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LESSON15Factor by GroupingLEARNING OBJECTIVES Today I am: examining Juan’s method for factoring trinomials. So that I can: factor by grouping. I’ll know I have it when I can: apply factoring to graphing quadratic functions.Exploration 1—Factor Trinomials by GroupingJuan found a pattern for factoring that works for all types of trinomials.1. Here is the beginning of Juan’s work. Discuss with your group, each step that Juan did. There arethought bubbles to help you identify each step.A. y 5 x2 1 8x 1 15OB. y 5 3x2 1 5x 2 2IMultiples of: 151 15Multiples of: 26Adds to: 83 5Adds to: 5The 15 is from3 5.x 3 CX 151 (26)The 26 is from.The 8 is fromGJ.The 5 is from.21 62 (23)22 3What is the purpose of these boxes?569

570Module 4 Quadratic Functions2. Juan’s next steps are shown below. Determine what he did at each stage.OA. y 5 x2 1 8x 1 15B. y 5 3x2 1 5x 2 21 15Multiples of: 15Adds to: 8The 8x became33 5511 .1 (26)Multiples of: 26The 5x becameAdds to: 5IX 1611.y 5 x2 1 3x 1 5x 1 15y 5 3x2 2 1x 1 6x 2 2y 5 x(x 1 3) 1 5(x 1 3)y 5 x(3x 2 1) 1 2(3x 2 1)What did Juan do at this step?What did Juan do at this step?21 62 (23)22 3Factor GCFFactor GCF3. What is Juan’s final equation for each problem?OTT X 13A. y 5 x(x 1 3) 1 5(x 1 3)YIB. y 5 x(3x 2 1) 1 2(3x 2 1)y5Cx 124. Use a generic rectangle to factor each of Juan’s equations.A.B.x23x2152231

Unit 9More with Quadratics—Factored FormLesson 15Factor by GroupingPractice Problems—Factor Trinomials by Groupingca5. y 51x2 1 7x 1 1031Xyac 5factors of ac:yfactors of ac:235.255Xt2y41 CX 114CXt2 CX 15J7. y 5 3x2 1 5x 1 2yac 5X X 5 t2 X 15y5106. y 5 4x2 1 5x 1 13ac 521 13 22factors of ac:3 2 1103 125X 11238. y 5 5x2 1 12x 1 42E53O46ac 50factors of ac:571

572Module 4 Quadratic FunctionsExploration 2—Looking for the GCFIn Lesson 13, we saw that many trinomials have a GCF. We can still use Juan’s method for thesetrinomials, but we’ll factor out the GCF first.9. Factor y 5 2x2 1 8x 1 8.First, we’ll factor out the GCF of 2x2 1 8x 1 8 to get y 5 2(x2 1 4x 1 4).Then use factor by grouping on the trinomial in the parenthesis.Focus on x2 1 4x 1 4y 5 2(x2 1 4x 1 4)ac 5 4factors of ac:Reflection10. Melissa said you don’t have to factor out the GCF right away. Melissa’s work is shown below.y 5 2x2 1 8x 1 8ac 5 16y 5 2x2 1 4x 1 4x 1 8factors of ac:y 5 2x(x 1 2) 1 4(x 1 2)1 16y 5 (x 1 2)(2x 1 4)2 8y 5 2(x 1 2)(x 1 2)4 4What are the advantages to factoring out the GCF right away? What are the advantages to waitingto factor out the GCF?

Unit 9More with Quadratics—Factored FormLesson 15Factor by Grouping573Practice Problems—Looking for the GCF11. y 5 3x2 1 12x 1 9ac 5sixfactors of ac:9 36 31y 41y 42X 3ac 5factors of ac:113. y 5 4x2 1 4x 2 48212. y 5 24x2 1 2x 2 212x14ac 5factors of ac:314. y 5 4x2 1 6x 2 18y2 22393Y 2factors of ac:2X X 32yac 512Gx33 X133 X 1315. Discussion What are the advantages to factoring by grouping (Juan’s method)? What are theadvantages to the generic rectangle method?

574Module 4 Quadratic FunctionsaZb2Exploration 3—Graphing Quadratic Equations Revisitedca b catbDetermine the key features of each quadratic and then graph the parabola.a IbX 13y5(Standard Form)( X1517. y 5 3x2 2 12342) Factored Formy53Key features:,54baaaa1Standard Form42 X 2 Factored Formdiff of two squares2x-intercepts:15y-intercept:3Key features:3x-intercepts:vertex: (3CX 05 128 C 4516. y 5 x2 1 8x 1 1512y-intercept:,I)vertex: (4OlsoUH2,,12)

Unit 9More with Quadratics—Factored FormNAME:Lesson 15PERIOD:Factor by GroupingDATE:Homework Problem SetUse Juan’s method to factor each equation.1. y 5 x2 1 14x 1 24ac 2. y 5 3x2 1 11x 1 6factors of ac:3. y 5 2x2 1 11x 1 14ac factors of ac:4. y 5 5x2 1 17x 1 6ac ac factors of ac:factors of ac:575

576Module 4 Quadratic FunctionsFactor each equation in Problems 5–10. Use any method.5. y 5 x2 1 8x 1 76. y 5 x2 2 11x 1 107. y 5 x2 1 3x 2 548. y 5 10x2 1 13x 2 309. y 5 12x2 2 43x 1 3510. y 5 14x2 1 19x 2 3

Unit 9More with Quadratics—Factored FormLesson 15Factor by GroupingFactor each equation in Problems 11–16.11. y 5 3x2 1 16x 2 3512. y 5 10x2 1 71x 2 7213. y 5 4x2 1 11x 2 2014. y 5 3x2 2 28x 1 915. y 5 7x2 1 6x 2 116. y 5 3x2 2 10x 1 7577

Discussion What are the advantages to factoring by grouping (Juan’s method)? What are the advantages to the generic rectangle method? ac 5 factors of ac: ac 5 factors of ac: ac 5 factors of ac: ac 5 factors of ac: six 9 36 31 1 y 41 2 2 12 y 2 2 2