# CP Algebra 2 Unit 2-1 Factoring And Solving Quadratics

1y ago
48 Views
2.55 MB
18 Pages
Last View : 2m ago
Transcription

CP Algebra 2Unit 2-1Factoring and Solving QuadraticsName:Period:1

Unit 2-1Factoring and Solving QuadraticsLearning Targets:1. I can factor using GCF.2. I can factor by grouping.FactoringQuadraticExpressions3. I can factor when a is one.4. I can factor when a is not equal to one.5. I can factor perfect square trinomials.6. I can factor using difference of squares.7. I can solve by factoring.8. I can solve by taking the square root.9. I can perform operations with imaginary numbers.SolvingQuadraticEquations10. I can solve by completing the square.11. I can solve equations using the quadratic formula (with rationalized denominators).12. I can use the discriminant to determine the number and type of solutions.13. I can write quadratic equations given the real solutions.2

Factoring BasicsDate:After this lesson and practice, I will be able to ! factor using GCF. (LT 1)factor by grouping. (LT 2)! factor when a is one. (LT 3)factor when a is not one. (LT hen we’ve modeled real-life situations with functions before, it’s been withfunctions (example: at-bats vs. runs in baseball, GB vs. cost of a data plan, etc.).Suppose we wanted to determine how high you could throw a tennis ball? What would the path of theball look like?As you can see, in order to model situations such as these, we require a function family that is not. For situations such as these, we can use the family offunctions.(Throughout the year, we will be studying several different families of functions. We will learn how tograph and solve each type of function. We’re going to start Quadratics with solving!)Quadratic Function: A function that can be written in the formwhere .Standard Form of a Quadratic Function:You might recognize these from previous math classes! Try to solve a quadratic using the usual methodwe used to solve linear equations 𝑥 ! 8𝑥 7 5It turns out we’ll need some new (and some not so new) skills in order to solve quadratic equations.One of those essential skills is factoring.3

To factor completely is to write a polynomial as a product of polynomialswith integer coefficient. Make sure your final answers contain only unfactorable pieces.Factor Using GCF (LT 1)This is the strategy you should use when factoring any expression.Example 1: Factor completely.A) 14 x 2 21xB) 80n7 8n 4 16n3C) 2x 3 y 2 yWhile the following technique does not necessarily apply exclusively to expressions,it will provide us the skills necessary to factor quadratics.Factor By Grouping (LT 2)Factor by grouping - pairing together binomials that have common monomial factors:Example 2: Factor completely.A) 5x 3 35x 2 2x 14B) 3n3 18n2 2n 12Example 3: Factor completely.A) x3 6 x 2 7 x 42B) 9r 3 15r 2 3r 5Factoring when a is one (LT 3)You probably remember doing this kind of factoring in Algebra 1 and/or Geometry.Example 4: Factor completely.A) x 2 11x 10B) x 2 4 x 21C) x3 2 x 2 15 x4

Example 5: Factor completely.A) x 2 17 x 72B) x 2 8 x 12C) 4 x 2 20 x 24D) 6 x5 30 x 4 36 x3Factoring when a is not one (LT 4)Example 6: Factor completely.A) 7 x 2 43x 6B) 12 x 2 37 x 11Example 7: Factor completely.A) 3x 2 11x 8B) 7 x 2 37 x 30C) 18 x 2 90 x 100D) 6 x 2 21x 275

More FactoringDate:After this lesson and practice, I will be able to ! factor perfect square trinomials. (LT 5)! factor using difference of squares. (LT hen factoring, there are types of polynomials that are so common that it is helpful to memorizefor how to factor them quickly.Factoring Perfect Square Trinomials (LT 5)Let’s start by factoring x 2 14 x 49 using our familiar methods:Observe:- The last term is a . The first term is a .- The middle term is the product of the quantities being squared in the andterms.- Algebraically:Example 1: Determine if the quadratic is a perfect square trinomial.A) x 2 50 x 100B) x 2 6 x 9C) 16 x 2 24 x 9Example 2: Factor completely the perfect square trinomials from above.Example 3: Factor completely using the perfect square trinomial pattern.A) 9 x 2 24 x 16C) 25r 2 30r 96

Factoring Using Difference of Two Squares (LT 6)Let’s start by factoring x 2 25 using our familiar methods:Observe:- The last term is a .The first term is a .- The two factors are identical except for the .- Algebraically:Example 4: Factor completely.A) x 2 121B) 4 x 2 49C) x 2 4Example 5: Factor completely.A) 16 x 2 9B) 27m2 75!C) 64𝑥 817

Solving Quadratic EquationsDate:After this lesson and practice, I will be able to ! solve by factoring. (LT 7)! solve by taking the square root. (LT t’s finally time to start solving! The first way of solving combines factoring and the zero productproperty. Here’s what that property states:If ab 0, then .You’ll first want to get your standard form equation set equal to . Then get it in factored form.To get an equation in factored form, just write it in standard form, then factor! Once the quadratic equation is in factored form, use the zero product property. Essentially to do this,you just try to find the value for x that makes the equation .Example 1: Solve by factoring.A)( x 4)(3x 2) 0B)x 2 x 12 0C)2n2 13n 23 3D)15x 6 12x 5 105x 4 84x 3 0E)2 x 2 12 x 80F)7r 2 20r 36 48

Sometimes you won’t be able to factor as nicely as you did above. What’s different about the equationsin Example 2?For these, you can solve by getting x2 alone, and then take the .Example 2: Solve. If necessary, simplify all radicals and give the radical rounded to 2 decimal places.A)5x 2 180 0B)3x 2 24C)4x 2 25 0D)x2 E)(𝑥 4)! 91 04Always make sure your radical answers are always simplified! Here’s a quick review A)180B)252C)54 p3D)481E)7169

Imaginary NumbersDate:After this lesson and practice, I will be able to ! perform operations with imaginary numbers. (LT arm Up: Solve the equation 𝑥 ! 121 0This quadratic equation has real solutions. However, there do exist solutions outside thenumber system. Let’s review that number system An imaginary number is any number whose square is -1. We use the letter 𝒊 to represent imaginarynumbers.Here’s the most important thing: 𝑖 ! 1That means that 𝑖 Here’s a quick example: 4 Example 1: Simplify each radical completely.A) 5B)! 36C)! 12D) 49E)5 54F)! 12010

So that’s imaginary numbers. When you combine a real number with an imaginary number, you get acomplex number. A complex number is a real number plus an imaginary number. The standard formof complex numbers looks like this:Example 2: Write each expression as a complex number in standard form.A) (12 11i ) ( 8 3i)B) (15 9i) (24 9i)Example 3: Write each expression as a complex number in standard form.A) 4i ( 6 i )B) (9 2i)( 4 7i)C) (4 i)(4 i)Example 4: Write each expression as a complex number in standard form.A) (9 i) ( 6 7i)B) 4 (1 i) (2 3i)C) (2 7i)(2 7i)D) 5i (8 9i )F) (3 4i) 2E) ( 8 2i)(4 7i)Let’s return to the equation from the beginning of class: !x 2 121 0 . You are now prepared to solveequations to find all solutions, whether they are or .Example 5: Solve !4x 2 72 011

Completing the SquareDate:After this lesson and practice, I will be able to ! solve by completing the square. (LT Do you remember perfect square trinomials? We learned how to factor them earlier in the unit. Let’suse that to help us solve an equation!Example 1: Solve by finding square roots x 2 8 x 16 25That’s a pretty great way to solve quadratic equations, but unfortunately, we won’t always have niceperfect square trinomials in our equations. Luckily for us, there’s a way to create perfect squaretrinomials. It’s called completing the square In your group, set up the following scenarios with your papers and fill in the table accordingly.EquationNumber of 1-tiles needed toExpression written as a squarecomplete the square𝑥 ! 2𝑥 𝑥 ! 4𝑥 𝑥 ! 6𝑥 𝑥 ! 8𝑥 𝑥 ! 10𝑥 Answer these questions with your group. Consider the general statement 𝑥 ! 𝑏𝑥 𝑐 (𝑥 𝑑)! .a. How is d related to b?b. How is c related to d?c. How can you obtain the numbers in the second column of the table directly from the coefficientsof x in the expressions from the first column?12

Let’s practice completing the square before we solve Example 2: Find the value of c that makes x 2 bx a perfect square trinomial. Then write the expressionas the square of a binomial.A) x 2 16 x cD) x 2 6 x cB) x 2 22 x cC) x 2 9 x cE) x 2 15 x cSolving quadratic equations of the form x 2 bx c 0 by completing the squareExample 3: Solve by completing the square:A) x 2 12 x 4 0B) x 2 8x 36 0Example 4: Solve each equation by completing the square.1.2!x 10x 8 02.2!x 4x 4 013

3.2!x 4x 1 04.2!x 6x 12 0Solving quadratic equations of the form ax 2 bx c 0, a 1 by completing the square.Example 5: Solve by completing the square:A) 2 x 2 8 x 14 0B) 3x 2 36 x 150 0C) 2x 2 x 614

Fill out the table below with your group (split up the work if you want).EquationValue of DiscriminantNumber and Type ofSolutions2!8x 2x 3 02!2x 8x 12 02!x 10x 25 0The discriminants of different quadratic equations are given below. Look for patterns in the table above,then predict how many real solutions each quadratic equation has.DiscriminantNumber of Real Solutions-5160Summary:Value ofDiscriminantNumber and Types of SolutionsPositiveZeroNegativeExample 2: Find the discriminant and give the number and type of solutions. Do not solve.A) x 2 10 x 23 0B) x 2 10 x 25 0C) x 2 10 x 27 016

We now know how to find the solutions of a quadratic equation. Next we will do the reverse: find thequadratic equation given the solutions. We will use this quite a bit in the next unit.Factored Form – , given r1 and r2 are zeros.!!Example 1: Write a quadratic equation in factored form with the following solutions:1 and -7A)2 and -4B)4C)! 2Example 2: Write a quadratic equation in factored form with the following solutions.A)-1 and 3B)-4 and 0Example 3: Write a quadratic equation in STANDARD form with the following solutions.A)-2 and 2B)-5 and 117

18

Unit 2-1 Factoring and Solving Quadratics Learning Targets: Factoring Quadratic Expressions 1. I can factor using GCF. 2. I can factor by grouping. 3. I can factor when a is one. 4. I can factor when a is not equal to one. 5. I can factor perfect square trinomials. 6. I can factor using difference of squares. Solving Quadratic Equations 7.

Related Documents:

Robert Gerver, Ph.D. North Shore High School 450 Glen Cove Avenue Glen Head, NY 11545 gerverr@northshoreschools.org Rob has been teaching at . Algebra 1 Financial Algebra Geometry Algebra 2 Algebra 1 Geometry Financial Algebra Algebra 2 Algebra 1 Geometry Algebra 2 Financial Algebra ! Concurrently with Geometry, Algebra 2, or Precalculus

So you can help us find X Teacher/Class Room Pre-Algebra C-20 Mrs. Hernandez Pre-Algebra C-14 . Kalscheur Accelerated Math C-15 Mrs. Khan Honors Algebra 2 Honors Geometry A-21 Mrs. King Math 7 Algebra 1 Honors Algebra 1 C-19 Mrs. Looft Honors Algebra C-16 Mr. Marsh Algebra 1 Honors Geometry A-24 Mrs. Powers Honors Pre-Algebra C-18 Mr. Sellaro .

McDougal Littell Algebra I 2004 McDougal Littell Algebra I: Concepts and Skills 2004 Prentice Hall Algebra I, Virginia Edition 2006 Algebra I (continued) Prentice Hall Algebra I, Virginia Edition Interactive Textbook 2006 CORD Communications, Inc. Algebra I 2004 Glencoe/McGraw Hill Algebra: Concepts and Applications, Volumes 1 and 2 2005

Feb 9, 2021 — Algebra 1 Unit 2 7 Unit 2 Worksheet 5 Write an algebraic word problem for the . [GET] Algebra 1 Unit 2 Part 1 Test Review Answers HOT. algebra 1 . 2 Linear Equations, Inequalities, and Systems. In this unit, students expand and deepen their prior understanding of expressions, equations, and inequalities. Geometry Unit 2 .

Trigonometry Unit 4 Unit 4 WB Unit 4 Unit 4 5 Free Particle Interactions: Weight and Friction Unit 5 Unit 5 ZA-Chapter 3 pp. 39-57 pp. 103-106 WB Unit 5 Unit 5 6 Constant Force Particle: Acceleration Unit 6 Unit 6 and ZA-Chapter 3 pp. 57-72 WB Unit 6 Parts C&B 6 Constant Force Particle: Acceleration Unit 6 Unit 6 and WB Unit 6 Unit 6

Boolean algebra is a system of mathematical logic. Any complex logic can be expressed by Boolean function. Boolean algebra is governed by certain rules and laws. Boolean algebra is different from ordinary algebra & binary number system. In ordinary algebra; A A 2A and AA A2, here

Algebra 1 Algebra 1 Practice TestPractice TestPractice Test 3. Solve the following inequality: -20 4 – 2x A. 8 x C. 12 x B. 8 x D. 12 x 4. Which inequality is graphed ? . Algebra 1 Algebra 1 Practice TestPractice TestPractice Test 5. Which equation is represented on the graph? A. y x2 13x 36 B. y x2-13x 36

all digital components available for the Pearson Algebra 1, Geometry, Algebra 2 Common Core Edition 2015. This includes access to the . Screening Test Benchmark Test . Algebra 1 Geometry Algebra 2 Easy access to all textbook answers and solutions

Title: Prentice Hall Algebra 1, Geometry, and Algebra 2 (Florida) : Program Components Author: Pearson Subject: Prentice Hall Algebra 1, Geometry, and Algebra 2 (Florida)

Algebra 2 - Midterm Exam Review The Algebra 2 Midterm Exam must be taken by ALL Algebra 2 students. An exemption pass may be used to exempt the score for the Algebra 2 Midterm Exam. It should be presented to your teacher prior to taking the exam. The Algebra 2 Midterm Exam will consist of 30 multiple choice questions.

MTH308A: Summit Algebra II MTH308B: Summit Algebra II MTH309A: Summit Honors Algebra II MTH309B: Summit Honors Algebra II . MTH113A: Pre-Algebra MTH113B: Pre-Algebra CS Essential Skills (Study Skills) [Elective] CS Essential Skills (Study Skills) [Elective] 11 WAVA Omak H

Algebra 2 Honors Summer Packet 1 Congratulations! You are going to be in Algebra 2 Honors! Here is a packet of pre-algebra/algebra topics that you are expected to know before you start Advanced Algebra Honors. Most problems are non-calculator. To prepare yourself for the year a

Algebra 1: The Florida State Assessments (FSA) includes an End-of-Course exam for Algebra I. Your grade on this exam will constitute 30% of your overall year-long grade in the course. It is required that you pass this test for graduation. Algebra 1A: Students in Algebra

ice cream Unit 9: ice cream ka bio Unit 3: say it again kaa Unit 10: car kakra Unit 3: a little Kofi Unit 5: a name (boy born on Fri.) Koforidua Unit 4: Koforidua kↄ Unit 9: go Kↄ so Unit 7: Go ahead. kↄↄp Unit 9: cup kube Unit 10: coconut Kumase Unit 4: Kumasi Labadi Beach Unit 10: Labadi Beach

CAPE Management of Business Specimen Papers: Unit 1 Paper 01 60 Unit 1 Paper 02 68 Unit 1 Paper 03/2 74 Unit 2 Paper 01 78 Unit 2 Paper 02 86 Unit 2 Paper 03/2 90 CAPE Management of Business Mark Schemes: Unit 1 Paper 01 93 Unit 1 Paper 02 95 Unit 1 Paper 03/2 110 Unit 2 Paper 01 117 Unit 2 Paper 02 119 Unit 2 Paper 03/2 134

v Algebra 1 Activities Page 14-5 Algebra Activity: Theoretical and Experimental Probability . . . . . . .208 14-5 Algebra Activity Recording Sheet . . .209 Algebra 2 .

Prentice Hall Algebra 1 Lesson 10 3 Pearson iBook Algebra 1 Lesson 10.3 Square Roots Dry Erase Activity Quiz Homework 8.2.1 -Add, subtract, multiply, and divide rational numbers (integers, fractions, and terminating decimals) in multi-step problems. Prentice Hall Algebra 1 Lesson 1 2, 1 4, 1 5, and 1 6-Pearson iBook Algebra 1 Lesson 1.2, 1.5 .

ALGEBRA 1 Course Rationale: Algebra 1 is the critical element in secondary mathematics education. Topics introduced in Algebra 1 provide the foundation students require for future success in high school mathematics, critical thinking, and problem solving. The primary goal in Algebra 1 is to help students transfer their

Algebra 1 Algebra 1 Practice TestPractice TestPractice Test Algebra 1 Practice Test Part 1: Directions: For questions 1-20, circle the correct answer on your answer sheet. 1. Solve for x: 2(x 7) – 3(2x-4) -18 A. x 5 B. x 11 C. x -11 D. x -5 2. Which system of equations is represented on the graph? A. y 2x – 2

How it fits into Linear Algebra Lots of “examples” of ranking in linear algebra texts but not many are realistic to students. This was a good way to introduce and work with matrix algebra. Using matrix algebra you can easily scale up to work with relatively large systems.