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Unit 3A Quadratic Functions – Factoring and Solving Table of Contents Title Page # Glossary.……..………………………………………………………….……… 2 Lesson 3A-1 Intro Quadratics and GCF…………….……………………….3 Lesson 3A-2 Factoring – The GCF Method..……………………….………. 5 Lesson 3A-3 Factoring – a = 1.………………………………………………. 6 Lesson 3A-4 Factoring – a ≠ 1.………………………………………………. 9 Lesson 3A-5 Factoring – Difference of square.……………………..……12 Lesson 3A-6 Factoring – Perfect Square Trinomial.…………………… 13 Lesson 3A-7 Solving Quadratics – Square Roots ………………………15 Lesson 3A-8 Solving Quadratics – Factoring……………………………16 Lesson 3A-9 Solving Quadratics – Completing the Square……………18 Lesson 3A-10 Solving Quadratics – Quadratic Formula.……………… 20 This packet belongs to: _________________________________ Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving) 1 Glossary Complete Factorization over the Integers: Completing the square: Difference of two squares: Parabola: Perfect square trinomial: Quadratic equation: Quadratic function: Root: Standard form of a quadratic function: Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving) 2 Write your Questions here! Lesson 3A – 1: Intro to Quadratics and Greatest Common Factor Learning Target: I can find the greatest common factor of a set of algebraic terms. A.SSE.3 Vocabulary: - Quadratic equation - Quadratic function - Parabola Guided Notes: Which of the following are quadratic? A) y = 3x + 2 B) y = -2x2 + x - 1 C) y = 3x4 - 3x2 + 2 The graph of the most basic quadratic function, 𝑓(𝑥) = 𝑥 2 , looks like this: The first tool you will need in your mathematician’s tool belt is finding the Greatest Common Factor (GCF) of a set of terms. A __________ is anything that is being multiplied by something else in math. There are 3 simple steps to finding the GCF: 1. Find the _______________ of each term. 2. Identify the ___________ that are present in each number. 3. ___________ the common factors together. Example 1: Find the GCF: 12v2 and 18v Example 2: Find the GCF: 2x2, 6x and 8 Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving) 3 Write your Questions here! 3A-1 Practice Identify whether or not the following are quadratic expressions. Justify your decision. 1) 𝑦 = 4𝑥 2 − 7𝑥 + 3 3) 𝑦 = 2𝑥 + 7𝑥 − 4 2) 𝑦 = 7𝑥 2 − 3 4) 𝑦 = 3𝑥 2 5) 𝑦 = 4𝑥 4 − 3𝑥 2 + 9 Find the greatest common factor of the given terms for each problem. 6. −17𝑥 2 19x 23x 3𝑥 7. 35𝑥 2 -15x 45 3. 3𝑥 2 7𝑥 8. −21𝑥 2 -36x -81 4. 32𝑥 2 8𝑥 9. −12𝑥 2 -28x -40 5. 14𝑥 2 21𝑥 10. 42𝑥 2 -108𝑥 2 240𝑥 2 1. 36 18 2. 27𝑥 2 42 63 Application 3A-1 There is not an application for this lesson. Please begin preparing for your mastery check. Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving) 4 Write your Questions here! Lesson 3A – 2: Factoring – The GCF Method Learning Target: I can factor the GCF from a polynomial. A.SSE.3a Expression 4x2y (x-2)(x+3) Vocabulary: - Complete Factorization over the Integers Factors 4, x2 (x and x), and y (x-2) and (x+3) YOU TRY: What are the factors of: -2x2 _______ and _________ x(x-1)(2x+3) _________ and __________ and ___________ ____________ is the process of breaking down algebraic expressions into the most simplified form of all of its factors. There are many strategies to factoring, and we will learn several of them in this unit. No matter what strategy we use, we will ALWAYS first look for a _______, greatest common factor, and __________ that out before proceeding with other strategies. GCF Method: Example 1: Factor 2x + 6x2 ______ ( ______ ( 1. Find the _______ of all terms. ) + 2. Write down your ________, then a set of parenthesis. ) This is the “factored form”. 3. To find out what goes in the parenthesis, you ______________ _______________________! FACTOR THESE EXAMPLES: Ex. 2 12a2 – 18a Ex. 3 -15 + 25m Ex. 4 4x3 – 24x2 + 12x Ex. 5 9x2y - 27xy2 Ex. 6 –3x2 – 7x – 9 EX. 7 49 – 7x Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving) 5 Write your Questions here! 3A-2 Practice Factor the following quadratic expressions. 1. 4𝑣 2 + 8𝑣 6. 9𝑥 3 − 3𝑥 + 12 2. 𝑥 2 − 3𝑥 7. −4𝑥 2 − 16𝑥 − 20 3. 2𝑏 2 − 18𝑏 8. −12𝑥 2 𝑦 − 18𝑥𝑦 − 28𝑥𝑦 2 4.6𝑏 2 + 12𝑏 2 9. 48𝑥 2 − 16𝑥 + 40 5. 8x+4 10. 64x-8 Application 3A-2 There is not an application for this lesson. Please begin preparing for your mastery check. Lesson 3A – 3: Factoring when a = 1 Learning Target: I can factor quadratic trinomials of the form 𝒙𝟐 + 𝒃𝒙 + 𝒄. A.SSE.3a We now need to learn a strategy for factoring polynomials that have 3 terms, ________________. After looking for and factoring out a ________, we will see if we can factor more using this method. WARMUP EXERCISE: 1. Think of 2 numbers whose sum = 8 and whose product = 12: _________ 2. Think of 2 numbers whose sum = 5 and whose product = -14:_________ 3. Think of 2 numbers whose sum = 16 and whose product = 15:________ 4. Think of 2 numbers whose sum = -14 and whose product = 40:________ Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving) 6 Write your Questions here! 5. MULTIPLY a) (x + 6)(x + 3) b) (x + 7)(x - 2) What do you notice about the sum of the constants in each factor? What do you notice about the product of the constants in each factor? FACTORING A TRINOMIAL is kind of like multiplying binomials in reverse!!! Example 1 x2 + 6x + 8 Factors of (x ) )( x Example 3 y2 – 10y + 16 Factors of Sum Sum Example 2 x2 + 12x – 45 Example 4 x2 – 3x – 28 Factors of Sum Factors of Sum Note: Not every trinomial is factorable. If you have tried every combination and none of them work, write “non factorable” and move on! 3A-3 Practice 1. 𝑚2 + 16𝑚 + 60 3. 𝑎2 + 3𝑎 − 40 5. 𝑥 2 + 𝑥 − 90 2. 𝑚2 − 10𝑚 + 24 4. 𝑣 2 + 13𝑣 + 40 6. 𝑟 2 − 12𝑚 + 20 Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving) 7 Write your Questions here! 7. 𝑥 2 + 10𝑥 + 21 12. 𝑛2 + 12𝑛 + 27 17. 𝑝2 − 2𝑝 + 48 8. 𝑟 2 + 8𝑟 + 12 13. 𝑎2 + 3𝑎 − 4 18. 𝑛2 + 3𝑛 + 10 9. 𝑥 2 − 3𝑥 − 54 14. 𝑝2 − 2𝑝 + 48 19. 𝑚2 − 7𝑚 + 6 10. 𝑛2 − 14𝑛 + 48 15. 𝑛2 − 16𝑛 + 60 20. 𝑛2 − 17𝑛 + 70 11. 𝑟 2 − 13𝑟 + 42 16. 𝑥 2 − 6𝑥 − 7 21. 𝑛2 + 17𝑛 + 70 Application 3A-3 There is not an application for this lesson. Please begin preparing for your mastery check. Lesson 3A – 4: Factoring when a≠1 Learning Target: I can factor quadratic trinomials of the form 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄. A.SSE.3a If the leading coefficient is a number other than 1, we need to consider how this will affect our first term in each parenthesis: Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving) 8 Write your Questions here! Think about where our numbers come from when we multiply binomials. Example 1 5x2 + 11x + 2 Factors of Factors of Sum of outer and inner products Example 3 2x2 + 4x – 6 Factors of Factors of Example 2 4x2 – 9x + 5 Factors of Factors of Example 4 Sum of outer and inner products Factors of Sum of outer and inner products 6n2 – 11n – 10 Factors of Sum of outer and inner products There is an additional consideration when factoring trinomials in which “a” is negative. When the leading coefficient is negative, ___________________ from each term before using other factoring methods. Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving) 9 Write your Questions here! Example 5 -6x2 + 13x – 2 Factors of Factors of Sum of outer and inner products 3A-4 Practice Factor the following quadratic expressions. 1. 3𝑥 2 − 5𝑥 − 28 6. 3𝑥 2 − 14𝑥 − 5 2. 7𝑥 2 + 60𝑥 − 27 7. 6𝑥 2 − 32𝑥 + 40 3. −5𝑥 2 − 16𝑥 − 12 8. 35𝑥 2 + 130𝑥 − 225 4. 6𝑥 2 + 𝑥 − 1 9. 3𝑥 2 + 17𝑥 + 24 5. 5𝑥 2 + 37𝑥 + 42 10. −7𝑥 2 + 27𝑥 − 18 Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving) 10 11. 5𝑥 2 + 46𝑥 + 48 16. 3𝑥 2 − 28𝑥 − 20 12. 7𝑥 2 + 67𝑥 − 30 17. 27𝑥 2 − 240𝑥 + 192 13. 7𝑥 2 − 2𝑥 + 14 18. −10𝑥 2 + 63𝑥 + 49 Write your Questions here! 14. 9𝑥 2 + 9𝑥 − 28 15. 3𝑥 2 + 19𝑥 − 14 19. 27𝑥 2 + 9𝑥 − 60 20. 18𝑥 2 − 44𝑥 − 30 Application 3A-4 There is not an application for this lesson. Please begin preparing for your mastery check. Vocabulary: - Difference of Two Squares Lesson 3A – 5: Factoring – Difference of Squares Learning Target: I can factor the difference of two squares. A.SSE.2 Sometimes patterns can be used to help the factoring process along. The next couple of lessons are going to focus on some special cases of trinomials. The difference of 2 squares has the form _________________. A polynomial is a difference of squares if: There are ____ terms, one ____________ from the other. Both terms are _____________________.* * A term is a perfect square when it is the product of a number and itself. Examples: 4, x2, 9y2 Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving) 11 Write your Questions here! Example 1: 8𝑥 2 − 18 Example 3: 𝑥 6 − 7𝑦 2 Example 2: 9𝑝4 − 16𝑞 2 Example 4: 1 − 25𝑘 8 3A-5 Practice Determine whether the binomial is a difference of squares. If so, factor. If not, explain why it is not. 1. 1 − 4𝑥 2 6. 81𝑥 2 − 1 2. 2𝑝8 − 18𝑞 6 7. 4𝑥 4 − 9𝑦 2 3. 16𝑥 2 − 4𝑦 5 8. 𝑥 8 − 50 4. 𝑥 2 − 81 9. 𝑥 6 − 9 5. 𝑠 2 − 42 10. 2𝑚2 − 8𝑟 6 Application 3A-5 There is not an application for this lesson. Please begin preparing for your mastery check. Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving) 12 Write your Questions here! Lesson 3A – 6: Factoring – Perfect-Square Trinomials Learning Target: I can factor perfect-square trinomials. A.SSE.2 Vocabulary: - Perfect Square Trinomials Another special case for factoring is when you are given a perfect-square trinomial. A trinomial is a perfect square if: The ________ and ________terms are ________________. The ____________ term is ______________ one factor from the _____________ and one factor from the _____________. Example 1: 𝑥 2 + 12𝑥 + 36 Factors of Example 3: 8𝑥 2 − 40𝑥 + 50 Factors of Sum Sum Example 2: 4𝑥 2 − 24𝑥 + 9 Factors of Sum Notice the pattern! 3A-6 Practice Determine whether each trinomial is a perfect square. If so, factor. If not, explain. 1. 4𝑥 2 − 12𝑥 + 9 3. 𝑥 2 + 4𝑥 + 4 2. 𝑥 2 + 9𝑥 + 16 4. 4𝑥 2 − 14𝑥 + 49 Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving) 13 Write your Questions here! 5. 9𝑥 2 − 6𝑥 + 4 8. 2𝑥 2 − 8𝑥 + 8 6. 9𝑥 2 − 12𝑥 + 4 9. 𝑥 2 − 6𝑥 + 9 7. 𝑥 2 − 4𝑥 − 4 10. 𝑥 2 − 6𝑥 − 9 Application 3A-6 There is not an application for this lesson. Please begin preparing for your mastery check. Lesson 3A – 7: Solving Quadratics – Square Roots Learning Target: I can solve simple quadratic equations by taking square roots. Vocabulary: A.REI.4b - Root When solving linear equations in Unit 2 we relied heavily on inverse operations. This will remain true when solving quadratics. Recall from Unit 1 that the inverse operation of squaring a number is taking the _____________________. So √𝑥 2 = 𝑥, √4 = 2 and √4𝑥 2 = 2𝑥 We will now use this fact to solve simple quadratic equations. Example 1 x2 = 49 Not every quadratic equation will be a one step solution. At times, we have to work to get the x2 ______________ before taking the ___________________. Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving) 14 Write your Questions here! Steps for Solving Quadratic Equations Using Square Roots 1. Simplify each side of the equation by __________ and ___________________. 2. Move all ____________ to one side of the equation. 2 3. Get x by itself using ___________________. 4. Take the __________________ of both sides of the equation. 5. There will ALWAYS be a ________________ AND a __________________. Example 3: 2(𝑥 2 − 5) = −𝑥 2 − 1 Example 2: 2𝑥 2 − 338 = 0 1 Example 5: (𝑥 + 6)2 = 49 Example 4: 2 𝑥 2 + 3 = 12 3A-7 Practice Solve each quadratic equation. 4. (𝑥 + 4)2 = 121 7. (𝑥 + 3)2 + 6 = 18 2. 3𝑥 2 − 7 = 47 5. (2𝑥 − 3)2 = 9 8. (2𝑥 + 6)2 − 8 = 24 3. 𝑥 2 + 11 = 16 6. (𝑥 − 7)2 = 99 9. 𝑥 2 − 21 = 5 1. 𝑥 2 + 4 = 29 Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving) 15 Write your Questions here! 10. 3(𝑥 + 4)2 = 9 11. 3(𝑥 2 − 4) = 2𝑥 2 − 1 2 12. 5 𝑥 2 − 3 = 7 Application 3A-7 Collect 3A-7 Solving Quadratics Application sheet from the Algebra Embassy. Lesson 3A – 8: Solving Quadratics – Factoring Learning Target: I can solve quadratic equations by factoring. A.REI.4b Isolating the x2 term is great when that is the only x term in the equation. Unfortunately for us, most quadratic functions are not that simple. For this reason, we need to look at several more tools for solving quadratic equations. This method requires use of the factoring skills we mastered in lessons 2 through 6. Steps for Solving Quadratic Equations Using Factoring 1. Make sure each equation is equal to ________. 2. Check to see if there is a _______ that can be factored out. 3. _________. 4. Set each factor equal to ________. 5. ________ each equation. Example 1: Solve: x2 + 3x – 18 = 0 Example 2: Solve: 4x2 + 20x – 24 = 0 Example 3: Solve: 5x2 + 10x = 0 Example 4: Solve: 4x2 + 12x + 9 = f(x) 3A-8 Practice Find the zeros of each function by factoring. 1. x2 x 2. x2 x Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving) 3. 𝑔(𝑥) = 𝑥 2 + 𝑥 − 20 16 Write your Questions here! Solve each quadratic function by factoring. 4. 9x2 4 = 12x 5. 16x2 9 6. 𝒚𝟐 + 𝟕𝒚 + 𝟏𝟐 = 𝟎 7.𝑥 2 − 7𝑥 − 44 = 0 9. 𝑧 2 − 12𝑧 + 27 = 0 10. 4𝑘 2 − 35𝑘 − 9 = 0 8. 𝑥 2 − 12𝑥 = −36 11.3x2 x=0 12. −2𝑥 2 − 5𝑥 + 10 = −2 Application 3A-8 Collect 3A-8 Solving Quadratics Application sheet from the Algebra Embassy. Lesson 3A – 9: Solving Quadratics – Completing the Square Learning Target: I can solve quadratic equations by completing the square. A.REI.4a & A.REI.4b Many quadratic equations contain expressions that are not _______ __________ AND cannot be easily ____________. These equations require us to learn yet another method of solving. Vocabulary: - Complete the Square In this lesson, we are going to be using algebraic properties to rewrite any quadratic expression as a perfect square. Steps for Solving Quadratic Equations by Completing the Square 1. 2. 3. Move the ___________ to the other side. Find the number that “completes the square” using the formula, ______ that number to ______ sides! 4. 5. 6. _________. NOTE: (𝑥 + 2) = # Take the ____________ of each side. Solve for x. 𝑏 2 Just cut the middle number in half and square it! Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving) 17 Write your Questions here! Example 1: Solve: 𝑥 2 = 6𝑥 + 4 Example 3: Solve: 4𝑥 2 + 32𝑥 + 16 = 0 Example 2: Solve: 𝑥 2 − 2𝑥 − 1 = 0 Example 4: Solve: 𝑥 2 + 4𝑥 − 4 = 0 3A-9 Practice Solve each equation. 1. 𝑥 2 + 8𝑥 − 20 = 0 3. 3𝑥 2 + 12𝑥 − 12 = 0 2. 𝑥 2 + 6𝑥 − 18 = 0 4. 4𝑥 2 − 8𝑥 − 4 = 0 Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving) 18 Write your Questions here! 5. 𝑥 2 − 8𝑥 − 36 = 0 8. 2𝑥 2 + 8𝑥 − 15 = 0 6. 𝑥 2 + 6𝑥 = 16 9. 𝑥 2 + 2𝑥 − 3 = 0 7. 2𝑥 2 = 3𝑥 + 4 10. 𝑥 2 = 6𝑥 + 4 Application 3A-9 There is not an application for this lesson. Please begin preparing for your mastery check. Lesson 3A – 10: Solving Quadratics – The Quadratic Formula Learning Target: I can solve quadratic equations using the quadratic formula. A.REI.4b Vocabulary: - Standard Form of a Quadratic Equation We have learned several methods for solving quadratic equations now. Now its time to show you one last method before moving on. This final method can be used on ________ quadratic equation as long as it is written in ________ _______. Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving) 19 Write your Questions here! When we say standard form of a quadratic, we mean it looks like this: By completing the square on the standard form of a quadratic equation, you can derive the Quadratic Formula. The quadratic formula, in case you can’t read the unicorn’s handwriting (or Mr. Thomaswick’s), is: This IS on the formula sheet! Steps for Solving Quadratic Equations by Using the Quadratic Formula 1. Write the equation in _______________. 2. Determine the value for _________________. 3. Plug these numbers into the ____________________. 4. Simplify. Example 1: Example 2: Find the zeros: 𝑥 2 + 9𝑥 − 14 = 0 Solve: −𝑥 2 + 3𝑥 = −4 Although we have come to expect two solutions for every quadratic formula, there are scenarios where this is not the case. It is possible for a quadratic equation to have only _____________ or even _____________. Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving) 20 Write your Questions here! Example 3: Solve: 2𝑥 2 − 12𝑥 = −18 Example 4: Solve: 𝑥 2 − 6𝑥 + 11 = 0 There will be one real solution when: There will be no real solutions when: 3A-10 Practice Solve each equation using the Quadratic Formula. 3. 2𝑥 2 − 16𝑥 = −32 2 1. 2𝑥 + 5 = 2𝑥 2. 2𝑥 2 − 3𝑥 = 8 4. 3𝑥 2 + 3 = 10𝑥 Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving) 21 Write your Questions here! 5. 𝑥 2 = −6𝑥 8. 𝑥 2 − 5 = 𝑥 6. 𝑥(𝑥 − 3) = 4 9. −3𝑥 2 + 4𝑥 = 4 7. 7𝑥 2 − 3 = 0 10. 4𝑥 2 − 28𝑥 = −49 Find the zeros of each function using the Quadratic Formula. 11. 𝑓(𝑥) = 𝑥 2 + 7𝑥 + 10 13. ℎ(𝑥) = 3𝑥 2 − 5𝑥 12. 𝑔(𝑥) = 3𝑥 2 − 4𝑥 − 1 14. 𝑟(𝑥) = 𝑥 2 + 6𝑥 + 12 Application 3A-10 Collect 3A-10 Solving Quadratics Application sheet from the Algebra Embassy. Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving) 22