a perfect square. 15. x 13x + c In this trinomial, b = 13. So, c must be to make the trinomial a perfect square. 16. x x + c In this trinomial, b =. So, c must be 11 to make the trinomial a perfect square. 17. x 15x + c In this trinomial, b = 15. So, c must be to make the trinomial a perfect square. 18. x + 4x + c Page 7

8 So, c must be 11 to make the trinomial a perfect square. 17. x 15x + c In this trinomial, b = 15. So, c must be to make the trinomial a perfect square. 18. x + 4x + c In this trinomial, b = 4. So, c must be 144 to make the trinomial a perfect square. Solve each equation by completing the square. Round to the nearest tenth if necessary. 19. x + 6x 16 = 0 The solutions are 8 and. 0. x x 14 = 0 Page 8

9 9-4 Solving Quadratic by Completing the Square The solutions are 8Equations and. 0. x x 14 = 0 The solutions are about.9 and x 8x 1 = 8 The solutions are 1 and 9.. x + 3x + 1 = Page 9

10 The solutions are 1 and 9.. x + 3x + 1 = The solutions are about 3.3 and x 11x + 3 = 5 The solutions are about 0. and x 10x = 3 Page 10

11 The solutions are about 0. and x 10x = 3 The solutions are about 1.4 and x x + 7 = 5 The square root of a negative number has no real roots. So, there is no solution. 6. 3x + 1x + 81 = 15 Page 11

12 The square root of a negative number has no real roots. So, there is no solution. 6. 3x + 1x + 81 = 15 The square root of a negative number has no real roots. So, there is no solution. 7. 4x + 6x = 1 The solutions are about.6 and x + 5 = 10x Page 1

13 The solutions are about.6 and x + 5 = 10x The solutions are about 0.7 and x + 10x = 14 Page 13

14 The solutions are about 0.7 and x + 10x = 14 The solutions are about 1.1 and x 1 = 14x Page 14

15 The solutions are about 1.1 and x 1 = 14x The solutions are about 3.5 and STOCK The price p in dollars for a particular stock can be modeled by the quadratic equation p = 3.5t 0.05t, wh the number of days after the stock is purchased. When is the stock worth $60? Let p = 60. Page 15

16 The solutions are about 3.5 and STOCK The price p in dollars for a particular stock can be modeled by the quadratic equation p = 3.5t 0.05t, wh the number of days after the stock is purchased. When is the stock worth $60? Let p = 60. The stock is worth $60 on the 30th and 40th days after purchase. GEOMETRY Find the value of x for each figure. Round to the nearest tenth if necessary. 3. area = 45 in Page 16

17 The stock is worth $60 on the 30th and 40th days after purchase. GEOMETRY Find the value of x for each figure. Round to the nearest tenth if necessary. 3. area = 45 in The height cannot be negative. So, x area = 110 ft Page 17

18 9-4 Solving Quadratic by xcompleting the Square The height cannot beequations negative. So, area = 110 ft The dimensions of the rectangle cannot be negative. So, x NUMBER THEORY The product of two consecutive even integers is 4. Find the integers. Let x = the first integer and x + = the second integer. The integers are 14 and 16 or 16 and CCSS PRECISION The product of two consecutive negative odd integers is 483. Find the integers. Let x = the first integer and x + = the second integer. Page 18

19 The integers are 14 and 16 or 16 and CCSS PRECISION The product of two consecutive negative odd integers is 483. Find the integers. Let x = the first integer and x + = the second integer. The integers must be negative. So, they are 3 and GEOMETRY Find the area of the triangle. Page 19 The dimensions of the triangle cannot be negative. So, x = 18. The base of the triangle is 18 meters and the height is or 4 meters.

20 The integers must be negative. So, they are 3 and GEOMETRY Find the area of the triangle. The dimensions of the triangle cannot be negative. So, x = 18. The base of the triangle is 18 meters and the height is or 4 meters. The area of the triangle is 16 square meters. Solve each equation by completing the square. Round to the nearest tenth if necessary x 0.x 0.4 = 0 Page 0

21 The area of the triangle is 16 square meters. Solve each equation by completing the square. Round to the nearest tenth if necessary x 0.x 0.4 = 0 The solutions are 1 and x = x 0.3 The solutions are about 0. and 3.8. Page 1

22 9-4 Solving Quadratic by Completing the Square The solutions are 1Equations and x = x 0.3 The solutions are about 0. and x The solutions are about 0. and 0.9. Page

23 The solutions are about 0. and x The solutions are about 0. and Page 3

24 The solutions are about 0. and The solutions are 0.5 and The solutions are about 8. and 0.. Page 4

25 The solutions are 0.5 and The solutions are about 8. and Page 5

26 The solutions are about 8. and The solutions are about 5.1 and ASTRONOMY The height of an object t seconds after it is dropped is given by the equation, where h 0 is the initial height and g is the acceleration due to gravity. The acceleration due to gravity near the surface of Mars is 3.73 m/s, while on Earth it is 9.8 m/s. Suppose an object is dropped from an initial height of 10 meters above the surface of each planet. a. On which planet would the object reach the ground first? b. How long would it take the object to reach the ground on each planet? Round each answer to the nearest tenth. c. Do the times that it takes the object to reach the ground seem reasonable? Explain your reasoning. a. The object on Earth will reach the ground first because it is falling at a faster rate. b. Mars: So, tmanual 8.0 -seconds. esolutions Powered by Cognero Earth: Page 6

27 The solutions are about 5.1 and ASTRONOMY The height of an object t seconds after it is dropped is given by the equation, where h 0 is the initial height and g is the acceleration due to gravity. The acceleration due to gravity near the surface of Mars is 3.73 m/s, while on Earth it is 9.8 m/s. Suppose an object is dropped from an initial height of 10 meters above the surface of each planet. a. On which planet would the object reach the ground first? b. How long would it take the object to reach the ground on each planet? Round each answer to the nearest tenth. c. Do the times that it takes the object to reach the ground seem reasonable? Explain your reasoning. a. The object on Earth will reach the ground first because it is falling at a faster rate. b. Mars: So, t 8.0 seconds. Earth: So, t 4.9 seconds. c. Sample answer: Yes; the acceleration due to gravity is much greater on Earth than on Mars, so the time to reach the ground should be much less. 44. Find all values of c that make x + cx a perfect square trinomial. So, c can be 0 or 0 to make the trinomial a perfect square. 45. Find all values of c that make x + cx + 5 a perfect square trinomial. Page 7

28 So, t 4.9 seconds. c. Sample 9-4 Solving Quadratic Equations by Completing the Square answer: Yes; the acceleration due to gravity is much greater on Earth than on Mars, so the time to reach the ground should be much less. 44. Find all values of c that make x + cx a perfect square trinomial. So, c can be 0 or 0 to make the trinomial a perfect square. 45. Find all values of c that make x + cx + 5 a perfect square trinomial. So, c can be 30 or 30 to make the trinomial a perfect square. 46. PAINTING Before she begins painting a picture, Donna stretches her canvas over a wood frame. The frame has a length of 60 inches and a width of 4 inches. She has enough canvas to cover 480 square inches. Donna decides to increase the dimensions of the frame. If the increase in the length is 10 times the increase in the width, what will the dimensions of the frame be? Let x = the increase in the width and let 10x = the increase in the length. So, x + 4 = the new width and 10x + 60 = the new length. Page 8

29 So, c can be 30 or 30 to make the trinomial a perfect square. 46. PAINTING Before she begins painting a picture, Donna stretches her canvas over a wood frame. The frame has a length of 60 inches and a width of 4 inches. She has enough canvas to cover 480 square inches. Donna decides to increase the dimensions of the frame. If the increase in the length is 10 times the increase in the width, what will the dimensions of the frame be? Let x = the increase in the width and let 10x = the increase in the length. So, x + 4 = the new width and 10x + 60 = the new length. The dimensions cannot be negative, so x =. The width is + 4 or 6 inches and the length is 10() + 60 or 80 inches. 47. MULTIPLE REPRESENTATIONS In this problem, you will investigate a property of quadratic equations. a. TABULAR Copy the table shown and complete the second column. Trinomial Number of b 4ac Roots 0 1 x 8x + 16 x 11x + 3 3x + 6x + 9 x x + 7 x + 10x + 5 x + 3x 1 b. ALGEBRAIC Set each trinomial equal to zero, and solve the equation by completing the square. Complete the last column of the table with the number of roots of each equation. c. VERBAL Compare the number of roots of each equation to the result in the b 4ac column. Is there a relationship between these values? If so, describe it. Page 9 d. ANALYTICAL Predict how many solutions x 9x + 15 = 0 will have. Verify your prediction by solving the equation.

30 x + 10x + 5 x + 3x 1 b. ALGEBRAIC Set each trinomial equal to zero, and solve the equation by completing the square. Complete the last column of the table with the number of roots of each equation. c. VERBAL Compare the number of roots of each equation to the result in the b 4ac column. Is there a relationship between these values? If so, describe it. d. ANALYTICAL Predict how many solutions x 9x + 15 = 0 will have. Verify your prediction by solving the equation. a. Trinomial x 8x + 16 x 11x + 3 3x + 6x + 9 x x + 7 x + 10x + 5 x + 3x 1 b 4ac ( 8) 4(1)(16) = 0 Number of Roots 1 ( 11) 4()(3) = 97 (6) 4(3)(9) = 7 ( ) 4(1)(7) = 4 (10) 4(1)(5) = 0 (3) 4(1)( 1) = 57 b. The trinomial x 8x + 16 has 1 root. + 3 has The Manual trinomial x by 11x esolutions - Powered Cognero roots. Page 30

31 The trinomial x 11x + 3 has roots. The square root of a negative number has no real roots. So, the trinomial 3x + 6x + 9 has 0 roots. The square root of a negative number has no real roots. So, the trinomial x x + 7. The trinomial x + 10x + 5 has 1 root. Page 31

32 9-4 Solving Quadratic Equations by Completing the Square The trinomial x + 10x + 5 has 1 root. The trinomial x + 3x 1 has roots. Trinomial 0 Number of Roots b 4ac x 8x + 16 x 11x + 3 3x + 6x + 9 x x + 7 x + 10x + 5 x + 3x 1 c. If b 4ac is negative, the equation has no real solutions. If b 4ac is zero, the equation has one solution. If b 4ac is positive, the equation has solutions. d. The equation x 9x + 15 = 0 has 0 real solutions because b 4ac is negative. Page 3

33 9-4 Solving Quadratic Equations by Completing the Square The equation x 9x + 15 = 0 has 0 real solutions because b 4ac is negative. The equation cannot be solved because the square root of a negative number has no real roots. 48. CCSS PERSEVERANCE Given y = ax + bx + c with a 0, derive the equation for the axis of symmetry by completing the square and rewriting the equation in the form y = a(x h) + k. Let and, then the equation becomes y = (x - h) + k. For an equation in the form y = ax + bx + c, the equation for the axis of symmetry is. So, for an equation in the form y = (x - h) + k, the equation for the axis of symmetry becomes x = h. 49. REASONING Determine the number of solutions x + bx = c has if. Explain. None; Sample answer: If you add to each side of the equation and each side of the inequality, you get Page 33. Since the left side of the last equation is a perfect square, it cannot

34 Let and, then the equation becomes y = (x - h) + k. For an equation in the form y = ax + bx + c, the equation for the axis of symmetry is. So, for an equation in the form y = (x - h) + k, the equation for the axis of symmetry becomes x = h. 49. REASONING Determine the number of solutions x + bx = c has if. Explain. None; Sample answer: If you add to each side of the equation and each side of the inequality, you get. Since the left side of the last equation is a perfect square, it cannot equal the negative number. So, there are no real solutions. 50. WHICH ONE DOESN T BELONG? Identify the expression that does not belong with the other three. Explain your reasoning. The first 3 trinomials are perfect squares. The trinomial is not a perfect square. So, it does not belong. 51. OPEN ENDED Write a quadratic equation for which the only solution is 4. Find a quadratic that has two roots of 4. Verify that the solution is 4. Page 34

35 9-4 Solving Quadratic Equations Completing The trinomial is notby a perfect square.the So,Square it does not belong. 51. OPEN ENDED Write a quadratic equation for which the only solution is 4. Find a quadratic that has two roots of 4. Verify that the solution is 4. So, the quadratic equation x 8x + 16 = 0 has a solution of WRITING IN MATH Compare and contrast the following strategies for solving x 5x 7 = 0: completing the square, graphing, and factoring. Sample answer: Because the leading coefficient is 1, solving the equation by completing the square is simpler and yields exact answers for the solutions. To solve the equation by graphing, use a graphing calculator to graph the related function y = x - 5x - 7. Select the zero option from the nd [CALC] menu to determine the roots. Page 35

36 So, the quadratic equation x 8x + 16 = 0 has a solution of WRITING IN MATH Compare and contrast the following strategies for solving x 5x 7 = 0: completing the square, graphing, and factoring. Sample answer: Because the leading coefficient is 1, solving the equation by completing the square is simpler and yields exact answers for the solutions. To solve the equation by graphing, use a graphing calculator to graph the related function y = x - 5x - 7. Select the zero option from the nd [CALC] menu to determine the roots. The roots are given as decimal approximations. So, for this equation the solutions would have to be given as estimations. There are no factors of 7 that have a sum of 5, so solving by factoring is not possible. 53. The length of a rectangle is 3 times its width. The area of the rectangle is 75 square feet. Find the length of the rectangle in feet. A 5 B 15 esolutions Page 36 C 10Manual - Powered by Cognero D5

37 The roots are given as decimal approximations. So, for this equation the solutions would have to be given as estimations. There are no factors of 7 that have a sum of 5, so solving by factoring is not possible. 53. The length of a rectangle is 3 times its width. The area of the rectangle is 75 square feet. Find the length of the rectangle in feet. A 5 B 15 C 10 D5 Let x = the width of the rectangle and let 3x = the length of the rectangle. The length of the rectangle is 3(5) or 15 feet. Choice B is the correct answer. 54. PROBABILITY At a festival, winners of a game draw a token for a prize. There is one token for each prize. The prizes include 9 movie passes, 8 stuffed animals, 5 hats, 10 jump ropes, and 4 glow necklaces. What is the probability that the first person to draw a token will win a movie pass? F G H J There are or 36 possible outcomes. P(movie pass) = or. Choice J is the correct answer. 55. GRIDDED RESPONSE The population of a town can be modeled by P =, t, where P represents the population and t represents the number of years from 000. How many years after 000 will the population be 6,000? In 3 years the population of the town will be 6, Percy delivers pizzas for Pizza King. He is paid $6 an hour plus $.50 for each pizza he delivers. Percy earned $80 last week. If he worked a total of 30 hours, how many pizzas did he deliver? A 50 pizzas B 184 pizzas C 40 pizzas Page 37 D 34 pizzas

38 In 3 years the population of the town will be 6, Percy delivers pizzas for Pizza King. He is paid $6 an hour plus $.50 for each pizza he delivers. Percy earned $80 last week. If he worked a total of 30 hours, how many pizzas did he deliver? A 50 pizzas B 184 pizzas C 40 pizzas D 34 pizzas Let p = the number of pizzas Percy delivered. Percy delivered 40 pizzas. Choice C is the correct answer. Describe how the graph of each function is related to the graph of f (x) = x. 57. g(x) = 1 + x The graph of f (x) = x + c represents a translation up or down of the parent graph. Since c = 1, the translation is down. So, the graph is shifted down 1 units from the parent function. 58. h(x) = (x + ) The graph of f (x) = (x c) represents a translation left or right from the parent graph. Since c =, the translation is to the left by units. 59. g(x) = x + 5 The Manual function can bebywritten f (x) esolutions - Powered Cognero = ax + c, where a = and c = 5. Since > 0 and is the graph of y = x vertically stretched and shifted up 5 units > 1, the graph of y = x Page

39 59. g(x) = x + 5 The function can be written f (x) = ax + c, where a = and c = 5. Since > 0 and > 1, the graph of y = x + 5 is the graph of y = x vertically stretched and shifted up 5 units. 60. The function can be written f (x) = a(x b), where a = and b = 6. Since > 0 and < 1, the graph of is the graph of y = x vertically compressed and shifted right 6 units. 61. g(x) = 6 + x The function can be written f (x) = ax + c, where a = and c = 6. Since > 0 and > 1, the graph of y = 6 + x is the graph of y = x vertically stretched and shifted up 6 units. = 1 - Powered x by Cognero 6. h(x)manual esolutions Page 39

40 6. h(x) = 1 x The function can be written f (x) = ax + c, where a = = 1 and c = 1. Since < 0 and > 1, the graph of y x is the graph of y = x vertically stretched, shifted down 1 unit and reflected across the x-axis. 63. RIDES A popular amusement park ride whisks riders to the top of a 50-foot tower and drops them. A function for the height of a rider is h = 16t + 50, where h is the height and t is the time in seconds. The ride stops the descent of the rider 40 feet above the ground. Write an equation that models the drop of the rider. How long does it take to fall from 50 feet to 40 feet? The ride stops the descent of the rider at 40 feet above ground, so h = 40. Then, the equation 40 = 16t + 50 models the drop of the rider. Time cannot be negative. So, it takes about 3.6 seconds to complete the ride. Simplify. Assume that no denominator is equal to zero Page 40

41 b (m )(b ) Solve each open sentence. 70. y > 7 Case 1 y is Page 41 and Case y is

42 Solve each open sentence. 70. y > 7 Case 1 y is positive. and Case y is negative. The solution set is {y y > 9 or y < 5}. 71. z + 5 < 3 Case 1 z +5 is positive. and Case z +5 is negative. The solution set is {z 8 < z < }. 7. b cannot be less than or equal to 6. The solution set is empty. cannot be negative. So, y 8 Case 1 3 y is positive. and Case 3 y is negative. The solution set is {y y 5.5 or y.5} m < 1 cannot be less than or equal to 1. The solution set is empty. cannot be negative. So, 75. 5c 13 Case 1 5c is positive. and Case 5c is negative. Page 4

43 m < 1 cannot be negative. So, cannot be less than or equal to 1. The solution set is empty c 13 Case 1 5c is positive. and Case 5c is negative. The solution set is {c. c 3}. Evaluate 76. a =, b = 5, c = for each set of values. Round to the nearest tenth if necessary. 77. a = 1, b = 1, c = a = 9, b = 10, c = a = 1, b = 7, c = a =, b = 4, c = 6 Page 43

44 80. a =, b = 4, c = a = 3, b = 1, c = This value is not a real number. Page 44