COMMON CORE Learning Standards HSN-CN.C.7 HSA-REI.B.4b 3.4 Using the Quadratic Formula Essential Question How can ou derive a general formula for solving a quadratic equation? Deriving the Quadratic Formula Work with a partner. Analze and describe what is done in each step in the development of the Quadratic Formula. Step Justification REASONING ABSTRACTLY To be proficient in math, ou need to create a coherent representation of the problem at hand. a + b + c = 0 a + b = c + b a = c a + b a + ( a) b = c a + ( a) b + b a + ( a) b ( + a) b = 4ac 4a + b 4a = b 4ac 4a + b a = ± b 4ac 4a = b a ± b 4ac a The result is the Quadratic Formula. = b ± b 4ac a Using the Quadratic Formula Work with a partner. Use the Quadratic Formula to solve each equation. a. 4 + 3 = 0 b. + = 0 c. + 3 = 0 d. + 4 + 4 = 0 e. 6 + = 0 f. + 4 + 6 = 0 Communicate Your Answer 3. How can ou derive a general formula for solving a quadratic equation? 4. Summarize the following methods ou have learned for solving quadratic equations: graphing, using square roots, factoring, completing the square, and using the Quadratic Formula. Section 3.4 Using the Quadratic Formula 11
3.4 Lesson Core Vocabular Quadratic Formula, p. 1 discriminant, p. 14 What You Will Learn Solve quadratic equations using the Quadratic Formula. Analze the discriminant to determine the number and tpe of solutions. Solve real-life problems. Solving Equations Using the Quadratic Formula Previousl, ou solved quadratic equations b completing the square. In the Eploration, ou developed a formula that gives the solutions of an quadratic equation b completing the square once for the general equation a + b + c = 0. The formula for the solutions is called the Quadratic Formula. Core Concept The Quadratic Formula Let a, b, and c be real numbers such that a 0. The solutions of the quadratic equation a + b + c = 0 are = b ± b 4ac. a COMMON ERROR Remember to write the quadratic equation in standard form before appling the Quadratic Formula. Solving an Equation with Two Real Solutions Solve + 3 = 5 using the Quadratic Formula. + 3 = 5 Write original equation. + 3 5 = 0 Write in standard form. = b ± b 4ac a = 3 ± 3 4(1)( 5) (1) = 3 ± 9 Quadratic Formula Substitute 1 for a, 3 for b, and 5 for c. Simplif. So, the solutions are = 3 + 9 1.19 and = 3 9 4.19. Check Graph = + 3 5. The -intercepts are about 4.19 and about 1.19. 7 5 Zero X=1.19584 Y=0 Monitoring Progress Solve the equation using the Quadratic Formula. Help in English and Spanish at BigIdeasMath.com 1. 6 + 4 = 0. + 4 = 7 3. 5 = + 8 1 Chapter 3 Quadratic Equations and Comple Numbers
ANOTHER WAY You can also use factoring to solve 5 0 + 4 = 0 because the left side factors as (5 ). Check 4 Solving an Equation with One Real Solution Solve 5 8 = 1 4 using the Quadratic Formula. 5 8 = 1 4 5 0 + 4 = 0 = ( 0) ± ( 0) 4(5)(4) (5) = 0 ± 0 50 Write original equation. Write in standard form. a = 5, b = 0, c = 4 Simplif. 0.5 Zero X=.4 Y=0 1 1.5 = 5 Simplif. So, the solution is = 5. You can check this b graphing = 5 0 + 4. The onl -intercept is 5. Solving an Equation with Imaginar Solutions Solve + 4 = 13 using the Quadratic Formula. COMMON ERROR Remember to divide the real part and the imaginar part b when simplifing. + 4 = 13 + 4 13 = 0 = 4 ± 4 4( 1)( 13) ( 1) = 4 ± 36 4 ± 6i = = ± 3i The solutions are = + 3i and = 3i. Write original equation. Write in standard form. a = 1, b = 4, c = 13 Simplif. Write in terms of i. Simplif. Check Graph = + 4 13. There are no -intercepts. So, the original equation has no real solutions. The algebraic check for one of the imaginar solutions is shown. 8 1 ( + 3i ) + 4( + 3i ) =? 13 5 1i + 8 + 1i =? 13 13 = 13 50 Monitoring Progress Solve the equation using the Quadratic Formula. Help in English and Spanish at BigIdeasMath.com 4. + 41 = 8 5. 9 = 30 + 5 6. 5 7 = 3 + 4 Section 3.4 Using the Quadratic Formula 13
Analzing the Discriminant In the Quadratic Formula, the epression b 4ac is called the discriminant of the associated equation a + b + c = 0. = b ± b 4ac a discriminant You can analze the discriminant of a quadratic equation to determine the number and tpe of solutions of the equation. Core Concept Analzing the Discriminant of a + b + c = 0 Value of discriminant b 4ac > 0 b 4ac = 0 b 4ac < 0 Number and tpe of solutions Two real solutions One real solution Two imaginar solutions Graph of = a + b + c Two -intercepts One -intercept No -intercept Analzing the Discriminant Find the discriminant of the quadratic equation and describe the number and tpe of solutions of the equation. a. 6 + = 0 b. 6 + 9 = 0 c. 6 + 8 = 0 Equation Discriminant Solution(s) a + b + c = 0 b 4ac = b ± b 4ac a a. 6 + = 0 ( 6) 4(1)() = 4 Two imaginar: 3 ± i b. 6 + 9 = 0 ( 6) 4(1)(9) = 0 One real: 3 c. 6 + 8 = 0 ( 6) 4(1)(8) = 4 Two real:, 4 Monitoring Progress Help in English and Spanish at BigIdeasMath.com Find the discriminant of the quadratic equation and describe the number and tpe of solutions of the equation. 7. 4 + 8 + 4 = 0 8. 1 + 1 = 0 9. 5 = 8 13. 7 3 = 6 11. 4 + 6 = 9 1. 5 + 1 = 6 14 Chapter 3 Quadratic Equations and Comple Numbers
Writing an Equation Find a possible pair of integer values for a and c so that the equation a 4 + c = 0 has one real solution. Then write the equation. In order for the equation to have one real solution, the discriminant must equal 0. b 4ac = 0 Write the discriminant. ANOTHER WAY Another possible equation in Eample 5 is 4 4 1 = 0. You can obtain this equation b letting a = 4 and c = 1. ( 4) 4ac = 0 Substitute 4 for b. 16 4ac = 0 Evaluate the power. 4ac = 16 Subtract 16 from each side. ac = 4 Divide each side b 4. Because ac = 4, choose two integers whose product is 4, such as a = 1 and c = 4. So, one possible equation is 4 + 4 = 0. Check Graph = 4 + 4. The onl -intercept is. You can also check b factoring. 8 4 + 4 = 0 ( ) = 0 = 3 Zero X= Y=0 7 Monitoring Progress Help in English and Spanish at BigIdeasMath.com 13. Find a possible pair of integer values for a and c so that the equation a + 3 + c = 0 has two real solutions. Then write the equation. The table shows five methods for solving quadratic equations. For a given equation, it ma be more efficient to use one method instead of another. Suggestions about when to use each method are shown below. Concept Summar Methods for Solving Quadratic Equations Method Graphing Using square roots Factoring Completing the square Quadratic Formula When to Use Use when approimate solutions are adequate. Use when solving an equation that can be written in the form u = d, where u is an algebraic epression. Use when a quadratic equation can be factored easil. Can be used for an quadratic equation a + b + c = 0 but is simplest to appl when a = 1 and b is an even number. Can be used for an quadratic equation. Section 3.4 Using the Quadratic Formula 15
Solving Real-Life Problems The function h = 16t + h 0 is used to model the height of a dropped object. For an object that is launched or thrown, an etra term v 0 t must be added to the model to account for the object s initial vertical velocit v 0 (in feet per second). Recall that h is the height (in feet), t is the time in motion (in seconds), and h 0 is the initial height (in feet). h = 16t + h 0 h = 16t + v 0 t + h 0 Object is dropped. Object is launched or thrown. As shown below, the value of v 0 can be positive, negative, or zero depending on whether the object is launched upward, downward, or parallel to the ground. V 0 > 0 V 0 < 0 V 0 = 0 Modeling a Launched Object A juggler tosses a ball into the air. The ball leaves the juggler s hand 4 feet above the ground and has an initial vertical velocit of 30 feet per second. The juggler catches the ball when it falls back to a height of 3 feet. How long is the ball in the air? Because the ball is thrown, use the model h = 16t + v 0 t + h 0. To find how long the ball is in the air, solve for t when h = 3. h = 16t + v 0 t + h 0 Write the height model. 3 = 16t + 30t + 4 Substitute 3 for h, 30 for v 0, and 4 for h 0. 0 = 16t + 30t + 1 Write in standard form. This equation is not factorable, and completing the square would result in fractions. So, use the Quadratic Formula to solve the equation. t = 30 ± 30 4( 16)(1) a = 16, b = 30, c = 1 ( 16) t = 30 ± 964 3 t 0.033 or t 1.9 Simplif. Use a calculator. Reject the negative solution, 0.033, because the ball s time in the air cannot be negative. So, the ball is in the air for about 1.9 seconds. Monitoring Progress Help in English and Spanish at BigIdeasMath.com 14. WHAT IF? The ball leaves the juggler s hand with an initial vertical velocit of 40 feet per second. How long is the ball in the air? 16 Chapter 3 Quadratic Equations and Comple Numbers
3.4 Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check 1. COMPLETE THE SENTENCE When a, b, and c are real numbers such that a 0, the solutions of the quadratic equation a + b + c = 0 are =.. COMPLETE THE SENTENCE You can use the of a quadratic equation to determine the number and tpe of solutions of the equation. 3. WRITING Describe the number and tpe of solutions when the value of the discriminant is negative. 4. WRITING Which two methods can ou use to solve an quadratic equation? Eplain when ou might prefer to use one method over the other. Monitoring Progress and Modeling with Mathematics In Eercises 5 18, solve the equation using the Quadratic Formula. Use a graphing calculator to check our solution(s). (See Eamples 1,, and 3.) 5. 4 + 3 = 0 6. 3 + 6 + 3 = 0 7. + 6 + 15 = 0 8. 6 + 1 = 0 9. 14 = 49. + 4 = 30 11. 3 + 5 = 1. 3 = 4 13. = 5 14. 5 6 = 4 15. 4 + 3 = 5 16. + 11 = 17. z = 1z + 6 18. 7w + 6 = 4w In Eercises 19 6, find the discriminant of the quadratic equation and describe the number and tpe of solutions of the equation. (See Eample 4.) 19. + 1 + 36 = 0 0. + 6 = 0 1. 4n 4n 4 = 0. + + 1 = 0 3. 4 = 5 4. 18p = p + 81 8. USING EQUATIONS Determine the number and tpe of solutions to the equation + 7 = 11. A B C D two real solutions one real solution two imaginar solutions one imaginar solution ANALYZING EQUATIONS In Eercises 9 3, use the discriminant to match each quadratic equation with the correct graph of the related function. Eplain our reasoning. 9. 6 + 5 = 0 30. 0 + 50 = 0 31. 3 + 6 9 = 0 3. 5 35 = 0 A. 8 4 4 B. 4 0 40 8 5. 4 = 48 3 6. 6 = C. D. 35 7. USING EQUATIONS What are the comple solutions of the equation 16 + 50 = 0? A 4 + 3i, 4 3i B 4 + 1i, 4 1i C 16 + 3i, 16 3i D 16 + 1i, 16 1i 4 0 4 8 5 15 5 6 Section 3.4 Using the Quadratic Formula 17
ERROR ANALYSIS In Eercises 33 and 34, describe and correct the error in solving the equation. 33. + + 74 = 0 = ± 4(1)(74) (1) = ± 196 ± 14 = = 1 or COMPARING METHODS In Eercises 47 58, solve the quadratic equation using the Quadratic Formula. Then solve the equation using another method. Which method do ou prefer? Eplain. 47. 3 1 = 3 48. 5 + 38 = 3 49. 54 = 1 50. = 3 + 15 51. 7 + 1 = 0 5. + 8 13 = 0 53. 5 50 = 135 54. 8 + 4 + 5 = 0 55. 3 = 4 + 9 56. 31 + 56 = 34. + 6 + 8 = = 6 ± 6 4(1)(8) (1) = 6 ± 4 = 6 ± = or 4 57. = 1 58. 9 + 36 + 7 = 0 MATHEMATICAL CONNECTIONS In Eercises 59 and 60, find the value for. 59. Area of the rectangle = 4 m ( + ) m ( 9) m OPEN-ENDED In Eercises 35 40, find a possible pair of integer values for a and c so that the quadratic equation has the given solution(s). Then write the equation. (See Eample 5.) 35. a + 4 + c = 0; two imaginar solutions 36. a + 6 + c = 0; two real solutions 37. a 8 + c = 0; two real solutions 38. a 6 + c = 0; one real solution 39. a + = c; one real solution 40. 4 + c = a ; two imaginar solutions 60. Area of the triangle = 8 ft (3 7) ft ( + 1) ft 61. MODELING WITH MATHEMATICS A lacrosse plaer throws a ball in the air from an initial height of 7 feet. The ball has an initial vertical velocit of 90 feet per second. Another plaer catches the ball when it is 3 feet above the ground. How long is the ball in the air? (See Eample 6.) USING STRUCTURE In Eercises 41 46, use the Quadratic Formula to write a quadratic equation that has the given solutions. 41. = 8 ± 176 43. = 4 ± 14 14 45. = 4 ± 6 4. = 15 ± 15 44. = 9 ± 137 4 46. = ± 4 6. NUMBER SENSE Suppose the quadratic equation a + 5 + c = 0 has one real solution. Is it possible for a and c to be integers? rational numbers? Eplain our reasoning. Then describe the possible values of a and c. 18 Chapter 3 Quadratic Equations and Comple Numbers
63. MODELING WITH MATHEMATICS In a volleball game, a plaer on one team spikes the ball over the net when the ball is feet above the court. The spike drives the ball downward with an initial velocit of 55 feet per second. How much time does the opposing team have to return the ball before it touches the court? 67. MODELING WITH MATHEMATICS A gannet is a bird that feeds on fish b diving into the water. A gannet spots a fish on the surface of the water and dives 0 feet to catch it. The bird plunges toward the water with an initial velocit of 88 feet per second. 64. MODELING WITH MATHEMATICS An archer is shooting at targets. The height of the arrow is 5 feet above the ground. Due to safet rules, the archer must aim the arrow parallel to the ground. 5 ft 3 ft a. How long does it take for the arrow to hit a target that is 3 feet above the ground? b. What method did ou use to solve the quadratic equation? Eplain. 65. PROBLEM SOLVING A rocketr club is launching model rockets. The launching pad is 30 feet above the ground. Your model rocket has an initial velocit of 5 feet per second. Your friend s model rocket has an initial velocit of 0 feet per second. a. Use a graphing calculator to graph the equations of both model rockets. Compare the paths. b. After how man seconds is our rocket 119 feet above the ground? Eplain the reasonableness of our answer(s). 66. PROBLEM SOLVING The number A of tablet computers sold (in millions) can be modeled b the function A = 4.5t + 43.5t + 17, where t represents the ear after 0. a. How much time does the fish have to swim awa? b. Another gannet spots the same fish, and it is onl 84 feet above the water and has an initial velocit of 70 feet per second. Which bird will reach the fish first? Justif our answer. 68. USING TOOLS You are asked to find a possible pair of integer values for a and c so that the equation a 3 + c = 0 has two real solutions. When ou solve the inequalit for the discriminant, ou obtain ac <.5. So, ou choose the values a = and c = 1. Your graphing calculator displas the graph of our equation in a standard viewing window. Is our solution correct? Eplain. 69. PROBLEM SOLVING Your famil has a rectangular pool that measures 18 feet b 9 feet. Your famil wants to put a deck around the pool but is not sure how wide to make the deck. Determine how wide the deck should be when the total area of the pool and deck is 400 square feet. What is the width of the deck? a. In what ear did the tablet computer sales reach 65 million? b. Find the average rate of change from 0 to 01 and interpret the meaning in the contet of the situation. 18 ft 9 ft c. Do ou think this model will be accurate after a new, innovative computer is developed? Eplain. Section 3.4 Using the Quadratic Formula 19
70. HOW DO YOU SEE IT? The graph of a quadratic function = a + b + c is shown. Determine whether each discriminant of a + b + c = 0 is positive, negative, or zero. Then state the number and tpe of solutions for each graph. Eplain our reasoning. a. b. 74. THOUGHT PROVOKING Describe a real-life stor that could be modeled b h = 16t + v 0 t + h 0. Write the height model for our stor and determine how long our object is in the air. 75. REASONING Show there is no quadratic equation a + b + c = 0 such that a, b, and c are real numbers and 3i and i are solutions. 76. MODELING WITH MATHEMATICS The Stratosphere Tower in Las Vegas is 91 feet tall and has a needle at its top that etends even higher into the air. A thrill ride called Big Shot catapults riders 160 feet up the needle and then lets them fall back to the launching pad. c. 71. CRITICAL THINKING Solve each absolute value equation. a. 3 14 = 4 b. = + 6 7. MAKING AN ARGUMENT The class is asked to solve the equation 4 + 14 + 11 = 0. You decide to solve the equation b completing the square. Your friend decides to use the Quadratic Formula. Whose method is more efficient? Eplain our reasoning. 73. ABSTRACT REASONING For a quadratic equation a + b + c = 0 with two real solutions, show b that the mean of the solutions is. How is this a fact related to the smmetr of the graph of = a + b + c? Maintaining Mathematical Proficienc Solve the sstem of linear equations b graphing. (Skills Review Handbook) 77. + = 6 78. = 1 + 4 = 4 = + 1 79. 3 + = 4 80. = + 6 + = 4 5 + 5 = Graph the quadratic equation. Label the verte and ais of smmetr. (Section.) 81. = + + 1 8. = + 3 83. = 0.5 + + 5 84. = 3 a. The height h (in feet) of a rider on the Big Shot can be modeled b h = 16t + v 0 t + 91, where t is the elapsed time (in seconds) after launch and v 0 is the initial velocit (in feet per second). Find v 0 using the fact that the maimum value of h is 91 + 160 = 81 feet. b. A brochure for the Big Shot states that the ride up the needle takes seconds. Compare this time to the time given b the model h = 16t + v 0 t + 91, where v 0 is the value ou found in part (a). Discuss the accurac of the model. Reviewing what ou learned in previous grades and lessons 130 Chapter 3 Quadratic Equations and Comple Numbers