Finding Complex Solutions of Quadratic Equations

Home , Quadratic formula, Quadratic function

CorrectionKey=NL-C;CA-C DO NOT EDIT--Changes mustbemadethrough “File info” LESSON esn 3 . Lesson 139 preview LessonPerformance the Task. go after will ithow is hit. abaseball high Then and how to solve aquadratic equation to determine View Engage the online. photo section the Discuss a quadratic formula. For thegeneral equation square); complete thesquare; orapplythe square ofcompleting root the (whichmay bepart real solutions; approximate from agraph; finda Possible answer: You canfactor, ifpossible, to find Essential Question: quadratic equations are real ornotreal andjustifyreasoning. Work orsmallgroup to determine withapartner whethersolutionsto Objective Language Practices Mathematical solutions. A-REI.B.4b Also N-CN.C.2, Solve quadratic equations withreal coefficients that have complex The student isexpectedto: Common Core Math Standards Equations ofQuadraticSolutions Finding Complex complex solutionsoftheequation. square oruse thequadratic formula to findthe quadratic equation? find thecomplex solutionsofany PERFORMANCE TASK PREVIEW: LESSON ENGAGE COMMON COMMON x CORE CORE 2

+ bx MP.2 Reasoning N-CN.C.7 +c=0,you musteithercomplete the 3 . 3 3. How canyou CorrectionKey=NL-A;CA-A DO NOT EDIT--Changes mustbemadethrough “File info” A2_MNLESE385894_U2M03L3.indd 139

Explore Complete table. the The graph of ƒ Repeat Steps Aand ƒ Bwhen - - 4 4 a -2 -2 -2 a - - 2 2 2 x x x x x 2 2 2

of Finding Complex Solutions x x x 2 2 2

+bx +bx +4x3=0 +4x2=0 +4x1=0 - 2 2 2

+4x-3=0 +4x-2=0 +4x-1=0 2 2 2 0 0 y ( y Quadratic Equations +c=0 x +c=0 ) Investigating ofQuadratic RealSolutions Equations

= 2 2 x 2

+ 4x 4 4 x x ( x is shown. Graph each g )

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= -2 2 2 2 -2 -2 -2

Name Finding Complex3. 3 Solutions x x x a

Essential Question: How can you find the a

Explore COMMON

CORE A2_MNLESE385894_U2M0 

2 2 2

Complete the table. x  x

N-CN.C.7 Solve quadratic equations with real coefficien Also N-CN.C.2, A-REI.B.4b

The graph of ƒ x x x © Houghton Mifflin Harcourt Publishing Company + 4x + 4x + 4x 2

of 2 2

2 +bx +bx Repeat Steps A and B when 2 2 4 x

2 2 2 2

x 2 + through “File info” x Module 3 Quadratic Equations 2 3L3.indd 139 + 4x + 1 = 0 2 + + 4x + 2 = 0

- + 4x + 3 = 0 bx

- ax Investigating Real Solutions of Quadratic Equations + 4x + 4x + 4x -2 +

-2 2 4 y -2 2 Class ( 0 c bx x + x

x 2 ) = 0 - x 2 + 4x - 1 = 0 2 = 2 + 4x - 2 = 0 2 + 4x - 3 = 0

c 0 4x. - + = 2 x 2

2 0 2

y + 4x

x = = =

complex solutions of any quadratic equation?

is shown. Graph each g

ƒ Class 2 =-c =-c 2 ax

( 2 x x x

2 x

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2 4x + x = -2 4x + + + 4x

bx 139 139 = 3 = 2 = 1 -3 -2 -1 -2 ax = -2 = x = -2 -2 -2 -2 = ts that have complex solutions. 2

2 -1 x c + 2 - + bx - x 2 -3 x 2

2 4x + 4x. 2 2 2 4x + + 4x

2 = -c 2 x 2 ( x 2 Equation x x = 1 2 x x x 2 + 4x + 1 = 0 ) = 2 + 4x + 2 = 0 . Complete the table. = 3 4x 3 0 + + = Date

-2 x f x x -2 f 2 2 2

-2 f ( 139 (

f x ( x x Equation ( x x 2 ) x ) x +4x3=0 +4x2=0 +4x1=0 Equation 2 ) Equation

+ 4x - 1 = 0 ax 2 =

2 ( ) = 2 2 2

+ 4x - 2 = 0 2 = + 4x - 3 = 0 = 2

f +4x-3=0 +4x-2=0 +4x-1=0 ( f x x f ( x x ( 2 2 ( x x x 2

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-2

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19/03/14 12:05 PM ( ( x x x ) ) ) x x

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=a = 2 = 2 = 2 =a Date -2 -2 -2 x x x x x x x x

2 2 2 2 2

2 2 2 +bx +bx + 4x + 4x + 4x

+ 4x + 4x + 4x Number ofReal Number ofReal Solutions Solutions PAGES 99108

2 0 1 0 1 2 g g g g g g g g ( ( ( ( ( ( ( ( x x x x x x x x ) ) ) ) ) ) ) ) =-c =-c

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= 3 = 2 = 1 -3 -2 -1 Resource Locker Lesson 3 Lesson 19/03/14 12:00PM

) x 140 (

You You to verify the number of real real of verify to the number ) x ( determines the of a determines sign The How do you determine where the graph of a of the graph where determine do you How the x-axis? function crosses quadratic If an equation is written in vertex form, what what form, in vertex written is equation an If has if it find use to out you can information EXPLORE EXPLORE INTEGRATE TECHNOLOGY INTEGRATE QUESTIONING STRATEGIES QUESTIONING direction or of the opening and the maximum real are whether there you tells minimum value solutions. -intercepts of the graph of a quadratic of a quadratic of the graph can find the x-intercepts factoring by the functionfunction form in standard Iffunction the form. is not get its intercept to using by can be found the x-intercepts factorable, of the the zeros find to formula the quadratic function. real solutions? solutions? real Investigating Real Solutions of Investigating Equations Quadratic f graph to calculator use a graphing can Students solutions to each equation. to solutions and each function g and DO NOT EDIT--Changes must be made through “File info” “File through made be must EDIT--Changes NOT DO CorrectionKey=NL-B;CA-B 5/22/14 10:47 AM

2 b 4a ___ is Equations Solutions of Quadratic Complex Finding ) is - ) x x ( Lesson 3 ( =

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) 0. ) b x x 2a ___ has real ( ( ≥ - ( 1 0 2 of 1 2 0 of 4ac = g = g ) ) is f = -c x -

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where a ) b 0, the sign of the discriminant the discriminant + cx + d = 0, the sign of 4a 140 ) _ =

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determines whether the equation has three real real three has whether determines the equation b 4a __

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2 x So, Substitute Substitute Add Add For For less than or equal to the maximum value of f the maximum value less than or equal to Whether Whether 4a, which is negative. Multiply both sides by Generalize the results of Reflect 2 in a similar way. What do you notice? you do What Reflect of way. Generalize 2 in a similar the results Substitute Add In other words, the equation a the equation words, other In solutions when g Multiply both sides by 4a, by both sides Multiply 2. Complete the table by by the table -2. Complete B is A and solutions real many how identifying ƒ the equation Look back at Steps A and B. Notice Notice B. A and Steps Look at back of value the minimum that ( Complete the table by identifying how how identifying by the table Complete ƒ the equation solutions real many ƒ Look back at Step C. Notice that the that C. Notice Step Look at back ƒ of value maximum given values of g(x). of values given You can generalize Reflect 1: For ƒ Reflect generalize can 1: For You So, ƒ of c

2 Reflect

PROFESSIONAL DEVELOPMENT PROFESSIONAL b The sign of the expression the expression signThe of In Algebra 1, students used the quadratic formula to find real solutions to a to find solutions to real formula used the quadratic 1, students Algebra In complex use to its extend to the formula revisit now Students equation. quadratic solutions. Math Background Math solutions, two real solutions, or one real solution. real one or solutions, real two solutions, equation has two real solutions, one real solution, or two nonreal solutions. For For solutions. nonreal two or solution, real one solutions, real two has equation a x the form of equations cubic

Module 3

2. 3.

A2_MNLESE385894_U2M03L3 140

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CorrectionKey=NL-C;CA-C DO NOT EDIT--Changes must be made through “File info” “File through be made must EDIT--Changes DO NOT CorrectionKey=NL-C;CA-C DO NOT EDIT--Changes mustbemadethrough “File info” calculators to check solutions. their problems algebraically and graphing their use then value of aquadratic Students function. can solve graphing calculator toamaximum find or minimum Focus on TechnologyFocus esn 3 . Lesson 141 MP.5 Completing theSquare Finding Complex by Solutions form f quadratic from standard functions form to vertex on a constant value to thesame. keep thefunction PRACTICES INTEGRATE MATHEMATICAL QUESTIONING STRATEGIES EXPLAIN 1 x 2

(

+ bx Discuss with students Discuss how to the use vertex form? Explain. How doyou convert quadratic functions to x ) =a . You have tothesame addandsubtract ( x -h ) 2

+ kby completing thesquare You canconvert CorrectionKey=NL-A;CA-A DO NOT EDIT--Changes mustbemadethrough “File info” A2_MNLESE385894_U2M03L3.indd 141

© Houghton Mifflin Harcourt Publishing Company how steps their for solving equation the were similar or different. repeating activity for the adifferent quadratic equation. Have students discuss nonreal the find solutions to equation. the Then have switch partners roles, written invarious forms. Have one student inhow partner the verballyinstruct to Have students work inpairs. Provide quadratic eachpair with several equations Peer-to-Peer Activity COLLABORATIVE LEARNING

 that completingRecall square the for expression the Explain1 oue 3 Module  3

trinomial trinomial

equation, adding

Example 1

Add 3. Add Add 3. Add 2. 2. Identify band 1. Write equation the form inthe 1. Write equation the form inthe 3 x x 2

Identify

x 2

x +

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+ x 9x ( ( x (

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_ 2 b 2 b 2 the Square 4 9 b , which you can factor as +

( ) )

_ = 3

= 6 = 2 = = 2+ 2 2 -2 . . 2 _ 1 4 9

)

2 = = = _ 4 9

-7 -2 1 + x x 2 2

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2 bx to other the side as well. bx bx

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2 ( √ x x __ . b 2 √

x x x

) + + 6 ―

17 2

2 (

to it, resulting square perfect inthe + +

x + 2x _ _ 2 2 3 3

x -

) ) _ _ 2 2 3 3 x x

2 2 -

+ = = = = 2+ = = 1 bx _ ± ±

- _ 17 -3 1 1 4 ) appears on one side of an

_  _ 2 √

2 _ = 17 = = = ± ± ―― _ _

17 4 9

2 4

± 1 -7 √

-6 √

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+ esn 3 Lesson i √ 6 ― 19/03/14 12:00PM

142 No, it gives the it gives No, by having having by numbers, real Does the discriminant give the solution of a of the solution give Does the discriminant Explain. equation? quadratic EXPLAIN 2 2 EXPLAIN QUESTIONING STRATEGIES QUESTIONING CONNECT VOCABULARY VOCABULARY CONNECT AVOID COMMON ERRORS COMMON AVOID and typenumber of solutions of solution, but it the actual solution. does not give students label the parts of a quadratic function a quadratic label the parts of students forms. in various written Review vocabulary related to quadratic functions, functions, quadratic Review to vocabulary related discriminant as such and Identifying Whether Solutions are Solutions are Whether Identifying Non-real Real or the quadratic write they must that students Remind the applying before form in standard equation formula. quadratic DO NOT EDIT--Changes must be made through “File info” “File through made be must EDIT--Changes NOT DO CorrectionKey=NL-C;CA-C 6/8/15 1:22 PM

2 _

√ 2 . _

4 √ 4 ± 32 _ - √ ± -5 -5 0 = 7 = = 32

= ) 0, you can obtain the quadratic obtain can 0, you 7

+ 5 = + 25 = 7 + 25 2 and _ + 5 = - + 10x

√ c x

2 4 ( + , and its value determines whether the solutions whether determines the solutions value its , and + = 196 - 196 = 0

+ 10x bx + 10x )

2 + x x x There are two non-real solutions: non-real two are There -5 x x

+ 2 = 12

2 + 10 = 0 -10 ( ) 6. 142 + 14t

+ 14t 2

-4.9 t 2 ( t - 4

-4.9 Number and Type of Solutions Type Number and 2 -4.9 Two real solutions real Two solution One real solutions non-real Two , is called, is the discriminant - 4ac

2 b

―― -1 2. Does the ball reach a height of 12 m? + 2. Does of the ball a height reach ± , which gives the solutions of the general quadratic equation. In the quadratic formula, the formula, the quadratic In equation. quadratic the general of the solutions , which gives

√ Identifying Whether Solutions Are Real or Non-real Solutions Are Whether Identifying - - 17 ± -1 -4 - 17 + 16 0 - 4ac + 14t

= = = = =

b __ 2 2a 2

√ ) 17 ± + 4 = equal to 12. equal to solutions of the equation are real or non-real. real or are the equation of solutions

+16 + 8x + -b ) ______

-4.9 + 4 t

2 i and -4 ( whether the determining and equation an writing by the question Answer x = = (

+ ) Value of Discriminant Value x t + 8x + 8x ,

( does reach a height of 12 m. does reach of a height 2

2 (in meters) h (in meters) height ball’s 2 m. The of initial height an 14 m/s from function time t (in seconds) the be quadratic can at modeled by h - 4ac > 0 - 4ac = 0 - 4ac < 0 x How many complex solutions do the equations in Parts A and B have? Explain. B have? A and in Parts the do equations solutions complex many How Each equation has two complex solutions, because the set of complex numbers includes because the set of complex solutions, complex two has Each equation numbers. as all non-real numbers as well all real x x x

x There are two non-real solutions: non-real two are There -4

2 2 2 initial vertical of an with velocity in the air A ball thrown is b b b

DIFFERENTIATE INSTRUCTION DIFFERENTIATE When students make errors, analyze their work carefully to see what part of part see of to what carefully their work analyze errors, make students When the part of that them extra practice on give and them trouble, giving is the process process. Some students have trouble completing the square because there are so many so many are because there the square completing trouble have students Some parts: three into the process break to them how Show steps. the square. completing needed for the form into (1) Get the equation the square. (2) Complete the results. simplifying both sides and (3) of roots taking square by the solution Finish Cognitive Strategies Cognitive Example 2

formula of the quadratic equation are real or non-real. or real are equation the quadratic of both sides. from 12 Subtract the discriminant. of the value Find 14 Set h expression under the radicalunder sign, expression Module Module 3  Explain 2 a x equation quadratic the general for the square completing By 5. Turn Your non-real. real or are whether the solutions State the square. completing by Solve the equation 4. Reflect so the ball solution, real one has the equation zero, is Because the discriminant

A2_MNLESE385894_U2M03L3 142

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CorrectionKey=NL-C;CA-C DO NOT EDIT--Changes must be made through “File info” “File through be made must EDIT--Changes DO NOT CorrectionKey=NL-C;CA-C DO NOT EDIT--Changes mustbemadethrough “File info” discriminant square. is aperfect solutions rational be value the will when of the irrational. Students should able be to explain why the related solutions to the whether are rational or conjecture about how value the of discriminant the is some with irrational solutions. Ask to them make a b students quadratic several equations for which rationalbetween and irrational solutions. Give Focus onReasoning esn 3 . Lesson 143 Quadratic Formula Finding Complex Using Solutions the MP.2 How are related? they they are complex conjugates. -b inthenumerator, resulting intwo solutions; its value willbebothaddedto andsubtractedfrom PRACTICES INTEGRATE MATHEMATICAL QUESTIONING STRATEGIES EXPLAIN 3

- 4ac The discriminant to distinguish used can be equation with only one solution? What is general the solution of aquadratic quadratic equation that has nonreal solutions? Why are there always two solutions to a is positive, some with rational solutions, and ic Since √ b ―――

- 4ac

isnotzero, x =- ___ 2a b

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© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©David Burton/Alamy game, players all inagroup must agree on card placement. turns and cards sort into piles according to kindof the solution. By end the of the real or not real. answer then They question the “How doyou Players know?” take solutions. In groups, small students draw acard and state solution the whether is quadratic equations; some have real number solutions, others nonreal or complex Students play “How doyou Give know?” students cards several containing Communicate Math LANGUAGE SUPPORT Write quadratic the formula. Simplify. Substitute values. 7. orreal non-real. Answer question the by writing an equation and determining solutions ifthe are Your Turn Subtract 700from sides. both Multiply on side. left the  oue 3 Module  form When using quadratic the formula to solve aquadratic equation, sure be equation the is inthe Explain3 discriminant the Because is [positive/zero/negative], equation the has [two real/one real/two non-real] Find value the of discriminant. the Set Set Example 3 Aperson wants to create avegetable garden and keep rabbits the

sail can be10i sail can n thediscriminant ispositive,Because theequation hastwo real solutions, sothearea ofthe Find thediscriminant. Write thearea A Substitute 10for A.10 Subtract 10fromSubtract bothsides. to be twice the length the ofb(ininches). twice to base be the area Canthe 10i of be sail the n A hobbyist is making atoy sailboat. For triangular the sail, she wants height the h(ininches) a x (in feet) of garden. the garden Canthe have an area of 700 ft is given by A function the out by enclosing it with 100feet of fencing. The area of garden the -5 solutions, garden the so [can/cannot] have an area of 700 ft 2

+ x A 2

Solve the equation the Solve using quadratic the formula. bx - 2x ( w

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= w ( x - 50 = = = w 50 50 - w 2

______-b 2 -

2 + 50w ±

( 4(-1)700) - -2 0 0 ) ± -10 where wis width the where  2

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( b ―――

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 (

1 ac - 4ac 700 - w w w -

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39 ― -5

_ b b 2 1 2

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b b = = 2 2

? -300 esn 3 Lesson 5/22/14 10:47AM 144

to be c to used You may wish to point out that quadratic quadratic that out point to wish may You AVOID COMMON ERRORS COMMON AVOID MATHEMATICAL INTEGRATE PRACTICES Students may have difficulty the remembering have may Students the copy to students Encourage formula. quadratic working. when they are hand on it have and formula form in standard the equation write themto Caution a, b, and of the identifying values before in the formula. Focus on Communication Focus MP.3 equations always have two roots. However, when the However, roots. two have always equations happen roots 0, the two is the discriminant of value to said is this case, the quadratic In beto the same. root. a double have DO NOT EDIT--Changes must be made through “File info” “File through made be must EDIT--Changes NOT DO CorrectionKey=NL-A;CA-A 19/03/14 11:59 AM © Houghton Mifflin Harcourt Publishing Company Equations Solutions of Quadratic Complex Finding Lesson 3

= 0 3

―― 0 = 0 √

+ 4 ≟ 0 + 4 ≟ 0 + 4 ≟ 0 + 4 ≟ 0 i

2 2

) ) ― 3 4

7 3

― 3

_ 7

― 3 ― 3 ) 7 √

7  - 7 7 + √ √ 6i ± _

4 3i

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7 1 2

_ 7 - 8 ≟ 0 - 8 ≟ 0 - 8 ≟ 0 - 8 ≟ 0 1 7 1 7 - x __

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(

5 - -

40 =

― 39

_ 39

( (

― 39 ― 39

7 5 ) 5 5 √ 27 √

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144 - 4ac i i 3 2

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√ 49 27

( _ ± - - 7 6

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 14 6i

√

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14 5

and - -b __ _

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― 3 ± ±

2 +

) 39 _ + =

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+ )

√

√ 49 x 1 7 i 2 2 1 7 39 25 _

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_ _

― 39 6i _

― 39

5 __ - - - - 5 √ (

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√ ______

_ 7

5 1 _

― 39 = = - 1

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( 2i 1 5

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2 x formula. the quadratic Write Simplify. Square. Distribute. So, the two solutions are and . are solutions the two So, the values. of one substituting Check by Substitute. Substitute values. Simplify. side. one 0 on with the equation Write Simplify. Square. Distribute. Substitute. So, the two solutions are - are solutions the two So, 7 Check by substituting one of the values. of one substituting Check by

Module 3

A2_MNLESE385894_U2M03L3.indd 144

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© Houghton Mifflin Harcourt Publishing Company 10. 8. equation the Solve using quadratic the asolution formula. by substitution. Check Module 3 Module 12. 11. Your Turn The the quadratic formula. Writing thex-intercepts asx x with onex-intercept oneachsideoftheline:x than completing square? the Discussion 6 Check-InEssential Question Discussion relative Explain. to axis the of symmetry? x solutions. What must other the solution How be? do you know? a formula andobtainthesolutionsofequation. quadratic equation a The quadratic formula istheresult ofcompleting thesquare onthegeneral form subtracted from –binthenumerator ofthequadratic formula, onesolutionwillhave the The othersolutionmustbep produces numberswhen imaginary

6 Check

So, thesolutionsare

and Elaborate x x ( ======2

2 3 4

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3 2 p __ 2a __ 2a

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- 5 - 11 12 12

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- √ 2a

- 4ac - 4ac 0 _ 2 1

( ――――――

-5 = 0 . ( 6

ontheotherside 0 0 x _ ) ) 5

+ 11 12 - 4 + bx

(

+ = Why is using quadratic the formula to solve aquadratic equation easier ) − ( c _

-4 3 4 . The radical qi. The 0. As long as any particular equation= 0.Aslongasany isintheform particular

)

(x) 2

. - 4ac

______√

= 0,whichare x b ―――

( 145 2a

x - 4ac Check x So, thesolutionsare = x

( and

) x -3

< 0.Since crosses x-axis, the where x-intercepts do the occur = = = = = 9.

2 √ + 6x

= + − =

-3

______-b -6 - -6 = b ――― +

,away from theaxisofsymmetry, - bx 6 qi.

2 x ( -

-3 i __ -6 - 4ac 2a

2 -

± b + 12=0 √

± ± ± 2i + __ 2a + 8x

2 -

6i 3 ―

2 ) i

- c

√ √

± √

)

0 has p has = 0 2a √ ± √

√ 2

______√ b ――― -12 ――

= inthequadratic formula (

+ 12=2x 3 ―

______+ 8 √ √ x b ―――

3 ―

√ 3 ―

b ―――

) 2

- 4ac

-b

______

b, andcinto thequadratic b ―――

3 ― - 24+

= 2a

- 4ac ―――――― ( - 4ac

( 2a

( 6 - 4ac -3

a ± 1 )

x )

2 +

√

- 4

2a + +

ononesideand

isbothaddedto and ――― b

shows that -3 -6 8i 2

bx i where q where

- 4ac ( √ √ 1

+ + + ) 3 ―

( 3 ―

)

has the vertical line c has vertical the 12 2i i

by the + 12≟ + 12≟ √ √ ) ≠ 0as one of its 3 ―

3 ―

= -6 2 -6 ( -3 + + Lesson 3 Lesson + 2i 2i √ i √ √ 10/17/14 4:53PM 3 ―

3 ―

)

146 Exercises 1–2 Exercises 3–8 Exercises 9–16 Exercises 17–20 Exercises Practice The value of the value The Concepts and Skills Concepts Explore Real Solutions of Investigating Equations Quadratic Example 1 Solutions by Complex Finding the Square Completing Example 2 Solutions are Whether Identifying Real or Non-real Example 3 Solutions Using Complex Finding Formula the Quadratic EVALUATE EVALUATE ASSIGNMENT GUIDE ASSIGNMENT CONNECT VOCABULARY VOCABULARY CONNECT discriminant indicates the number and types of indicates discriminant roots. What information does the value of the discriminant the discriminant does of the value information What equation? a quadratic about give DO NOT EDIT--Changes must be made through “File info” “File through made be must EDIT--Changes NOT DO CorrectionKey=NL-B;CA-B 5/22/14 10:47 AM

― 7 . y

― 7 √

√ 0 i 4 4

― 7 12 -

―― -7 - ± + 1 √ - 4 i Mathematical Practices Mathematical √ -1 - -1 ± ±

-7 -8 -8 ) Modeling Reasoning Logic Reasoning Logic Logic x 8 = = =

( , so the 2

- ) CORE )

― 7 and x , and

COMMON ( 9 2 + 1 = + 1 = - 3 = 0, + 1 = _ MP.4 MP.2 MP.3 MP.2 MP.3 MP.3 + 8 = 0 √ + 2x 12

+ 1 i 2

= - x + ( + 3x + 2x + 2x

2 2

x + 3x two non-real solutions: non-real two -1 x x x x

x 1 2

2 0 -9 x -12.

146 1 2 _ 4. = =

) x 0 has two real real - 3 = 0 has two ( + 6 = 0 has one real solution. = 0 has one real 0 has no

9 2 _ + 6x = 3, -

3 intersects the graph of = 3 intersects the graph

- + 6 = 0, x + 3x )

intersects the graph of f intersects the graph

- 6 = 0. Explain. + 6x

x 9 2 _

2 x

(

+ 3x

x + 12 = 2 1 _

2 = + 6x

+ 3x x

+ 3x ) once, so the equation so the equation once,

2 + 12 = 0. Explain. 1 2 _ )

2 x x x

( x x

-9, and 1 2 _

+ 6x ( 1 2 _

― 3 .

is shown. Use the graph to determine how how determine to the graph Use shown. + 3x is

― 3

2 = + 6x x √ x

0 has no real solutions. - 6 = 0 has no real 6 intersects the graph -6 intersects the graph is shown. Use the Use shown. + 6x is √ 2

1 2 _

2 - 6 doesn’t intersect the graph of f intersect the graph = 6 doesn’t -

― 3 x = ±

+ 4

) + 6x )

= √ = x x 2

+ 3x -2

( ) ( x ± 0 has one real solution. The graph graph The solution. 0 has one real -2 -1 -1

2 Recall of Information Thinking Strategic Thinking Strategic Recall of Information Recall of Information Skills/Concepts 0, and - = 0, and x

x (

Depth of Knowledge (D.O.K.) Depth of Knowledge 1 3 3 1 1 2 9 2 _ = 3 = = 1 2 _

6. The graph of g graph The = 6. 2

x - -6, - 12 doesn’t intersect the graph of intersect the graph -12 doesn’t )

― 3 and + 2 = + 4 = = + 9 = = + 1 = 0 0, and + 9 = 0, and

√ + 4x twice, so the equation so the equation twice, x + 2

) + 3x ) 2

+ 3x x

2 x x + x

2 twice, so the equation - so the equation twice, ( , so the equation , so the equation ( ( x

21 22 ) x

) g f + 4x + 6x + 6x 1–8 + 4x + 6x 1 2 _

1 2

x _

9–16 x

2 2 2

2 2

17–20 23–24 ( ( Evaluate: Homework and Practice and Homework Evaluate: -2 - equation equation f once, so the equation - so the equation once, The graph of g graph The For each equation, subtract the constant from both sides to both sides to from subtract the constant each equation, For - obtain these equations: solutions. The graph of g graph The solutions. x - graph to determine how many real solutions solutions real many how determine to graph have: equations the following many real solutions the following equations have: - have: equations the following solutions real many The graph of ƒ of graph The x The graph of ƒ (x) ƒ of graph The two real solutions: real two

For each equation, subtract the constant subtract the constant each equation, For obtain these equations: both sides to from x The graph of g graph The x has two real solutions. The graph of g graph The solutions. real has two of of real solutions. real intersects the graph of f intersects graph the f Exercise

2. Module Module 3 3. Solve the equation by completing the square. State whether the State the square. completing by Solve the equation non-real. real or are solutions 1.

A2_MNLESE385894_U2M03L3 146

CorrectionKey=NL-B;CA-B

DO NOT EDIT--Changes must be made through “File info” “File through made be must EDIT--Changes NOT DO

CorrectionKey=NL-C;CA-C DO NOT EDIT--Changes must be made through “File info” “File through be made must EDIT--Changes DO NOT CorrectionKey=NL-C;CA-C DO NOT EDIT--Changes mustbemadethrough “File info” esn 3 . Lesson 147 variables table. Ask students to create atable for eachof the discriminant, have organize them variables the ina If students have evaluating the difficulty the variablesthe list table. inthe they predict number the and of type solutions on based VISUAL CUES ( a, bc b 2

, 4ac , , b 2

- 4ac ) and have them CorrectionKey=NL-A;CA-A DO NOT EDIT--Changes mustbemadethrough “File info” A2_MNLESE385894_U2M03L3.indd 147

© Houghton Mifflin Harcourt Publishing Company 7. 11. 5. Module 3 Module 9. are or real they non-real. Without equation, the solving state number the of solutions and whether the equation hastwo real solutions. thediscriminant ispositive,Because 4 Find thediscriminant.

two non-real solutions:

_ x

2 5

two real solutions: x

the equation hastwo non-real solutions. thediscriminant isnegative,Because ( Find thediscriminant. -13 ______7 - x -16 -

x 2

2 2

2 - 5x - 5x - 4 + _

5 2

2 + 13x

____ x

i ) - + ( √

__ 14 13 - 2 2

x 7 (

2 x

+ 4x - 4 √ _

( 55 ―

5 2 x

x x x = 2

-16 +

+ x x 309 ――

+ - - +

and

_ = 5 2 5 = 1 -20 + + (

__ 25

__ ) 4 14 13 -1

__ 13

_ _ x

___ 2 5 2 5 + 13=0 196 169 2 7

_ and 5 2 )

__ __

14 13 14 13

= = = = )

(

x x 2 13 _ 2 5 )

- 1=0 = = ( = =

= =

-20 - 2 5 ± ± - -1

) -13 ______± __ 55 √

___ - ± ± 7 5 7 5 196 309

____ i 4 = 16+832848

____ i √

√ + )

____

i __

14 13 √ 2 - ――

_____

2 √ √ +

55 ―

___ __

14 196 169 55 14

___ ――

196 309 55 ―

309 ――

± 4

__ 25

25 . 4 √

_____

√

309 ――

- 4= 14 309 ――

- __ 25 96

10. 8. 6. 12. 147 x

2 and-0.8. two real solutions: the equation hastwo non-real solutions. thediscriminant isnegative,Because ( Find thediscriminant. 7 - 5 4 Find thediscriminant. 4 has onereal solution. thediscriminant iszero,Because theequation (

-3 t

-11 x x x

2 -12 wo non-real solutions: x

2 x

2 2 2

2 + 6x

2 - 6x + 9=12x - 11x

- 12x + - 6x

+ 6x9= ( 1.2x ) ) x 2 i 2 ( √

- 4 x + 3 x x - 4 x + 11=0 = 8 2

- 11=0 2 ―

+ 10=0 + 3= + 3= - x x

+ 0.36=1.6 - 1.2x + 9=0

and-3 - 0.6= - 0.6= ( 0.6 ) ( x 7 2 4

) = = ) ( ( ) 10 x 9 2 -3 -2 ± ± -11

) = 0.6±1.4 = 1.96 = 1.6

) √ i = 144-0

- √ = 121-280 ± ±1.4 ± -2 ――

+ 9 ― 2 i

√ √ i √

1.96 ――

2 ―

-159 Lesson 3 Lesson 19/03/14 11:59AM 148 DO NOT EDIT--Changes must be made through “File info” “File through made be must EDIT--Changes NOT DO CorrectionKey=NL-B;CA-B 10/17/14 4:56 PM © Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Aflo Foto Agency/Alamy Equations Solutions of Quadratic Complex Finding Lesson Lesson 3 − 300 -1 ? 2 = 26 - = 4900 − 5200 = 0 0

25 148 ) = 256 − 240 = 16 = = = 60

= )

) 60 15 15 ( - + are measured in feet and and in feet measured are ) −1300 + 64t

( 4

2 -6.5 equal to 1300. Then rewrite the equation the equation rewrite Then 1300. equal to ) ( ( ) 64t 16t ) 0 1300 w −1 ( − 4 + - (

= -1 -16t 2

2 2

( 2

) 4t w − 4

-4 2

2 -16 -16t , where x and y ( ) − 1300 = x 0 6.5 + 64t models the the golf height of h (in feet)

2 - = 6.5 = =

2 5 ) ( (in feet). Does the gardener Does w (in feet). width the gardener any for feet) A (in square area the garden’s gives x x

2 x 6.5 + 70w -16t

w - -

2 = - = 5 -

( ) 5x t 5x ( h +

2 = 70w -w

) equation y equation -x w ( Find the discriminant. the discriminant. Find equal to 60. Then rewrite the rewrite Then 60. h (t) equal to setting by an equation Write with 0 on one side. equation Because the discriminant is positive, the equation has two real real has two the equation Because is positive, the discriminant of 60 ft. a height so the golf ball does reach solutions, As a decoration for a school dance, the student council creates a creates council the student a school dance, for a decoration As walk to students for it to balloons with attached arch parabolic modeled is the arch by of shape The the dance. they as enter through the (in seconds). Does the golf ball reach a height of 60 ft? of a height Does ball time t (in seconds). ball the golf reach at A gardener has 140 feet of fencing to put around a rectangular vegetable garden. The function The garden. vegetable a rectangular around put to fencing of 140 feet has A gardener A where the origin is at one end of the arch. Can a student who is 6 feet who 6 feet is Can a student the arch. of end one at the is origin where ducking? without tall the arch 6 inches walk through A golf ball is hit with an initial vertical velocity of 64 ft/s. The vertical of initial an with velocity ball hit A golf is function have enough fencing for the area of the garden to be 1300 ft ft be to 1300 the garden of the area for fencing enough have Because the discriminant is negative, the equation has two non-real solutions, so a solutions, non-real has two the equation Because is negative, the discriminant without ducking. the arch through 6 inches tall cannot walk who is 6 feet student Find the discriminant. 5 the discriminant. Find equal to 6.5. Then rewrite the equation with 0 on the equation rewrite Then 6.5. setting y equal to by an equation Write x one side. Because the discriminant is negative, the equation has two non-real non-real has two the equation Because is negative, the discriminant enough fencing. not have does so the gardener solutions, Write an equation by setting A by equation an Write with 0 on one side. 70w with 0 on one side.

Find the discriminant. 70 the discriminant. Find

15. Module Module 3 13. Answer the question by writing an equation and determining whether the solutions of of whether the solutions determining and equation an writing by the question Answer non-real. real or are the equation 14.

A2_MNLESE385894_U2M03L3 148

CorrectionKey=NL-B;CA-B

DO NOT EDIT--Changes must be made through “File info” “File through made be must EDIT--Changes NOT DO

CorrectionKey=NL-C;CA-C DO NOT EDIT--Changes must be made through “File info” “File through be made must EDIT--Changes DO NOT CorrectionKey=NL-C;CA-C DO NOT EDIT--Changes mustbemadethrough “File info” esn 3 . Lesson 149 x-intercept. vertexthe away from axis the and have no parabola with two nonreal solutions open from will x-intercept with vertex the on x-axis; the and that a that aparabola with one solution have will one open fromparabola will vertex the toward x-axis; the two solutions have will two x-intercepts, and the solutions. Students should say that aparabola with real solution, and aparabola with two nonreal parabola with two real solutions, aparabola with one of following the three parabolas would a look: Ask students how to with apartner discuss graphs the PEER TO PEER DISCUSSION CorrectionKey=NL-B;CA-B DO NOT EDIT--Changes mustbemadethrough “File info” A2_MNLESE385894_U2M03L3 149

17. equation the Solve using quadratic the asolution formula. by substitution. Check 16. Module 3 Module (

x 4 So, thesolutionsare Check company to generate $25,600inrevenue by increasing price the of ticket? aseason where x projected revenue R(indollars) from season-ticket subscriptions is R increase cost inthe of ticket, aseason theater the company Amodelfor 10subscribers. lose will the revenue from season-ticket subscriptions is $24,000.Market research indicates that for each $10 theaterA small company currently has each pay who 200subscribers $120for ticket. aseason The one side. Write anequation by settingR generate $25,600inrevenue by increasing theprice ofaseasonticket. thediscriminant is zero,Because theequation hasonereal solution,soitispossibleto Find thediscriminant. 4 2

5 ======+

5 - 8x + + 8 + 8i _ _ i 8 8 ___

i √ - __ √ ± p is number the of $10price increases. According to model, this is it possible for theater the ±

± -b i ( 11 ―

+ 27=0 √ -8 11 ―

√ 2i

i √ -100p 2 √ 2 11 ―

and4-

( ± √

11 ―

) 120 ) ―― -44

-100p 11 ―

√ 11 ―

- 8 - 8

2a - 328i

√ b ―――

+ 10p

2 2

( (

( ――――――

+ 4 4 - -8 2 2

( + + √ + 800p 4ac 1 ) ) ) i i 5 2 11 ―

p 800p

√ √

( ( - 2

√ - 32+27

200 -8

. - 11 ― 11 ―

+ 4 11 ―

( ) 8p ) ) - 1 24, 000 2 - 10p

+ 27 + 27 + 27 ) - 4 ( 1600 + ( 27 p 0 16 ( ) ) 1 equalto 25,600. Then rewrite theequation with0on ) ≟ ≟ ≟ ≟ =

)

= 25,600 = = =

( 0 0 0 0 0

16 25,600 0 0 )

= 64 18.

149 400 x 15 So, thesolutionsare Check x 400 - 64=0 (

2 = = = = =

15 - 30x+50=0 + 5 15

+ 150 ______-b 30 30 - + 150 + 5 ( √ ± 5 -30 ± 10 ± ± √ 7 ―

2 2

√

and15-5 √ √

√ 7 ―

) 2a 700 ――

√ )

7 ―

b ―――

± 7 ―

7 ―

(

- 30 - 30 7 ―

- 4ac √ )

= 450 ( ―――――――

2 -30 ( ( (

( 120 15 15 1

400

- ) √

) + + 5 + 10p 2 150 7 ―

- 4

. - 150+50 5 √ √ √ ( ) 7 ― 7 ―

1 ( 7 ―

200 ) ) )

( + + 50 + 50 50 - 10p 50 ) 0

≟ ≟ = ≟ ≟ ) 0 0 0 0 0 , Lesson 3 Lesson 5/22/14 10:47AM 150

AVOID COMMON ERRORS COMMON AVOID Students need to be careful to avoid making sign making avoid need be to to careful Students that out Point the square. when completing errors simplified, is form vertex when the rule representing be in the original rule should written the result use this can fact a perform to Students form. standard and their results, of the reasonableness of check quick made. have they may sign errors any catch to in order DO NOT EDIT--Changes must be made through “File info” “File through made be must EDIT--Changes NOT DO CorrectionKey=NL-A;CA-A 19/03/14 11:59 AM © Houghton Mifflin Harcourt Publishing Company Equations Solutions of Quadratic Complex Finding Lesson 3 0 = 0 ≟ 0 ≟ 0 0 7 7

+ 7 ≟ + + )

7 ) )

(

― 10 )

- 4 + 7 ≟ 0

― 10 ― 10 2 2 2 √ ( 3

― 10 √ √ 4 i i _

- √ - 2 . i

+ + 2

X X X

)

10 ― 1 1 ± ) 2 ( ( 2 2 - 4 - 2i √ _

4 4

-4 i (

____ Solutions

( ―――――― 2 = - -

― 10

- 4ac

Two Non-Real Two

√ 2

)

2 )

√ = 0

――― b

― 10 ±

― 10

2a 7

―― -40

― 10 ) √

√

2 √ 4 4 and 1 -

+ 2i + i √ √

± i -4 _

― 10 ( + 2 ± 2i ±

-3 - √

3 2 -b 4 4 __ i ____

______

1+ - 4x 4x + 7 =

(

2 2

X X - + = = = = ( x 2 2 2x x

Check.

2 So, the two solutions are solutions are the two So, 1

Solution One Real 150 20. 0 0

= 0 ≟ ) ≟ ≟ 0 3 3 3 0 -3 - - 3 ≟ 0 X X (

- -

)

) )

12 ( _ Two Real Two Solutions

― 13

√

- 4

√

4 √

. 2

2 ) ) + 1

― 13 + + 2 (

1 _

2 1 √ 2 -1 _

2 _

(

―――――― ( (

- 4ac

(

Use the quadratic formula. quadratic the Use -

√

- - ______1

――― b

2 2a ±

― 13 )

√

― 13 )

2 ― 13

√

x and

√ ±

13 ―

√ -1

4 -3 ( ± Equation

― 13 √ 4

2 x

1 -b - __ + 2 _ ___

√

+ 3x + 1 = 0 + x + 1 = 0 + 2x + 1 = 0 - x +1 = 0 + 1 = 0 - 3x + 1 = 0 - 2x + 1 = 0 2 + 2 +

2 2 2 2 2 2 2 - + 3 =

x x x x x x x = = = + 1

2 14 _

__ Place an X in the appropriate column of the table to classify each equation by the by classify equation each to the table of column X in the appropriate an Place solutions. type its of and number x

14 ______1

So, the two solutions are solutions are the two So,

( Rewrite the equation with 0 on one side. with 0 on one equation the Rewrite x Check

21. Module 3 19.

A2_MNLESE385894_U2M03L3.indd 150

CorrectionKey=NL-A;CA-A

DO NOT EDIT--Changes must be made through “File info” “File through made be must EDIT--Changes NOT DO

CorrectionKey=NL-C;CA-C DO NOT EDIT--Changes must be made through “File info” “File through be made must EDIT--Changes DO NOT CorrectionKey=NL-C;CA-C DO NOT EDIT--Changes mustbemadethrough “File info” esn 3 . Lesson 151 solutions. with one or two real solutions and with two nonreal Have include them examples of quadratic equations discriminant to help solve any quadratic equation. Have students summarize how to the use JOURNAL CorrectionKey=NL-A;CA-A DO NOT EDIT--Changes mustbemadethrough “File info” A2_MNLESE385894_U2M03L3.indd 151

23.

24.

H.O.T. x to 0iny so When thevertex isonthex-axis, they-coordinate ofthevertex mustbe0, when thediscriminant isnegative, sosolving64-4c is zero, sosolving64-4c 64 The equation hastwo real solutionswhenthediscriminant ispositive, sosolving b Find thevalue ofthediscriminant. equation Analyze Relationships Make aConjecture Explain theError Suppose vertex the of graph the of y -x x situation. how coordinates the of vertex the and quadratic the formula are inagreement inthis completing square, the you obtain coordinates these for vertex: the -x

So, thetwo solutionsare 1+

The student didnotdividebothsidesby –1firstto make thecoefficient ofthe

x The correct solutionisasfollows. 2

So, two the solutions are -1

2 =

2 2

+ 8x

- 4ac c 2

- 4c - 2x - 2x 2

Focus onHigherOrder Thinking

- + 2x -x - + 2x ( x ( x __

2 2a x b

4a x x b -1 2

- 2x +

x x + 1 2

-x > ,whichisthex-coordinate ofthevertex. + 2x + 1= + 3= - 1= - 1= =

= - + 1= + 1= + 1=3 0, which can berewritten= 0,whichcan as c ) 0 x = 0has two real solutions, one real solution, and two non-real solutions. ax 2

) 2 8 3 x

2 + 2x for = = = 1±

2 = = 3 = 4 =

- 4

+ -2 -3 -3 0 ± ± ±2 ± -1 0 c gives c

bx 0. Describe and correct error. the - 3=0.Describe

√ i √ ( A student of method the completing used square the to solve the √ + 1 1 _

4 ± 2 i

-2 ―― Describe the values the of cforDescribe whichequation the )

+ √

― 2

2 ―

c andsolve for x,you getonereal solution,namely, = 64-4c

When you rewrite y

< + 2=1and -1 = 16. The equation hasonereal solutionwhenthediscriminant 0 for cgivesc i √ = 2 ―

and1- 2

+ bx b = - 2=

2 + cis located on x-axis. the Explain

= 151 - 4ac ax i √ 16. The equation hastwo non-real solutions 2

2 ―

+ -3.

. bx = 0. When you setyequal

+ cinvertex form by < 0 for cgivesc ( - __ 2a b

, c - __

4a b 2

)

. > 16. x 2

-term be1. Lesson 3 Lesson 19/03/14 11:59AM 152 is less than 0. Remind them that them 0. Remind that than less is -coordinate of the maximum or of the maximum or x-coordinate The How can the symmetry of a parabola help you you the symmetry can help a parabola of How if you minimum or findto the maximum INTEGRATE TECHNOLOGY INTEGRATE STRATEGIES QUESTIONING AVOID COMMON ERRORS COMMON AVOID know two different points on the graph with the same the same with the graph on points different two know y-coordinate? Students may sometimes make a mistake in sign a mistake make sometimes may Students particularly when when calculating the discriminant, 4ac the quantity a graph to utility use a graphing can Students value. find the maximum and parabola subtracting a negative number is the same as adding adding as the same is number a negative subtracting a and c have If number. positive, or the opposite, be will always the discriminant signs, opposite positive. minimum will be halfway between the between minimum will be halfway on the graph. points of the two x-coordinates Scoring Rubric his/her reasoning. and explains the problem solves correctly Student 2 points: but does not fully good understanding of the problem shows Student 1 point: his/her reasoning. or explain solve understanding of the problem. does not demonstrate Student 0 points: 6/8/15 1:23 PM

Lesson 3

max

256. + 4 - + 80t

2 t 0, using the quadratic = 0, using the quadratic - 106 = 0, 100 = 6400 - 6144 =

) - 4. Matt’s claim + 4. Matt’s + 80t 384. Since the discriminant is is the discriminant -384. Since

= 0, or -16 -96

+ 4 = 100, or ( ) + 80t + 80t

max

-16t 2

h -16 = + 80t (

-16t must equal 100. Applying the discriminant the discriminant must equal 100. Applying -16t ) t 2.7 and 27.3 s, so the ball does t = 2.7 and 27.3 s, + 4 - 4 ( =

) -16t t 152 80t ( Earth’s gravity, so the equation for simple simple for so the equation gravity, Earth’s 80

= 6400 - 6784 =

1 6 _ ) + =

+ 4 = 110, or 2 using the discriminant of the quadratic the quadratic of the discriminant using is the projectile’s initial vertical velocity the projectile’s is t 0

v max

h -106 + 4 = 104, or ( - 4ac + 80t 2.5. So, the ball reached its maximum height 2.5 seconds 2.5 seconds its maximum height the ball reached = 2.5. So,

)

2 2

b where where

0 -80 -32

___ + 80t h -16

gives -16 gives (

2 -16t =

+ is the projectile’s initial height (in feet). Use the model Use (in feet). initial height the projectile’s is -6400 -100 -104 0

max

― 0 t

0 h ) h - v = 0 = = 0 = = = 104 = 0 √

2 ) -16t )

) + ± 0, gives = 0, gives

-16 max max max

2 80 ( max

max max

t h h h 2 h h h 96 = traveling traveling if a baseball determine upward hit students + 4. Have - - 4ac ______-80

- - - - -16 4 equal to equal to show how you can find can you how show = b , . Applying the discriminant of the quadratic of the quadratic the discriminant some time t. Applying = 110 at 4 4 4 )

= ( t ( ( )

( )

max t ) + vt - 4ac t

( h

h 64 (

2 2

be the ball’s maximum height. By setting the projectile motion model motion the projectile setting By height. maximum be the ball’s + 80t h

t b

2 + 64

max

-16 ( h 6 16 _ equal to equal to (in seconds) using the projectile motion motion the projectile using time t (in seconds) at h (in feet) height the ball’s Model model Did the ball reach a height of 100 feet? Explain. 100 feet? of Did the ball a height reach Let to write an equation based on Matt’s claim, and then determine whether Matt’s whether then determine Matt’s and claim, based Matt’s on equation an write to correct. is claim (in feet per second) and per(in feet and second) formula. height. maximum which the ball its the reached time at Find - 4 -16t is that h is that formula. You already know that the discriminant is 0 when the ball reached its maximum is 0 when the ball reached the discriminant know that already You formula. so t height, , the discriminant of the of t, the discriminant value a single real for occurs the maximum height Since must equal 0. equation quadratic of the quadratic formula to the equation the equation to formula of the quadratic So, the ball reached a maximum height of 104 feet. a maximum height the ball reached So, the equation Solve is given by h by time t is given h at height ball’s The after hit. it was that solve the equation, the equation, solve of t that values real two are there is positive, the discriminant Since its reaching before times (once different two at of 100 feet a height so the ball did reach after). and once maximum height Setting formula to the equation the equation to formula gives gives that solve the equation, so Matt’s claim is incorrect. so Matt’s the equation, solve of t that values no real are there negative, h of 100 feet, of height reach the ball to For 6400

c. a. b. d. a. c. b. d.

EXTENSION ACTIVITY EXTENSION reach a height of 200 feet. a height reach is gravity of the force On the moon, 80 ft/s on the moon reaches a height of 200 of a height reaches the moon v = 80 ft/s on initial vertical of an velocity at find the solutions. real Theyfeet. should projectile motion for a ball hit at a height of 4 feet above the ground is is the ground above 4 feet of a height a ball at hit for motion projectile y = -

Module 3 Module Lesson Lesson Task Performance

Matt and his friends are enjoying an afternoon at a baseball game. A batter hits a towering a towering hits a baseball A batter at game. afternoon an enjoying friends his are and Matt been ball The high!” off 110 feet 4 feet was have must that “Wow, shouts, Matt run, and home per feet 80 vertically at traveling the ball the off bat and came it, hit when the batter the ground second.

A2_MNLESE385894_U2M03L3 152

CorrectionKey=NL-C;CA-C

DO NOT EDIT--Changes must be made through “File info” “File through made be must EDIT--Changes NOT DO

CorrectionKey=NL-C;CA-C DO NOT EDIT--Changes must be made through “File info” “File through be made must EDIT--Changes DO NOT