Finding Complex Solutions of Quadratic Equations

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1 y - y x x Locker LESSON.3 Finding Complex Solutions of Quadratic Equations Texas Math Standards The student is expected to: A..F Solve quadratic and square root equations. Mathematical Processes A..F The student is expected to analyze mathematical relationships to connect and communicate mathematical ideas. Language Ojective.D.,.H.3,.F.,.K 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 complex solutions of any quadratic equation? Possile answer: You can factor, if possile, to find real solutions; approximate from a graph; find a square root (which may e part of completing the square); complete the square; or apply the quadratic formula. For the general equation a x + x + c, you must either complete the square or use the quadratic formula to find the complex 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 aseall will go after it is hit. Then preview the Lesson Performance Task. Name Class Date.3 Finding Complex Solutions of Quadratic Equations Essential Question: How can you find the complex solutions of any quadratic equation? Explore A..F: Solve quadratic equations. Complete the tale y - - y x x Resource Locker Investigating Real Solutions of Quadratic Equations ax + x + c ax + x -c f (x) ax + x g (x) -c x + x + x + x + x + x + 3 The graph of ƒ (x) x + x is shown. Graph each g (x). Complete the tale. Repeat Steps A and B when ƒ (x) - x + x. Equation x + x + x + x + x + x + 3 Numer of Real Solutions ax + x + c ax + x -c f (x) ax + x g (x) -c - x + x - - x + x - - x + x - 3 x + x - x + x - x + x -3 - x + x - x + x - x + x 3 Equation - x + x - - x + x - - x + x - 3 f (x) x + x g (x) - f (x) x + x f (x) x + x g (x) - g (x) -3 f (x) - x + x g (x) f (x) - x + x f (x) - x + x g (x) g (x) 3 Numer of Real Solutions Module 3 Lesson 3 DO NOT EDIT--Changes must e made through File info CorrectionKeyTX-A Name Class Date.3 Finding Complex Solutions of Quadratic Equations Essential Question: How can you find the complex solutions of any quadratic equation? A..F: Solve quadratic equations. - Resource Explore Investigating Real Solutions of Quadratic Equations Complete the tale. g (x) -c ax + x -c f (x) f (x) x + x g (x) - c - ax + x + + x - x g (x) x f (x) x x - x g (x) x f (x) x + x + x -3 x + x + x + x + x + x + 3 The graph of ƒ (x) x + x is shown. Graph each g (x). Complete the tale. Repeat Steps A and B when ƒ (x) - x + x. Equation x + x + x + x + x + x + 3 Numer of Real Solutions g (x) -c ax + x -c f (x) f (x) - x + x g (x) c ax + x + x + x x g (x) - f (x) - x + 3 x + x x g (x) - f (x) - x + - x + x 3 Numer of Real Solutions Equation - x + x - - x + x - - x + x x + x - - x + x - - x + x - 3 Module 3 Lesson 3 AMTXESE35393UML3 3 // 3:5 AM HARDCOVER PAGES 56 Turn to these pages to find this lesson in the hardcover student edition. 3 Lesson.3

2 Reflect. Look ack at Steps A and B. Notice that the minimum value of f(x) in Steps A and B is -. Complete the tale y identifying how many real solutions the equation ƒ (x) g (x) has for the given values of g(x). Value of g (x) g (x) - g (x) > - Numer of Real Solutions of f (x) g (x) EXPLORE Investigating Real Solutions of Quadratic Equations. Look ack at Step C. Notice that the maximum value of ƒ (x) in Step C is. Complete the tale y identifying how many real solutions the equation ƒ (x) g (x) has for the given values of g (x). g (x) < - Value of g (x) g (x) g (x) > g (x) < Numer of Real Solutions of f (x) g (x) 3. You can generalize Reflect : For ƒ (x) a x + x where a >, ƒ (x) g (x) where g (x) -c has real solutions when g (x) is greater than or equal to the minimum value of ƒ (x). The minimum value of ƒ (x) is + a) ( - a) ( a - a a - a a - a - a. ƒ ( - a ) a ( - a ) So, ƒ (x) g (x) has real solutions when g (x) - a. Since g (x) -c, you otain: -c - a Add to oth sides. a a - c Multiply oth sides y a, which is positive. - ac In other words, the equation a x + x + c where a > has real solutions when - ac. Generalize the results of Reflect in a similar way. What do you notice? For f (x) a x + x where a <, f (x) g (x) where g (x) -c has real solutions when g (x) is less than or equal to the maximum value of f (x). The maximum value of f (x) is f - a) a. So, f (x) g (x) has real solutions when g (x) - a. Since g (x) -c, you otain: -c - a Add a to oth sides. a - c Multiply oth sides y a, which is negative. - ac Whether a > or a <, - ac tells when a x + x + c has real solutions. INTEGRATE TECHNOLOGY Students can use a graphing calculator to graph f (x) and each function g (x) to verify the numer of real solutions to each equation. QUESTIONING STRATEGIES If an equation is written in vertex 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 maximum or minimum value tells you whether there are real solutions. How do you determine where the graph of a quadratic function crosses the x-axis? You can find the x-intercepts of the graph of a quadratic function in standard form y factoring the function to get its intercept form. If the function is not factorale, the x-intercepts can e found y using the quadratic formula to find the zeros of the function. Module Lesson 3 PROFESSIONAL DEVELOPMENT Math Background In Algera, students used the quadratic formula to find real solutions to a quadratic equation. Students now revisit the formula to extend its use to complex solutions. The sign of the expression - ac determines whether the quadratic equation has two real solutions, one real solution, or two nonreal solutions. For cuic equations of the form ax 3 + x + cx + d, the sign of the discriminant c - ac 3-3 d - a d determines whether the equation has three real solutions, two real solutions, or one real solution. Finding Complex Solutions of Quadratic Equations

3 EXPLAIN Finding Complex Solutions y Completing the Square QUESTIONING STRATEGIES How do you convert quadratic functions to vertex form? Explain. You can convert quadratic functions from standard form to vertex form f (x) a (x - h) + k y completing the square on a x + x. You have to add and sutract the same constant to keep the function value the same. INTEGRATE MATHEMATICAL PROCESSES Focus on Technology Discuss with students how to use the graphing calculator to find a maximum or minimum value of a quadratic function. Students can solve prolems algeraically and then use their graphing calculators to check their solutions. Explain Finding Complex Solutions y Completing the Square Recall that completing the square for the expression x + x requires adding ( to it, resulting in the perfect square ) ) trinomial x + x + (, which you can factor as (x +. Don t forget that when x + x appears on one side of an equation, adding ( ) Example 3 x + 9x - 6 ) to it requires adding ( to the other side as well. Solve the equation y completing the square. State whether the solutions are real or non-real.. Write the equation in the form x + x c. 3 x + 9x x + 9x 6 x + 3x. Identify and ( ). 3 ( 3 ) ( 9 ) 3. Add ( to oth sides of the equation. ) x + 3x + x - x Write the equation in the form x + x c. x - x -. Identify and ( ). ( ) - ( ) - 3. Add ( ) to oth sides. x - x ). Solve for x. ( x ) 9 ( x + 3 ) x + 3 ± x + 3 ± x - 3 ± x -3 ± There are two real solutions: and.. Solve for x. x + x - + (x - ) -6 x - ± -6 x ± -6 There are two real/non-real solutions: and - i 6. + i 6 Module 5 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 verally 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. 5 Lesson.3

4 Reflect. How many complex solutions do the equations in Parts A and B have? Explain. Each equation has two complex solutions, ecause the set of complex numers includes all real numers as well as all non-real numers. Your Turn Solve the equation y completing the square. State whether the solutions are real or non-real. 5. x + 8x + 6. x + x - x + 8x - x + x x + 8x Explain Identifying Whether Solutions Are Real or Non-real By completing the square for the general quadratic equation a x + x + c, you can otain the quadratic - ± formula, x - ac, which gives the solutions of the general quadratic equation. In the quadratic formula, the a expression under the radical sign, - ac, is called the discriminant, and its value determines whether the solutions of the quadratic equation are real or non-real. Value of Discriminant - ac > - ac - ac < (x + ) - x + ± - x - ± i There are two non-real solutions: - + i and - - i. Numer and Type of Solutions Two real solutions One real solution Two non-real solutions x + x (x + 5) 3 x + 5 ± 3 x + 5 ± x -5 ± There are two non-real solutions: -5 + and EXPLAIN Identifying Whether Solutions are Real or Non-real QUESTIONING STRATEGIES Does the discriminant give the solution of a quadratic equation? Explain. No, it gives the numer of solutions and type of solution, ut it does not give the actual solution. AVOID COMMON ERRORS Remind students that they must write the quadratic equation in standard form efore applying the quadratic formula. CONNECT VOCABULARY Review vocaulary related to quadratic functions, such as discriminant and real numers, y having students lael the parts of a quadratic function written in various forms. Module 6 Lesson 3 DIFFERENTIATE INSTRUCTION Cognitive Strategies Some students have troule completing the square ecause there are so many steps. Show them how to reak the process into three parts: () Get the equation into the form needed for completing the square. () Complete the square. (3) Finish the solution y taking square roots of oth sides and simplifying the results. When students make errors, analyze their work carefully to see what part of the process is giving them troule, and give them extra practice on that part of the process. Finding Complex Solutions of Quadratic Equations 6

5 INTEGRATE MATHEMATICAL PROCESSES Focus on Reasoning The discriminant can e used to distinguish etween rational and irrational solutions. Give students several quadratic equations for which - ac is positive, some with rational solutions, and some with irrational solutions. Ask them to make a conjecture aout how the value of the discriminant is related to whether the solutions are rational or irrational. Students should e ale to explain why the solutions will e rational when the value of the discriminant is a perfect square. Example Answer the question y writing an equation and determining whether the solutions of the equation are real or non-real. A all is thrown in the air with an initial vertical velocity of m/s from an initial height of m. The all s height h (in meters) at time t (in seconds) can e modeled y the quadratic function h (t) -.9 t + t +. Does the all reach a height of m? Set h (t) equal to. -.9 t + t + Sutract from oth sides. -.9 t + t + Find the value of the discriminant. -.9) ) Because the discriminant is zero, the equation as one real solution, so the all does reach a height of m. A person wants to create a vegetale garden and keep the raits out y enclosing it with feet of fencing. The area of the garden is given y the function A (w) w (5 - w) where w is the width (in feet) of the garden. Can the garden have an area of ft? Set A (w) equal to. w (5 - w) Multiply on the left side. 5w - w Sutract from oth sides. - w + 5w - Image Credits: (t) Fred Fokkelman/Shutterstock; () David Burton/Alamy Find the value of the discriminant. 5 - )) 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 ft. Your Turn Answer the question y writing an equation and determining if the solutions are real or non-real.. A hoyist is making a toy sailoat. For the triangular sail, she wants the height h (in inches) to e twice the length of the ase (in inches). Can the area of the sail e i n? Write the area A of the sail as a function of. A () Sustitute for A. Sutract from oth sides. - Find the discriminant. - () ) + Because the discriminant is positive, the equation has two real solutions, so the area of the sail can e i n. Module Lesson 3 LANGUAGE SUPPORT Communicate Math Students play How do you know? Give students several cards containing quadratic equations; some have real numer solutions, others nonreal or complex 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. Lesson.3

6 Explain 3 Finding Complex Solutions Using the Quadratic Formula When using the quadratic formula to solve a quadratic equation, e sure the equation is in the form ax + x + c. Example 3 Solve the equation using the quadratic formula. Check a solution y sustitution. EXPLAIN 3 Finding Complex Solutions Using the Quadratic Formula -5 x - x ± Write the quadratic formula. x - ac a Sustitute values. -) ± ) - 5)8) 5) ± -56 ± i 39 Simplify So, the two solutions are i and i 5. Check y sustituting one of the values. Sustitute i 5 ) i 5 ) - 8 Square. -5 ( 5 + i ) 5 - i 5 ) - 8 Distriute i i Simplify. 5-8 QUESTIONING STRATEGIES Why are there always two solutions to a quadratic equation that has nonreal solutions? How are they related? Since - ac is not zero, its value will e oth added to and sutracted from - in the numerator, resulting in two solutions; they are complex conjugates. What is the general solution of a quadratic equation with only one solution? x - a AVOID COMMON ERRORS Students may have difficulty rememering 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 efore identifying the values of a,, and c to e used in the formula. Module 8 Lesson 3 Finding Complex Solutions of Quadratic Equations 8

7 INTEGRATE MATHEMATICAL PROCESSES Focus on Communication You may wish to point out that quadratic equations always have two roots. However, when the value of the discriminant is, the two roots happen to e the same. In this case, the quadratic is said to have a doule root. B x + x Write the equation with on one side. x + x + Write the quadratic formula. x - ac a - ± ( ) Sustitute values. - ± - ( )( ) ( ) Simplify. - ± ± i - ± i So, the two solutions are + 3i 3 and - 3i 3. Check y sustituting one of the values. Sustitute. + 3i 3 ) + + 3i 3 ) + Square. ( 9-6i ) + + 3i 3 ) + Distriute. Simplify. - 6i i Module 9 Lesson 3 9 Lesson.3

8 Your Turn Solve the equation using the quadratic formula. Check a solution y sustitution x - 5x - 9. x + 8x + x - ± - ac a -5) ± 5) - (6)) (6) 5 ± 5 ± 3 x So, the solutions are 5 + and 5 - Elaorate -. Check 6 ( 3 ) - 5 ( 3 ) x + 6x +. Discussion Suppose that the quadratic equation a x + x + c has p + qi where q as one of its solutions. What must the other solution e? How do you know? The other solution must e p qi. The radical - ac in the quadratic formula produces imaginary numers when - ac <. Since - ac is oth added to and. Discussion You know that the graph of the quadratic function ƒ (x) a x + x + c has the vertical line x - as its axis of symmetry. If the graph of ƒ (x) crosses the x-axis, where do the x-intercepts occur a relative to the axis of symmetry? Explain. - The x-intercepts are the solutions of f (x), which are x ± - ac y the a quadratic formula. Writing the x-intercepts as x - a ± - ac shows that a the x-intercepts are the same distance, - ac, away from the axis of symmetry, a with one x-intercept on each side of the line: x - a - - ac on one side and a x - a + - 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 x + x + c. As long as any particular equation is in the form x - ± - 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 ) i 3 ) i 3 ) 6-6i i i i i 3 sutracted from 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 x + x + c, you can simply sustitute the values of a,, and c into the quadratic formula and otain the solutions of the equation. ELABORATE INTEGRATE MATHEMATICAL PROCESSES Focus on Critical Thinking Emphasize that choosing which method to use to solve a quadratic equation is as important as eing ale to use each method. Have students discuss when each method might e preferred. AVOID COMMON ERRORS Students may sometimes make a mistake in sign when calculating the discriminant, particularly when the quantity ac is less than. Remind them that sutracting a negative numer is the same as adding the opposite, or positive, numer. If a and c are opposite signs, the discriminant will always e 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 y using the quadratic formula to solve the equation, and then writing the solutions as a pair of complex conjugates of the form a±i. Module Lesson 3 Finding Complex Solutions of Quadratic Equations