COMMON CORE y - 0 y - - 0 - Locker LESSON 3.3 Finding Comple Solutions of Quadratic Equations Name Class Date 3.3 Finding Comple Solutions of Quadratic Equations Essential Question: How can you find the comple solutions of any quadratic equation? Resource Locker Common Core Math Standards The student is epected to: COMMON CORE N-CN.C. Solve quadratic equations with real coefficients that have comple solutions. Also N-CN.C., A-REI.B.b Mathematical Practices COMMON CORE MP. Reasoning A Eplore Investigating Real Solutions of Quadratic Equations Complete the table. a + b + c 0 a + b -c f () a + b g () -c + + 0 + + 0 + + 3 0 + - + - + -3 f () + f () + f () + g () - g () - g () -3 Language Objective Work with a partner or small group to determine whether solutions to quadratic equations are real or not real and justify reasoning. ENGAGE Essential Question: How can you find the comple solutions of any quadratic equation? Possible answer: You can factor, if possible, to find real solutions; approimate from a graph; find a square root (which may be part of completing the square); complete the square; or apply the quadratic formula. For the general equation a + b + c 0, you must either complete the square or use the quadratic formula to find the comple solutions of the equation. PREVIEW: LESSON PERFORMANCE TASK View the Engage section online. Discuss the photo and how to solve a quadratic equation to determine how high a baseball will go after it is hit. Then preview the Lesson Performance Task. B The graph of ƒ () + is shown. Graph each g (). Complete the table. - - 0 y C Repeat Steps A and B when ƒ () - +. - Equation + + 0 + + 0 + + 3 0 Number of Real Solutions a + b + c 0 a + b -c f () a + b g () -c - + - 0 - + - 0 - + - 3 0-0 - y - + - + - + 3 Equation - + - 0 - + - 0 - + - 3 0 f () - + g () f () - + f () - + 0 g () g () 3 Number of Real Solutions Module 3 39 Lesson 3 DO NOT EDIT--Changes must be made through File info CorrectionKeyNL-A;CA-A Name Class Date 3.3 Finding Comple Solutions of Quadratic Equations Essential Question: How can you find the comple solutions of any quadratic equation? N-CN.C. Solve quadratic equations with real coefficients that have comple solutions. Also N-CN.C., A-REI.B.b Resource Eplore Investigating Real Solutions of Quadratic Equations Complete the table. g () -c a + b -c f () f () + g () - c 0 - a + b + + - g () f () + -3 + - g () f () + + -3 + + 0 + + 0 + + 3 0 The graph of ƒ () + is shown. Graph each g (). Complete the table. - Repeat Steps A and B when ƒ () - +. Equation + + 0 + + 0 + + 3 0 Number of Real Solutions g () -c 0 a + b -c f () f () - + g () c a + b + + g () - f () - + 3 + g () - f () - + - + 3 Number of Real Solutions Equation - + - 0 - + - 0 - + - 3 0 - + - 0 - + - 0 - + - 3 0 Module 3 39 Lesson 3 AMNLESE38589UM03L3.indd 39 9/03/ :05 PM 0 HARDCOVER PAGES 99 08 Turn to these pages to find this lesson in the hardcover student edition. 0 39 Lesson 3.3
Reflect. Look back at Steps A and B. Notice that the minimum value of f() in Steps A and B is -. Complete the table by identifying how many real solutions the equation ƒ () g () has for the given values of g(). Value of g () g () - g () > - Number of Real Solutions of f () g () EXPLORE Investigating Real Solutions of Quadratic Equations. Look back at Step C. Notice that the maimum value of ƒ () in Step C is. Complete the table by identifying how many real solutions the equation ƒ () g () has for the given values of g (). 3. You can generalize Reflect : For ƒ () a + b where a > 0, ƒ () g () where g () -c has real solutions when g () is greater than or equal to the minimum value of ƒ (). The minimum value of ƒ () is + b a) ( - b a) ( a b - b a b a - b a b a - b a - b a. ƒ ( - b a ) a ( - b a ) So, ƒ () g () has real solutions when g () - b a. Substitute -c for g (). -c - b a Add b to both sides. b a a - c 0 Multiply both sides by a, which is positive. b - ac 0 g () < - Value of g () g () g () > g () < In other words, the equation a + b + c 0 where a > 0 has real solutions when b - ac 0. Generalize the results of Reflect in a similar way. What do you notice? 0 Number of Real Solutions of f () g () For f () a + b where a < 0, f () g () where g () -c has real solutions when g () is less than or equal to the maimum value of f (). The maimum value of f () is f b (- - b a) a. b So, f () g () has real solutions when g () - a. b Substitute -c for g (). -c - a Add b a to both sides. b a - c 0 Multiply both sides by a, which is negative. b - ac 0 Whether a > 0 or a < 0, b - ac 0 tells when a + b + c 0 has real solutions. 0 INTEGRATE TECHNOLOGY Students can use a graphing calculator to graph f () and each function g () to verify the number of real solutions to each equation. QUESTIONING STRATEGIES If an equation is written in verte form, what information can you use to find out if it has real solutions? The sign of a determines the direction of the opening and the maimum or minimum value tells you whether there are real solutions. How do you determine where the graph of a quadratic function crosses the -ais? You can find the -intercepts of the graph of a quadratic function in standard form by factoring the function to get its intercept form. If the function is not factorable, the -intercepts can be found by using the quadratic formula to find the zeros of the function. Module 3 0 Lesson 3 PROFESSIONAL DEVELOPMENT Math Background In Algebra, students used the quadratic formula to find real solutions to a quadratic equation. Students now revisit the formula to etend its use to comple solutions. The sign of the epression b - ac determines whether the quadratic equation has two real solutions, one real solution, or two nonreal solutions. For cubic equations of the form a 3 + b + c + d 0, the sign of the discriminant b c - ac 3 - b 3 d - a d determines whether the equation has three real solutions, two real solutions, or one real solution. Finding Comple Solutions of Quadratic Equations 0
EXPLAIN Finding Comple Solutions by Completing the Square QUESTIONING STRATEGIES How do you convert quadratic functions to verte form? Eplain. You can convert quadratic functions from standard form to verte form f () a ( - h) + k by completing the square on a + b. You have to add and subtract the same constant to keep the function value the same. INTEGRATE MATHEMATICAL PRACTICES Focus on Technology MP.5 Discuss with students how to use the graphing calculator to find a maimum or minimum value of a quadratic function. Students can solve problems algebraically and then use their graphing calculators to check their solutions. Eplain Finding Comple Solutions by Completing the Square Recall that completing the square for the epression + b requires adding b ( to it, resulting in the perfect square ) ) trinomial + b + ( b, which you can factor as ( + b. Don t forget that when + b appears on one side of an equation, adding b ( ) Eample 3 + 9-6 0 ) to it requires adding b ( to the other side as well. Solve the equation by completing the square. State whether the solutions are real or non-real.. Write the equation in the form + b c. 3 + 9-6 0 3 + 9 6 + 3. Identify b and b ( ). b 3 b ( 3 ) ( 9 ) 3. Add b ( to both sides of the equation. ) + 3 + - + 0 9 + 9. Write the equation in the form + b c. - -. Identify b and b ( ). ( b ) b - ( ) - 3. Add b ( ) to both sides. - + - + ). Solve for. ( + 3 + ) 9 ( + 3 ) + 3 ± + 3 ± - 3 ± -3 ± There are two real solutions: -3 + -3 - and.. Solve for. + - + ( - ) -6 - ± -6 ± -6 There are two real/non-real solutions: and - i 6. + i 6 Module 3 Lesson 3 COLLABORATIVE LEARNING Peer-to-Peer Activity Have students work in pairs. Provide each pair with several quadratic equations written in various forms. Have one student verbally instruct the partner in how to find the nonreal solutions to the equation. Then have partners switch roles, repeating the activity for a different quadratic equation. Have students discuss how their steps for solving the equation were similar or different. Lesson 3.3
Reflect. How many comple solutions do the equations in Parts A and B have? Eplain. Each equation has two comple solutions, because the set of comple numbers includes all real numbers as well as all non-real numbers. Your Turn Solve the equation by completing the square. State whether the solutions are real or non-real. 5. + 8 + 0 6. + 0-0 + 8 - + 0 + 8 +6 - + 6 ( + ) - + ± - Eplain - ± i There are two non-real solutions: - + i and - - i. Identifying Whether Solutions Are Real or Non-real By completing the square for the general quadratic equation a + b + c 0, you can obtain the quadratic b formula, - ac, which gives the solutions of the general quadratic equation. In the quadratic formula, the a epression under the radical sign, b - ac, is called the discriminant, and its value determines whether the solutions of the quadratic equation are real or non-real. -b ± + 0 + 5 + 5 ( + 5) 3 + 5 ± 3-5 ± There are two real solutions: -5 + and -5 -. EXPLAIN Identifying Whether Solutions are Real or Non-real QUESTIONING STRATEGIES Does the discriminant give the solution of a quadratic equation? Eplain. No, it gives the number of solutions and type of solution, but it does not give the actual solution. AVOID COMMON ERRORS Remind students that they must write the quadratic equation in standard form before applying the quadratic formula. Value of Discriminant b - ac > 0 b - ac 0 b - ac < 0 Eample Number and Type of Solutions Two real solutions One real solution Two non-real solutions Answer the question by writing an equation and determining whether the solutions of the equation are real or non-real. A ball is thrown in the air with an initial vertical velocity of m/s from an initial height of m. The ball s height h (in meters) at time t (in seconds) can be modeled by the quadratic function h (t) -.9 t + t +. Does the ball reach a height of m? Set h (t) equal to. -.9 t + t + Subtract from both sides. -.9 t + t + 0 0 Image Credits: Fred Fokkelman/Shutterstock CONNECT VOCABULARY Review vocabulary related to quadratic functions, such as discriminant and real numbers, by having students label the parts of a quadratic function written in various forms. Find the value of the discriminant. - (-.9) (-0) 96-96 0 Because the discriminant is zero, the equation has one real solution, so the ball does reach a height of m. Module 3 Lesson 3 DIFFERENTIATE INSTRUCTION Cognitive Strategies Some students have trouble completing the square because there are so many steps. Show them how to break the process into three parts: () Get the equation into the form needed for completing the square. () Complete the square. (3) Finish the solution by taking square roots of both sides and simplifying the results. When students make errors, analyze their work carefully to see what part of the process is giving them trouble, and give them etra practice on that part of the process. Finding Comple Solutions of Quadratic Equations
INTEGRATE MATHEMATICAL PRACTICES Focus on Reasoning MP. The discriminant can be used to distinguish between rational and irrational solutions. Give students several quadratic equations for which b - ac is positive, some with rational solutions, and some with irrational solutions. Ask them to make a conjecture about how the value of the discriminant is related to whether the solutions are rational or irrational. Students should be able to eplain why the solutions will be rational when the value of the discriminant is a perfect square. EXPLAIN 3 Finding Comple Solutions Using the Quadratic Formula QUESTIONING STRATEGIES Why are there always two solutions to a quadratic equation that has nonreal solutions? How are they related? Since b - ac is not zero, its value will be both added to and subtracted from -b in the numerator, resulting in two solutions; they are comple conjugates. What is the general solution of a quadratic equation with only one solution? - b a Image Credits: David Burton/Alamy A person wants to create a vegetable garden and keep the rabbits out by enclosing it with 00 feet of fencing. The area of the garden is given by the function A (w) w (50 - w) where w is the width (in feet) of the garden. Can the garden have an area of 00 ft? Set A (w) equal to 00. w (50 - w) Multiply on the left side. 50w - w Subtract 00 from both sides. - w + 50w - 00 0 Find the value of the discriminant. 50 - (-)(-00) 500-800 -300 Because the discriminant is [positive/zero/negative], the equation has [two real/one real/two non-real] solutions, so the garden [can/cannot] have an area of 00 ft. Your Turn Answer the question by writing an equation and determining if the solutions are real or non-real.. A hobbyist is making a toy sailboat. For the triangular sail, she wants the height h (in inches) to be twice the length of the base b (in inches). Can the area of the sail be 0 i n? Write the area A of the sail as a function of b. A b (b) b Substitute 0 for A. 0 b Eplain 3 Finding Comple Solutions Using the Quadratic Formula When using the quadratic formula to solve a quadratic equation, be sure the equation is in the form a + b + c 0. Eample 3 Solve the equation using the quadratic formula. Check a solution by substitution. -5 - - 8 0 -b ± b Write the quadratic formula. - ac a Substitute values. 00 00 Subtract 0 from both sides. 0 b - 0 Find the discriminant. 0 - () (-0) 0 + 0 0 Because the discriminant is positive, the equation has two real solutions, so the area of the sail can be 0 i n. -(-) ± (-) - (-5)(-8) (-5) ± -56 ± i 39 Simplify. -0-5 Module 3 3 Lesson 3 LANGUAGE SUPPORT Communicate Math Students play How do you know? Give students several cards containing quadratic equations; some have real number solutions, others nonreal or comple solutions. In small groups, students draw a card and state whether the solution is real or not real. They then answer the question How do you know? Players take turns and sort cards into piles according to the kind of solution. By the end of the game, all players in a group must agree on card placement. 3 Lesson 3.3
So, the two solutions are - 5 - i 39 and - 5 5 + i 39 5. Check by substituting one of the values. Substitute. -5 (- 5 - i 39 5 ) - (- 5 - i 39 5 ) - 8 0 Square. -5 ( 5 + i 39-39 - 5 5) (- 5 - i 39 5 ) - 8 0 Distribute. - 5 - i 39 + 39 5 5 + 5 + i 39-8 0 5 Simplify. B + + 3-0 5-8 0 0 0 Write the equation with 0 on one side. + + 0 Write the quadratic formula. b - ac a -b ± ( ) Substitute values. - ± - ( )( ) ( ) Simplify. - ± - 08 - ± i - ± i 6 3 3 3 - - So, the two solutions are + 3i 3 and - 3i 3. Check by substituting one of the values. Substitute. (- + 3i 3 ) + (- + 3i 3 ) + 0 Square. ( 9-6i 3 9-9 ) + (- + 3i 3 ) + 0 Distribute. Simplify. - 6i 3 - - + 6i 3 + 0-8 + 0 0 0 AVOID COMMON ERRORS Students may have difficulty remembering the quadratic formula. Encourage students to copy the formula and have it on hand when they are working. Caution them to write the equation in standard form before identifying the values of a, b, and c to be used in the formula. INTEGRATE MATHEMATICAL PRACTICES Focus on Communication MP.3 You may wish to point out that quadratic equations always have two roots. However, when the value of the discriminant is 0, the two roots happen to be the same. In this case, the quadratic is said to have a double root. Module 3 Lesson 3 Finding Comple Solutions of Quadratic Equations
ELABORATE INTEGRATE MATHEMATICAL PRACTICES Focus on Critical Thinking MP.3 Emphasize that choosing which method to use to solve a quadratic equation is as important as being able to use each method. Have students discuss when each method might be preferred. AVOID COMMON ERRORS Students may sometimes make a mistake in sign when calculating the discriminant, particularly when the quantity ac is less than 0. Remind them that subtracting a negative number is the same as adding the opposite, or positive, number. If a and c are opposite signs, the discriminant will always be positive. SUMMARIZE THE LESSON When does a quadratic equation have nonreal solutions, and how do you find them? When the value of the discriminant is negative, the quadratic equation will have two nonreal solutions. You find the solutions by using the quadratic formula to solve the equation, and then writing the solutions as a pair of comple conjugates of the form a±bi. Your Turn Solve the equation using the quadratic formula. Check a solution by substitution. 8. 6-5 - 0 9. + 8 + -b ± b - ac a -(-5) ± (-5) - (6)(-) (6) 5 ± 5 ± 3 So, the solutions are 5 + and 5 - Elaborate -. Check 6 ( 3 ) - 5 ( 3 ) - 0 3 3-0 3-0 0 0 + 6 + 0 0. Discussion Suppose that the quadratic equation a + b + c 0 has p + qi where q 0 as one of its solutions. What must the other solution be? How do you know? The other solution must be p qi. The radical b - ac in the quadratic formula produces imaginary numbers when b - ac < 0. Since b - ac is both added to and. Discussion You know that the graph of the quadratic function ƒ () a + b + c has the vertical line - b as its ais of symmetry. If the graph of ƒ () crosses the -ais, where do the -intercepts occur a relative to the ais of symmetry? Eplain. -b The -intercepts are the solutions of f () 0, which are ± b - ac by the a quadratic formula. Writing the -intercepts as - b a ± b - ac shows that a the -intercepts are the same distance, b - ac, away from the ais of symmetry, a with one -intercept on each side of the line: - b a - b - ac on one side and a - b a + b - ac on the other side. a. Essential Question Check-In Why is using the quadratic formula to solve a quadratic equation easier than completing the square? The quadratic formula is the result of completing the square on the general quadratic equation a + b + c 0. As long as any particular equation is in the form -b ± b - ac a -(6) ± (6) - ()() () -6 ± - -6 ± i 3-3 ± i 3 So, the solutions are -3 + i 3 and -3 - i 3. Check (-3 + i 3 ) + 8 (-3 + i 3 ) + (-3 + i 3 ) 6-6i 3 - + 8i 3 + -6 + i 3-6 + i 3-6 + i 3 subtracted from b in the numerator of the quadratic formula, one solution will have the form p + qi, and the other will have the form p qi. a + b + c 0, you can simply substitute the values of a, b, and c into the quadratic formula and obtain the solutions of the equation. Module 3 5 Lesson 3 5 Lesson 3.3
Evaluate: Homework and Practice. The graph of ƒ () + 6 is shown. Use the graph to determine how many real solutions the following equations have: + 6 + 6 0, + 6 + 9 0, and + 6 + 0. Eplain. For each equation, subtract the constant from both sides to obtain these equations: + 6-6, + 6-9, and + 6 -. The graph of g () -6 intersects the graph of f () twice, so the equation + 6 + 6 0 has two real solutions. The graph of g () -9 intersects the graph of f () once, so the equation + 6 + 9 0 has one real solution. The graph of g () - doesn t intersect the graph of f (), so the equation + 6 + 0 has no real solutions. - - 0 -. The graph of ƒ () - + 3 is shown. Use the graph to determine how many real solutions the following equations have: - + 3-3 0, - + 3-9 0, and - + 3-6 0. Eplain. For each equation, subtract the constant from both sides to obtain these equations: - + 3 3, - + 3 9, and - + 3 6. The graph of g () 3 intersects the graph of f () twice, so the equation - + 3-3 0 has two real solutions. The graph of g () intersects the graph of f () once, so the equation - + 3-9 0 has one real solution. The graph of g () 6 doesn t intersect the graph of f (), so the equation - + 3-6 0 has no real solutions. Solve the equation by completing the square. State whether the solutions are real or non-real. + ± 3 - ± 3 two real solutions: - + 3 and - - 3. 9-8 - 3. + + 0. + + 8 0 + - + -8 + + - + + + -8 + ( + ) 3 ( + ) - + ± - + ± i - ± i two non-real solutions: - + i and - - i. y 6 y 5 3 0 3 Online Homework Hints and Help Etra Practice 5 6 EVALUATE ASSIGNMENT GUIDE Concepts and Skills Eplore Investigating Real Solutions of Quadratic Equations Eample Finding Comple Solutions by Completing the Square Eample Identifying Whether Solutions are Real or Non-real Eample 3 Finding Comple Solutions Using the Quadratic Formula CONNECT VOCABULARY Practice Eercises Eercises 3 8 Eercises 9 6 Eercises 0 What information does the value of the discriminant give about a quadratic equation? The value of the discriminant indicates the number and types of roots. Module 3 6 Lesson 3 Eercise Depth of Knowledge (D.O.K.) COMMON CORE Mathematical Practices 8 Recall of Information MP. Reasoning 9 6 Recall of Information MP.3 Logic 0 Skills/Concepts MP. Reasoning Recall of Information MP.3 Logic 3 Strategic Thinking MP.3 Logic 3 3 Strategic Thinking MP. Modeling Finding Comple Solutions of Quadratic Equations 6
VISUAL CUES If students have difficulty evaluating the discriminant, have them organize the variables in a table. Ask students to create a table for each of the variables (a, b, c, b, ac, b - ac) and have them predict the number and type of solutions based on the variables they list in the table. 5. - 5-0 6. 5-6 8-5 + 5-0 + 5 -..6 ( - 5 ) 55 - - 5 ± -. + 0.36.6 + 0.36 55 - ( - 0.6).96-5 ± i - 0.6 ±.96 55 5-0.6 ±. ± i 55 0.6 ±. two non-real solutions: 5 + i 55 and 5 two real solutions: - i 55. and -0.8.. + 3 5 8. - - 6-0 + 3 5 + 6 + 0 + 3 + 69 96 5 + 69 + 6 + 9 - + 9 96 ( + 3 ) ( + 3) - 309 96 + 3 ± - + 3 + 3 two real solutions: -3 + ± 309 96 ± 309 3 - ± 309-3 - + 3 ± i -3 ± i two non-real solutions: -3 + i and -3 - i. 309 309 and. Without solving the equation, state the number of solutions and whether they are real or non-real. 9. -6 + + 3 0 0. - + 0 0 Find the discriminant. Find the discriminant. - (-6)(3) 6 + 83 88 (-) - ()(0) - 80-59 Because the discriminant is positive, the equation has two real solutions.. - - 5. + 9 - - 5-0 - + 9 0 Find the discriminant. (- 5) - (-) (-) 5 - - 96 5 Because the discriminant is negative, the equation has two non-real solutions. Because the discriminant is negative, the equation has two non-real solutions. Find the discriminant. (-) - () (9) - 0 Because the discriminant is zero, the equation has one real solution. Module 3 Lesson 3 Lesson 3.3
Answer the question by writing an equation and determining whether the solutions of the equation are real or non-real. 3. A gardener has 0 feet of fencing to put around a rectangular vegetable garden. The function A (w) 0w - w gives the garden s area A (in square feet) for any width w (in feet). Does the gardener have enough fencing for the area of the garden to be 300 ft? Write an equation by setting A (w) equal to 300. Then rewrite the equation with 0 on one side. 0w - w 300 -w + 0w 300 0 Find the discriminant. 0 ( ) ( 300) 900 500 300 Because the discriminant is negative, the equation has two non-real solutions, so the gardener does not have enough fencing.. A golf ball is hit with an initial vertical velocity of 6 ft/s. The function h (t) -6t + 6t models the height h (in feet) of the golf ball at time t (in seconds). Does the golf ball reach a height of 60 ft? Write an equation by setting h (t) equal to 60. Then rewrite the equation with 0 on one side. -6t + 6t 60-6t + 6t - 60 0 t - 6t + 5 0 Find the discriminant. (-6) () (5) 56 0 6 Because the discriminant is positive, the equation has two real solutions, so the golf ball does reach a height of 60 ft. 5. As a decoration for a school dance, the student council creates a parabolic arch with balloons attached to it for students to walk through as they enter the dance. The shape of the arch is modeled by the equation y (5 - ), where and y are measured in feet and where the origin is at one end of the arch. Can a student who is 6 feet 6 inches tall walk through the arch without ducking? Write an equation by setting y equal to 6.5. Then rewrite the equation with 0 on one side. (5 - ) 6.5 5-6.5 - + 5-6.5 0 Find the discriminant. 5 - (-) (-6.5) 5-6 - Because the discriminant is negative, the equation has two non-real solutions, so a student who is 6 feet 6 inches tall cannot walk through the arch without ducking. Image Credits: Aflo Foto Agency/Alamy Module 3 8 Lesson 3 Finding Comple Solutions of Quadratic Equations 8
PEER-TO-PEER DISCUSSION Ask students to discuss with a partner how the graphs of the following three parabolas would look: a parabola with two real solutions, a parabola with one real solution, and a parabola with two nonreal solutions. Students should say that a parabola with two solutions will have two -intercepts, and the parabola will open from the verte toward the -ais; that a parabola with one solution will have one -intercept with the verte on the -ais; and that a parabola with two nonreal solutions will open from the verte away from the ais and have no -intercept. 6. A small theater company currently has 00 subscribers who each pay $0 for a season ticket. The revenue from season-ticket subscriptions is $,000. Market research indicates that for each $0 increase in the cost of a season ticket, the theater company will lose 0 subscribers. A model for the projected revenue R (in dollars) from season-ticket subscriptions is R (p) (0 + 0p) (00-0p), where p is the number of $0 price increases. According to this model, is it possible for the theater company to generate $5,600 in revenue by increasing the price of a season ticket? Write an equation by setting R (p) equal to 5,600. Then rewrite the equation with 0 on one side. (0 + 0p) (00-0p) 5,600-00p + 800p +, 000 5,600-00p + 800p - 600 0 p - 8p + 6 0 Find the discriminant. (-8) - () (6) 6-6 0 Because the discriminant is zero, the equation has one real solution, so it is possible to generate $5,600 in revenue by increasing the price of a season ticket. Solve the equation using the quadratic formula. Check a solution by substitution.. - 8 + 0 8. - 30+ 50 0 -b ± b - ac a -(-8) ± (-8) - ()() () 8 ± - 8 ± i ± i So, the solutions are + i and - i. Check ( + i ) - 8 ( + i ) + 0 5 + 8i - 8 ( + i ) + 0 5 + 8i - 3-8i + 0 5-3 + 0 0 0 -b ± b - ac a -(-30) ± (-30) - ()(50) () 30 ± 00 30 ± 0 5 ± 5 So, the solutions are 5 + 5 and 5-5. Check (5 + 5 ) - 30 (5 + 5 ) + 50 0 00 + 50-30 (5 + 5 ) + 50 0 00 + 50-50 - 50 + 50 0 00-50 + 50 0 0 0 Module 3 9 Lesson 3 9 Lesson 3.3
9. + 3 0. + Rewrite the equation with 0 on one side. - -3 0 Use the quadratic formula. -b ± b - ac a -(-) ± (-) - ()(-3) () ± 3 So, the two solutions are + 3 and - 3. Check ( + 3 ) - ( + 3 + 3 - ( + 3 + 3 - ( + 3 ) - 3 0 ) - 3 0 ) - 3 0-3 0 0 0 - + 0 -b ± b - ac a -(-) ± (-) - ()() () ± -0 0 0 ± i ± i So, the two solutions are + i 0 and - i 0. Check. (+ i 0 ) - ( + i 0 ) + 0 ( 3 - + i 0 ) - ( + i 0 ) + 0-3 + i 0 - - i 0 + 0-3 - + 0 0 0 AVOID COMMON ERRORS Students need to be careful to avoid making sign errors when completing the square. Point out that when the rule representing verte form is simplified, the result should be the original rule written in standard form. Students can use this fact to perform a quick check of the reasonableness of their results, and in order to catch any sign errors they may have made.. Place an X in the appropriate column of the table to classify each equation by the number and type of its solutions. Equation Two Real Solutions One Real Solution Two Non-Real Solutions - 3 + 0 X - + 0 - + 0 + 0 + + 0 + + 0 + 3 + 0 X X X X X X Module 3 50 Lesson 3 Finding Comple Solutions of Quadratic Equations 50
JOURNAL Have students summarize how to use the discriminant to help solve any quadratic equation. Have them include eamples of quadratic equations with one or two real solutions and with two nonreal solutions. H.O.T. Focus on Higher Order Thinking. Eplain the Error A student used the method of completing the square to solve the equation - + - 3 0. Describe and correct the error. - + - 3 0 - + 3 - + + 3 + ( + ) + ± + ± - ± So, the two solutions are - + and - - -3. The student did not divide both sides by first to make the coefficient of the -term be. The correct solution is as follows. - + 3 0 - -3 - + -3 + ( -) - - ± - - ± i ± i So, the two solutions are + i and - i. 3. Make a Conjecture Describe the values of c for which the equation + 8 + c 0 has two real solutions, one real solution, and two non-real solutions. Find the value of the discriminant. b - ac 8 - () c 6 - c The equation has two real solutions when the discriminant is positive, so solving 6 - c > 0 for c gives c < 6. The equation has one real solution when the discriminant is zero, so solving 6 - c 0 for c gives c 6. The equation has two non-real solutions when the discriminant is negative, so solving 6 - c < 0 for c gives c > 6.. Analyze Relationships When you rewrite y a + b + c in verte form by completing the square, you obtain these coordinates for the verte: (- b a, c - b a ). Suppose the verte of the graph of y a + b + c is located on the -ais. Eplain how the coordinates of the verte and the quadratic formula are in agreement in this situation. When the verte is on the -ais, the y-coordinate of the verte must be 0, so c - b a 0, which can be rewritten as b - ac 0. When you set y equal to 0 in y a + b + c and solve for, you get one real solution, namely, - b, which is the -coordinate of the verte. a Module 3 5 Lesson 3 5 Lesson 3.3
Lesson Performance Task Matt and his friends are enjoying an afternoon at a baseball game. A batter hits a towering home run, and Matt shouts, Wow, that must have been 0 feet high! The ball was feet off the ground when the batter hit it, and the ball came off the bat traveling vertically at 80 feet per second. a. Model the ball s height h (in feet) at time t (in seconds) using the projectile motion model h (t) -6 t + v 0 t + h 0 where v 0 is the projectile s initial vertical velocity (in feet per second) and h 0 is the projectile s initial height (in feet). Use the model to write an equation based on Matt s claim, and then determine whether Matt s claim is correct. b. Did the ball reach a height of 00 feet? Eplain. c. Let h ma be the ball s maimum height. By setting the projectile motion model equal to h ma, show how you can find h ma using the discriminant of the quadratic formula. d. Find the time at which the ball reached its maimum height. AVOID COMMON ERRORS Students may sometimes make a mistake in sign when calculating the discriminant, particularly when the quantity ac is less than 0. Remind them that subtracting a negative number is the same as adding the opposite, or positive, number. If a and c have opposite signs, the discriminant will always be positive. INTEGRATE TECHNOLOGY Students can use a graphing utility to graph a parabola and find the maimum value. a. The ball s height h at time t is given by h (t) -6t + 80t +. Matt s claim is that h (t) 0 at some time t. Applying the discriminant of the quadratic formula to the equation -6t + 80t + 0, or -6t + 80t - 06 0, gives b - ac 80 - (-6) (-06) 600-68 -38. Since the discriminant is negative, there are no real values of t that solve the equation, so Matt s claim is incorrect. b. For the ball to reach of height of 00 feet, h (t) must equal 00. Applying the discriminant of the quadratic formula to the equation -6t + 80t + 00, or -6t + 80t - 96 0, gives b - ac 80 - (-6)(-96) 600-6 56. Since the discriminant is positive, there are two real values of t that solve the equation, so the ball did reach a height of 00 feet at two different times (once before reaching its maimum height and once after). c. Setting h (t) equal to h ma gives -6 t + 80t + h ma 0, or -6 t + 80t + - h ma 0. Since the maimum height occurs for a single real value of t, the discriminant of the quadratic equation must equal 0. b - ac 0 80 - (-6) ( - h ma ) 0 600 + 6 ( - h ma ) 0 6 ( - h ma ) -600 - h ma -00 - h ma -0 h ma 0 So, the ball reached a maimum height of 0 feet. d. Solve the equation -6t + 80t + 0, or -6t + 80t - 00 0, using the quadratic formula. You already know that the discriminant is 0 when the ball reached its maimum height, so t -80 ± 0-80.5. So, the ball reached its maimum height.5 seconds (-6) -3 after it was hit. QUESTIONING STRATEGIES How can the symmetry of a parabola help you to find the maimum or minimum if you know two different points on the graph with the same y-coordinate? The -coordinate of the maimum or minimum will be halfway between the -coordinates of the two points on the graph. Module 3 5 Lesson 3 EXTENSION ACTIVITY On the moon, the force of gravity is Earth s gravity, so the equation for simple 6 projectile motion for a ball hit at a height of feet above the ground is 6 y - 6 t + vt +. Have students determine if a baseball hit upward traveling at an initial vertical velocity of v 80 ft/s on the moon reaches a height of 00 feet. They should find the real solutions. t. and.3 s, so the ball does reach a height of 00 feet. Scoring Rubric points: Student correctly solves the problem and eplains his/her reasoning. point: Student shows good understanding of the problem but does not fully solve or eplain his/her reasoning. 0 points: Student does not demonstrate understanding of the problem. Finding Comple Solutions of Quadratic Equations 5