Lesson 8 Solving Quadratic Equations
|
|
- Julian Harmon
- 6 years ago
- Views:
Transcription
1 Lesson 8 Solving Quadratic Equations Lesson 8 Solving Quadratic Equations We will continue our work with quadratic equations in this lesson and will learn the classic method to solve them the Quadratic Formula. We begin by viewing the Quadratic Formula then using it to solve a quadratic equation. Deriving the Quadratic Formula from the standard form quadratic equation follows this. The remainder of the lesson focuses on using the Quadratic Formula to solve equations, introducing the idea of solutions outside the real number system, then combining all our solution methods together. Lesson Topics Section 8.1: A Case for the Quadratic Formula Section 8.: Solving Quadratic Equations with the Quadratic Formula Section 8.3: Complex Numbers Section 8.: Complex Solutions to Quadratic Equations Section 8.5: Combining Solution Methods to Solve Any Quadratic Equation 315
2 316 Lesson 8 Solving Quadratic Equations
3 Lesson 8 - MiniLesson Section 8.1 A Case for the Quadratic Formula Number and Type of Solutions to a Quadratic Equation Remember that there are three possible cases for number of solutions to a quadratic equation in standard form as shown below. Graphing just the quadratic part of the equation in each case (i.e. the left side) would give you the accompanying graph. You should always begin your work with any quadratic equation (or function for that matter) by graphing. CASE 1: One, repeated, real number solution The parabola touches the x-axis once at the vertex. Example: x x + 0 Solutions: x 1, x CASE : Two unique, real number solutions The parabola crosses the x-axis at two unique locations. Example: 3x + x 0 Solutions: x 1 1,x CASE 3: No real number solutions but two unique, complex number solutions The parabola does NOT cross the x-axis. Solutions: Example: x + x x i, x 1 3 i 317
4 How could we determine the solutions for each of the examples listed on the previous page? Case 1: The solutions for the example equation in Case 1 are nice numbers (i.e. integers). Factoring as follows can solve the example equation: x x + 0 ( x ) x ( ) 0 x 0 x repeatedsolution x 1 and x Case : One of the solutions in Case is not a nice number meaning that factoring is not the preferred method. We could use the Graphing/Intersection Method, which would work, but is there an algebraic method that can be used? Case 3: These solutions do not look like anything we have seen to this point and are definitely not nice numbers. Graphing/intersection will not help us since the graph does not cross the x-axis. We will need some other kind of method to solve this kind of equation. The method we will learn that would solve the example equations above is the Quadratic Formula. The Quadratic Formula can be used to solve any quadratic equations written in standard form: x b± b ac a To solve a quadratic equation using the Quadratic Formula: Step 1: Make sure the quadratic equation is in standard form: ax + bx + c 0 and write down the coefficients a, b and c. Step : Graph the quadratic part of the equation and see how many times the graph crosses the x-axis. Identify the number and type of solutions. If the graph crosses the x-axis, enter 0 as Y then find the intersection point(s) and round to the appropriate number of decimal places. Write the x- values down. These will be your approximate solutions. Step 3: Substitute the coefficient values into the Quadratic Formula Step : Simplify your result completely leaving in the form indicated by the directions. 318
5 Section 8. Solving Quadratic Equations with the Quadratic Formula Problem 1 WORKED EXAMPLE Solve Quadratic Equations Using the Quadratic Formula Solve the equation 3x + x 0 using the steps as listed on page 318. Leave your solution(s) in exact form and in approximate form rounded to the thousandths place. Step 1: The given equation is in standard form with a 3, b 1 and c Step : Enter Y1 3x + x on the graphing calculator and graph on a standard window obtaining the graph below. This graph crosses the x-axis in places meaning there are two unique real number solutions to the original equation. Enter Y 0 and use the Graphing/Intersection Method to obtain the approximate solutions for x as follows: x 1 1, x Step 3: x (1)± (1) (3)( ) 1± 1 ( ) 1± 1+ 1± 5 (3) Step : Break up the final fraction into two parts and make computations for x 1 and x as below noting the complete simplification process: 1 5 x x Our final solutions in the requested forms are: Exact Form Approximate Form to the Thousandths Place x 1 1 andx 3 x 1, x x 1is exact to begin with so we use the same form for the approximation. 1 We use the symbol to indicate the approximation for x. Note this notation is not needed for x since exact and approximate forms are equal
6 How to Derive the Quadratic Formula The Quadratic Formula can be derived directly from a quadratic equation in standard form using a process called completing the square. This process is shown below. ax + bx + c 0 Start with a quadratic equation in standard form. x + bx c Subtract c from both sides. b c x + x Divide both sides by a. a a b a b x + x + a + a a c b Divide the coefficient of x by to get b a, square it to get b a, then add to both sides b ac b x + + a a a Factor the left side. On the right side, get a common denominator of a x + b a b ac a Combine the right side to one fraction, reorder the terms in the numerator, then take square root of both sides x + b a ± b ac a Simplify the square roots x b a ± x b ± b ac a b ac a Solve for x by subtracting b a from both sides. Combine over the common denominator to obtain the final form for the Quadratic Formula Let s look at more examples using the Quadratic Formula to solve equations. 30
7 Problem MEDIA EXAMPLE Solve Quadratic Equations Using the Quadratic Formula Solve the equation x 3 3x using the Quadratic Formula using the steps as listed on page 318. Leave your solution(s) in exact form and in approximate form rounded to the thousandths place. 31
8 Step in the previous problems requires skills in simplifying expressions involving radicals. The topic of radicals and radical expressions will be discussed more in a future lesson, but knowing some of the information related to square roots now will help you work through the Quadratic Formula problems in this lesson. Problem 3 WORKED EXAMPLE Simplifying Square Roots Simplifying Square Roots of Perfect Squares Problem MEDIA EXAMPLE Simplifying Square Roots Simplify each of the following as much as possible. Leave answers in exact form. Simplifying Square Roots of Composite Numbers: a) Simplifying Square Roots of Prime Numbers: b) 3 7 Simplifying Square Roots with Fractions (I): d) Simplifying Square Roots with Fractions (II): e) f)
9 Problem 5 YOU TRY Simplifying Square Roots Simplify each of the following as much as possible. Leave final answers in exact form. a) 0 d) b) 19 e) 8 c) 8 f) Exact vs. Approximate Form You have seen language asking you to leave answers in either exact form and/or approximated to a given decimal. The number 3 is in exact form. If 3 is rounded to three decimal places and written as 1.73, then it is in approximate form to the thousandths place. Numbers that don t need to be rounded to represent them fully (for example the number 1 or the number 0.5) are in exact form. 33
10 Problem 6 MEDIA EXAMPLE Solve Quadratic Equations Using Quadratic Formula Solve each quadratic equation by using the Quadratic Formula and the steps illustrated earlier in this lesson. Leave your solution(s) in exact form and in approximate form rounded to two decimals places. Quadratic Formula: x b± b ac a a) Solve x +3x b) Solve x x 3 3
11 Problem 7 YOU TRY Solve Quadratic Equations Using Quadratic Formula Solve each quadratic equation by using the Quadratic Formula and the steps illustrated earlier in this lesson. Leave your solution(s) in exact form and in approximate form rounded to the thousandths place. Label solution(s) on your graph. Quadratic Formula: x b± b ac a a) x x 5 0 b) 3x 7x
12 Section 8.3 Complex Numbers Suppose we are asked to solve the quadratic equation x 1. Any real number times itself always gives a positive result. Therefore, there is no real number x such that x 1. Approach this using the Quadratic Formula and see what happens. To solve x 1 using the Quadratic Formula, first write the equation in standard form as: x Then, identify the coefficients a, b and c to get: a 1, b 0 and c 1. Substitute these into the Quadratic Formula to get: x 0 ± 0 (1)(1) ± ± ( 1) ± 1 ± 1 ± 1 (1) Again, the number + 1 does not live in the real number system nor does the number 1, yet these are the two solutions to the equation x The way mathematicians have handled this problem is to define a number system that is an extension of the real number system. This system is called the Complex Number System and it allows equations such as x to be solved in this system. To do so, a special definition is used and that is the definition that: i 1 With this definition, then, the solutions to x are just x i and x i which is a lot simpler than the notation with negative numbers under the radical. 36
13 You can see in the image below how the Complex Number System interacts with the other types of numbers we are familiar with. The Complex Number System Each set in the image below is a ComplexNumbers subset of the larger set in which it is Allnumbersoftheforma + biwherea, bare contained. Complex numbers contain all the sets of numbers. Natural realnumbers,i 1, i 1 numbers only contain themselves. Examples:3+ i, +( 3)i, 0+ i, 3+ 0i RealNumbers When Will We See Complex Solutions? Allnumbersontherealnumberline. IncludesRationalandIrrationalNumbers. Examples:, 1, π, 17, RationalNumbers Ratiosofintegerswithdecimalsthatterminate orrepeat Examples: 1, 3, 0.3 Integers Includes0,thenaturalnumbersand theirnegatives. {... 3,, 1, 0,1,, 3,...} NaturalNumbers CountingNumbers {...1,,3,,5,...} We will see solutions that involve the complex number i when we solve quadratic equations whose graphs never cross the x-axis. Standard Form for a Complex Number Standard form for a complex number is: a + bi where a and b are real numbers 37
14 Problem 8 WORKED EXAMPLE Writing Complex Numbers in Exact Form Simplify each of the following leaving your final result in exact a + bi form. a) i b) i c) i i Problem 9 MEDIA EXAMPLE Writing Complex Numbers in Exact Form Simplify each of the following leaving your final result in exact a + bi form. a) 5 b) 3 c) Problem 10 YOU TRY Writing Complex Numbers in Exact Form Simplify each of the following leaving your final result in exact a + bi form. a) 11 b) 3 c)
15 Problem 11 WORKED EXAMPLE Writing Complex Numbers in Approximate Form Each of the following complex numbers is written in exact a + bi form. Change each of them to an approximate a+bi form rounded to the nearest thousandths place, if possible. a) 1 7i 1.66i b) i i Problem 1 MEDIA EXAMPLE Writing Complex Numbers in Approximate Form Each of the following complex numbers is written in exact a + bi form. Change each of them to an approximate a+bi form rounded to the nearest thousandths place, if possible. a) 3+ 15i b) i Problem 13 YOU TRY Writing Complex Numbers in Approximate Form Each of the following complex numbers is written in exact a + bi form. Change each of them to an approximate a+bi form rounded to the nearest thousandths place, if possible. a) 1 11i b) i 39
16 Section 8. Complex Solutions to Quadratic Equations Work through the following to see how to deal with equations that can only be solved in the Complex Number System. Problem 1 WORKED EXAMPLE Solving Quadratic Equations with Complex Solutions Solve x + x for x. Leave results in the form of a complex number, a+bi. Include both exact form and approximate form rounded to the thousandths place. The graph below shows that y x + x + 1 does not cross the x-axis at all. This is an example of our Case 3 possibility and will result in no Real Number solutions but two unique Complex Number Solutions. To find the solutions, make sure the equation is in standard form (check). Identify the coefficients a, b 1 and c 1. Insert these into the Quadratic Formula and simplify as follows: 1± 1 ()(1) 1± 1 8 1± 7 x () Break this into two solutions and use the a+bi form to get x i i and x i i The final exact solutions are x i, x 1 1 The final approximate solutions are x i,x i 1 7 i 330
17 Problem 15 MEDIA EXAMPLE Solving Quadratic Equations with Complex Solutions Solve x + x for x. Leave results in the form of a complex number, a+bi. 331
18 Problem 16 YOU TRY Solving Quadratic Equations with Complex Solutions Solve x 3x 5 for x. Leave results in the form of a complex number, a+bi. 33
19 Section 8.5 Combining Solution Methods to Solve any Quadratic Equation We have learned several methods to solve quadratic equations. Which one should you use when? The pros and cons of each method are presented below and the problems in this lesson will provide practice for all solution methods. Solving Quadratic Equations by Graphing Pros Good starting point for all quadratic equations as graphing will provide information about the number and types of solutions Not necessary to put the equation in standard form to use this method Cons Non-integer solution values will be approximate, not exact Window adjustments are often necessary in order to view necessary parts of the graph Solving Quadratic Equations by Factoring Pros Can be the fastest solution path if you quickly recognize how to factor the quadratic part of your equation Cons Equation must be in standard form to apply this method Does not work well for non-integer solutions and large numbers Can be time consuming if there are many choices for the Trial and Error Method Solving Quadratic Equations by Quadratic Formula Pros Can be used to solve any quadratic equation. Best way to find exact form solutions. Cons Equation must be in standard form to apply this method Must remember the formula. Must pay close attention to details in order to avoid computational errors 333
20 Problem 17 WORKED EXAMPLE Solving Quadratic Equations Given the quadratic equation x x 3 0, solve using the methods indicated below leaving all solutions in exact form. If solutions are complex, leave them in the form a + bi. a) Solve by graphing (if possible). Sketch the graph on a good viewing window (the vertex, vertical intercept and any horizontal intercepts should appear on the screen). Mark and label the solutions on your graph. b) Solve by factoring (if possible). Show all steps. Clearly identify your final solutions. (x+1)(x-3)0 x+1 0 or x 3 0 x -1 or x 3 c) Solve using the Quadratic Formula. Clearly identify your final solutions. x ( )± ( ) ( 1) ( 3) ± +1 ± 16 ( 1) ± Break up the final fraction into two parts and make computations for x 1 and x as below: Our final solutions in the requested forms are: x 1 1 x Exact Form x 1 1 andx 3 33
21 Problem 18 MEDIA EXAMPLE Solving Quadratic Equations Given the quadratic equation x x, solve using the methods indicated below, leaving all solutions in exact form and in approximate form rounded to the thousandths place. If solutions are complex, leave them in the form a + bi. a) Solve by graphing (if possible). Sketch the graph on a good viewing window (the vertex, vertical intercept and any horizontal intercepts should appear on the screen). Mark and label the solutions on your graph. b) Solve by factoring (if possible). Show all steps. c) Solve using the Quadratic Formula. Show all steps. 335
22 Problem 19 YOU TRY SOLVING QUADRATIC EQUATIONS Given the quadratic equation x + 3x 7 3, solve using the methods indicated below leaving all solutions in exact form. If solutions are complex, leave them in the form a + bi. Clearly identify your solutions in all cases. a) Solve by graphing (if possible). Sketch the graph on a good viewing window (the vertex, vertical intercept, and any horizontal intercepts should appear on the screen). Mark and label the solutions on your graph. b) Solve by factoring (if possible). Show all steps. c) Solve using the Quadratic Formula. Show all steps. 336
23 Problem 0 MEDIA EXAMPLE Solving Quadratic Functions Given the quadratic equations below, solve for x using an appropriate method of your choice leaving all solutions in exact form and approximate form to the thousandths place. Clearly identify each method that you attempt. If solutions are complex, leave them in the form a + bi. a) x 3x 10 0 b) x 6x
24 Problem 1 YOU TRY SOLVING QUADRATIC EQUATIONS Given the quadratic equations below, solve for x using an appropriate method of your choice leaving all solutions in exact form and approximate form to the thousandths place. Clearly identify each method that you attempt. If solutions are complex, leave them in the form a + bi. a) x +5x 3 0 b) x +3x 5 338
Lesson 5b Solving Quadratic Equations
Lesson 5b Solving Quadratic Equations In this lesson, we will continue our work with Quadratics in this lesson and will learn several methods for solving quadratic equations. The first section will introduce
More informationLearning Packet. Lesson 5b Solving Quadratic Equations THIS BOX FOR INSTRUCTOR GRADING USE ONLY
Learning Packet Student Name Due Date Class Time/Day Submission Date THIS BOX FOR INSTRUCTOR GRADING USE ONLY Mini-Lesson is complete and information presented is as found on media links (0 5 pts) Comments:
More informationIn this lesson, we will focus on quadratic equations and inequalities and algebraic methods to solve them.
Lesson 7 Quadratic Equations, Inequalities, and Factoring Lesson 7 Quadratic Equations, Inequalities and Factoring In this lesson, we will focus on quadratic equations and inequalities and algebraic methods
More informationPolynomials: Adding, Subtracting, & Multiplying (5.1 & 5.2)
Polynomials: Adding, Subtracting, & Multiplying (5.1 & 5.) Determine if the following functions are polynomials. If so, identify the degree, leading coefficient, and type of polynomial 5 3 1. f ( x) =
More informationLesson #33 Solving Incomplete Quadratics
Lesson # Solving Incomplete Quadratics A.A.4 Know and apply the technique of completing the square ~ 1 ~ We can also set up any quadratic to solve it in this way by completing the square, the technique
More informationLesson 7.1 Polynomial Degree and Finite Differences
Lesson 7.1 Polynomial Degree and Finite Differences 1. Identify the degree of each polynomial. a. 3x 4 2x 3 3x 2 x 7 b. x 1 c. 0.2x 1.x 2 3.2x 3 d. 20 16x 2 20x e. x x 2 x 3 x 4 x f. x 2 6x 2x 6 3x 4 8
More informationUnit 2-1: Factoring and Solving Quadratics. 0. I can add, subtract and multiply polynomial expressions
CP Algebra Unit -1: Factoring and Solving Quadratics NOTE PACKET Name: Period Learning Targets: 0. I can add, subtract and multiply polynomial expressions 1. I can factor using GCF.. I can factor by grouping.
More informationAlgebra II Unit #2 4.6 NOTES: Solving Quadratic Equations (More Methods) Block:
Algebra II Unit # Name: 4.6 NOTES: Solving Quadratic Equations (More Methods) Block: (A) Background Skills - Simplifying Radicals To simplify a radical that is not a perfect square: 50 8 300 7 7 98 (B)
More informationP.1 Prerequisite skills Basic Algebra Skills
P.1 Prerequisite skills Basic Algebra Skills Topics: Evaluate an algebraic expression for given values of variables Combine like terms/simplify algebraic expressions Solve equations for a specified variable
More informationHONORS GEOMETRY Summer Skills Set
HONORS GEOMETRY Summer Skills Set Algebra Concepts Adding and Subtracting Rational Numbers To add or subtract fractions with the same denominator, add or subtract the numerators and write the sum or difference
More information1 Quadratic Functions
Unit 1 Quadratic Functions Lecture Notes Introductory Algebra Page 1 of 8 1 Quadratic Functions In this unit we will learn many of the algebraic techniques used to work with the quadratic function fx)
More informationLesson 10.1 Solving Quadratic Equations
Lesson 10.1 Solving Quadratic Equations 1. Sketch the graph of a quadratic equation with each set of conditions. a. One -intercept and all nonnegative y-values b. The verte in the third quadrant and no
More informationChapter 9: Roots and Irrational Numbers
Chapter 9: Roots and Irrational Numbers Index: A: Square Roots B: Irrational Numbers C: Square Root Functions & Shifting D: Finding Zeros by Completing the Square E: The Quadratic Formula F: Quadratic
More informationCommon Core Algebra 2. Chapter 3: Quadratic Equations & Complex Numbers
Common Core Algebra 2 Chapter 3: Quadratic Equations & Complex Numbers 1 Chapter Summary: The strategies presented for solving quadratic equations in this chapter were introduced at the end of Algebra.
More information2017 SUMMER REVIEW FOR STUDENTS ENTERING GEOMETRY
2017 SUMMER REVIEW FOR STUDENTS ENTERING GEOMETRY The following are topics that you will use in Geometry and should be retained throughout the summer. Please use this practice to review the topics you
More informationFoundations of Math II Unit 5: Solving Equations
Foundations of Math II Unit 5: Solving Equations Academics High School Mathematics 5.1 Warm Up Solving Linear Equations Using Graphing, Tables, and Algebraic Properties On the graph below, graph the following
More informationCP Algebra 2. Unit 3B: Polynomials. Name: Period:
CP Algebra 2 Unit 3B: Polynomials Name: Period: Learning Targets 10. I can use the fundamental theorem of algebra to find the expected number of roots. Solving Polynomials 11. I can solve polynomials by
More informationCP Algebra 2. Unit 2-1 Factoring and Solving Quadratics
CP Algebra Unit -1 Factoring and Solving Quadratics Name: Period: 1 Unit -1 Factoring and Solving Quadratics Learning Targets: 1. I can factor using GCF.. I can factor by grouping. Factoring Quadratic
More informationQuadratic Functions. and Equations
Name: Quadratic Functions and Equations 1. + x 2 is a parabola 2. - x 2 is a parabola 3. A quadratic function is in the form ax 2 + bx + c, where a and is the y-intercept 4. Equation of the Axis of Symmetry
More informationApplied 30S Unit 1 Quadratic Functions
Applied 30S Unit 1 Quadratic Functions Mrs. Kornelsen Teulon Collegiate Institute Learning checklist Quadratics Learning increases when you have a goal to work towards. Use this checklist as guide to track
More informationCh. 7.6 Squares, Squaring & Parabolas
Ch. 7.6 Squares, Squaring & Parabolas Learning Intentions: Learn about the squaring & square root function. Graph parabolas. Compare the squaring function with other functions. Relate the squaring function
More information1. Definition of a Polynomial
1. Definition of a Polynomial What is a polynomial? A polynomial P(x) is an algebraic expression of the form Degree P(x) = a n x n + a n 1 x n 1 + a n 2 x n 2 + + a 3 x 3 + a 2 x 2 + a 1 x + a 0 Leading
More information3.3 Real Zeros of Polynomial Functions
71_00.qxp 12/27/06 1:25 PM Page 276 276 Chapter Polynomial and Rational Functions. Real Zeros of Polynomial Functions Long Division of Polynomials Consider the graph of f x 6x 19x 2 16x 4. Notice in Figure.2
More informationSolving a Linear-Quadratic System
CC-18 Solving LinearQuadratic Systems Objective Content Standards A.REI.7 Solve a simple system consisting of a linear equation and a quadratic equation in two variables... A.REI.11 Explain why the x-coordinates
More informationUNIT 5 QUADRATIC FUNCTIONS Lesson 2: Creating and Solving Quadratic Equations in One Variable Instruction
Prerequisite Skills This lesson requires the use of the following skills: simplifying radicals working with complex numbers Introduction You can determine how far a ladder will extend from the base of
More informationUnit 8 - Polynomial and Rational Functions Classwork
Unit 8 - Polynomial and Rational Functions Classwork This unit begins with a study of polynomial functions. Polynomials are in the form: f ( x) = a n x n + a n 1 x n 1 + a n 2 x n 2 +... + a 2 x 2 + a
More informationFinding the Equation of a Graph. I can give the equation of a curve given just the roots.
National 5 W 7th August Finding the Equation of a Parabola Starter Sketch the graph of y = x - 8x + 15. On your sketch clearly identify the roots, axis of symmetry, turning point and y intercept. Today
More informationInstructor Quick Check: Question Block 12
Instructor Quick Check: Question Block 2 How to Administer the Quick Check: The Quick Check consists of two parts: an Instructor portion which includes solutions and a Student portion with problems for
More informationComplex Numbers. Essential Question What are the subsets of the set of complex numbers? Integers. Whole Numbers. Natural Numbers
3.4 Complex Numbers Essential Question What are the subsets of the set of complex numbers? In your study of mathematics, you have probably worked with only real numbers, which can be represented graphically
More information3.4 The Fundamental Theorem of Algebra
333371_0304.qxp 12/27/06 1:28 PM Page 291 3.4 The Fundamental Theorem of Algebra Section 3.4 The Fundamental Theorem of Algebra 291 The Fundamental Theorem of Algebra You know that an nth-degree polynomial
More informationThe Graph of a Quadratic Function. Quadratic Functions & Models. The Graph of a Quadratic Function. The Graph of a Quadratic Function
8/1/015 The Graph of a Quadratic Function Quadratic Functions & Models Precalculus.1 The Graph of a Quadratic Function The Graph of a Quadratic Function All parabolas are symmetric with respect to a line
More informationMATH College Algebra Review for Test 2
MATH 4 - College Algebra Review for Test Sections. and.. For f (x) = x + 4x + 5, give (a) the x-intercept(s), (b) the -intercept, (c) both coordinates of the vertex, and (d) the equation of the axis of
More informationRoots are: Solving Quadratics. Graph: y = 2x 2 2 y = x 2 x 12 y = x 2 + 6x + 9 y = x 2 + 6x + 3. real, rational. real, rational. real, rational, equal
Solving Quadratics Graph: y = 2x 2 2 y = x 2 x 12 y = x 2 + 6x + 9 y = x 2 + 6x + 3 Roots are: real, rational real, rational real, rational, equal real, irrational 1 To find the roots algebraically, make
More informationCHAPTER EIGHT: SOLVING QUADRATIC EQUATIONS Review April 9 Test April 17 The most important equations at this level of mathematics are quadratic
CHAPTER EIGHT: SOLVING QUADRATIC EQUATIONS Review April 9 Test April 17 The most important equations at this level of mathematics are quadratic equations. They can be solved using a graph, a perfect square,
More informationMATH College Algebra Review for Test 2
MATH 34 - College Algebra Review for Test 2 Sections 3. and 3.2. For f (x) = x 2 + 4x + 5, give (a) the x-intercept(s), (b) the -intercept, (c) both coordinates of the vertex, and (d) the equation of the
More informationSolving Quadratic & Higher Degree Equations
Chapter 7 Solving Quadratic & Higher Degree Equations Sec 1. Zero Product Property Back in the third grade students were taught when they multiplied a number by zero, the product would be zero. In algebra,
More informationA Level Summer Work. Year 11 Year 12 Transition. Due: First lesson back after summer! Name:
A Level Summer Work Year 11 Year 12 Transition Due: First lesson back after summer! Name: This summer work is compulsory. Your maths teacher will ask to see your work (and method) in your first maths lesson,
More informationALGEBRA. COPYRIGHT 1996 Mark Twain Media, Inc. ISBN Printing No EB
ALGEBRA By Don Blattner and Myrl Shireman COPYRIGHT 1996 Mark Twain Media, Inc. ISBN 978-1-58037-826-0 Printing No. 1874-EB Mark Twain Media, Inc., Publishers Distributed by Carson-Dellosa Publishing Company,
More informationUNIT 4: RATIONAL AND RADICAL EXPRESSIONS. 4.1 Product Rule. Objective. Vocabulary. o Scientific Notation. o Base
UNIT 4: RATIONAL AND RADICAL EXPRESSIONS M1 5.8, M2 10.1-4, M3 5.4-5, 6.5,8 4.1 Product Rule Objective I will be able to multiply powers when they have the same base, including simplifying algebraic expressions
More informationLesson 9 Exploring Graphs of Quadratic Functions
Exploring Graphs of Quadratic Functions Graph the following system of linear inequalities: { y > 1 2 x 5 3x + 2y 14 a What are three points that are solutions to the system of inequalities? b Is the point
More informationChapter 1A -- Real Numbers. iff. Math Symbols: Sets of Numbers
Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 1A! Page 1 Chapter 1A -- Real Numbers Math Symbols: iff or Example: Let A = {2, 4, 6, 8, 10, 12, 14, 16,...} and let B = {3, 6, 9, 12, 15, 18,
More informationLesson 6b Rational Exponents & Radical Functions
Lesson 6b Rational Exponents & Radical Functions In this lesson, we will continue our review of Properties of Exponents and will learn some new properties including those dealing with Rational and Radical
More informationSolving Quadratic Equations
Concepts: Solving Quadratic Equations, Completing the Square, The Quadratic Formula, Sketching Quadratics Solving Quadratic Equations Completing the Square ax + bx + c = a x + ba ) x + c Factor so the
More informationHerndon High School Geometry Honors Summer Assignment
Welcome to Geometry! This summer packet is for all students enrolled in Geometry Honors at Herndon High School for Fall 07. The packet contains prerequisite skills that you will need to be successful in
More informationStudent Instruction Sheet: Unit 3, Lesson 3. Solving Quadratic Relations
Student Instruction Sheet: Unit 3, Lesson 3 Solving Quadratic Relations Suggested Time: 75 minutes What s important in this lesson: In this lesson, you will learn how to solve a variety of quadratic relations.
More informationSTANDARDS OF LEARNING CONTENT REVIEW NOTES ALGEBRA II. 2 nd Nine Weeks,
STANDARDS OF LEARNING CONTENT REVIEW NOTES ALGEBRA II 2 nd Nine Weeks, 2016-2017 1 OVERVIEW Algebra II Content Review Notes are designed by the High School Mathematics Steering Committee as a resource
More informationSolving Equations by Factoring. Solve the quadratic equation x 2 16 by factoring. We write the equation in standard form: x
11.1 E x a m p l e 1 714SECTION 11.1 OBJECTIVES 1. Solve quadratic equations by using the square root method 2. Solve quadratic equations by completing the square Here, we factor the quadratic member of
More informationMATHEMATICAL METHODS UNIT 1 CHAPTER 4 CUBIC POLYNOMIALS
E da = q ε ( B da = 0 E ds = dφ. B ds = μ ( i + μ ( ε ( dφ 3 MATHEMATICAL METHODS UNIT 1 CHAPTER 4 CUBIC POLYNOMIALS dt dt Key knowledge The key features and properties of cubic polynomials functions and
More informationTwitter: @Owen134866 www.mathsfreeresourcelibrary.com Prior Knowledge Check 1) Simplify: a) 3x 2 5x 5 b) 5x3 y 2 15x 7 2) Factorise: a) x 2 2x 24 b) 3x 2 17x + 20 15x 2 y 3 3) Use long division to calculate:
More informationTHE LANGUAGE OF FUNCTIONS *
PRECALCULUS THE LANGUAGE OF FUNCTIONS * In algebra you began a process of learning a basic vocabulary, grammar, and syntax forming the foundation of the language of mathematics. For example, consider the
More informationAlgebra II (Common Core) Summer Assignment Due: September 11, 2017 (First full day of classes) Ms. Vella
1 Algebra II (Common Core) Summer Assignment Due: September 11, 2017 (First full day of classes) Ms. Vella In this summer assignment, you will be reviewing important topics from Algebra I that are crucial
More informationName Period Date. QUADRATIC FUNCTIONS AND EQUATIONS Student Pages for Packet 2: Solving Quadratic Equations 1
Name Period Date QUAD2.1 QUAD2.2 QUAD2.3 The Square Root Property Solve quadratic equations using the square root property Understand that if a quadratic function is set equal to zero, then the result
More informationLooking Ahead to Chapter 10
Looking Ahead to Chapter Focus In Chapter, you will learn about polynomials, including how to add, subtract, multiply, and divide polynomials. You will also learn about polynomial and rational functions.
More informationSolving Quadratic & Higher Degree Equations
Chapter 9 Solving Quadratic & Higher Degree Equations Sec 1. Zero Product Property Back in the third grade students were taught when they multiplied a number by zero, the product would be zero. In algebra,
More informationChapter 4: Quadratic Functions and Factoring 4.1 Graphing Quadratic Functions in Stand
Chapter 4: Quadratic Functions and Factoring 4.1 Graphing Quadratic Functions in Stand VOCAB: a quadratic function in standard form is written y = ax 2 + bx + c, where a 0 A quadratic Function creates
More informationAccel Alg E. L. E. Notes Solving Quadratic Equations. Warm-up
Accel Alg E. L. E. Notes Solving Quadratic Equations Warm-up Solve for x. Factor. 1. 12x 36 = 0 2. x 2 8x Factor. Factor. 3. 2x 2 + 5x 7 4. x 2 121 Solving Quadratic Equations Methods: (1. By Inspection)
More informationPractice Test - Chapter 2
Graph and analyze each function. Describe the domain, range, intercepts, end behavior, continuity, and where the function is increasing or decreasing. 1. f (x) = 0.25x 3 Evaluate the function for several
More informationAlgebra & Trig Review
Algebra & Trig Review 1 Algebra & Trig Review This review was originally written for my Calculus I class, but it should be accessible to anyone needing a review in some basic algebra and trig topics. The
More informationOBJECTIVES UNIT 1. Lesson 1.0
OBJECTIVES UNIT 1 Lesson 1.0 1. Define "set," "element," "finite set," and "infinite set," "empty set," and "null set" and give two examples of each term. 2. Define "subset," "universal set," and "disjoint
More information2-7 Solving Quadratic Inequalities. ax 2 + bx + c > 0 (a 0)
Quadratic Inequalities In One Variable LOOKS LIKE a quadratic equation but Doesn t have an equal sign (=) Has an inequality sign (>,
More informationSolving Quadratic & Higher Degree Equations
Chapter 9 Solving Quadratic & Higher Degree Equations Sec 1. Zero Product Property Back in the third grade students were taught when they multiplied a number by zero, the product would be zero. In algebra,
More informationMath 1302 Notes 2. How many solutions? What type of solution in the real number system? What kind of equation is it?
Math 1302 Notes 2 We know that x 2 + 4 = 0 has How many solutions? What type of solution in the real number system? What kind of equation is it? What happens if we enlarge our current system? Remember
More informationConcept Category 4. Quadratic Equations
Concept Category 4 Quadratic Equations 1 Solving Quadratic Equations by the Square Root Property Square Root Property We previously have used factoring to solve quadratic equations. This chapter will introduce
More informationQuadratic Formula: - another method for solving quadratic equations (ax 2 + bx + c = 0)
In the previous lesson we showed how to solve quadratic equations that were not factorable and were not perfect squares by making perfect square trinomials using a process called completing the square.
More information4.1 Graphical Solutions of Quadratic Equations Date:
4.1 Graphical Solutions of Quadratic Equations Date: Key Ideas: Quadratic functions are written as f(x) = x 2 x 6 OR y = x 2 x 6. f(x) is f of x and means that the y value is dependent upon the value of
More informationEquations and Inequalities
Equations and Inequalities 2 Figure 1 CHAPTER OUTLINE 2.1 The Rectangular Coordinate Systems and Graphs 2.2 Linear Equations in One Variable 2.3 Models and Applications 2.4 Complex Numbers 2.5 Quadratic
More informationMAT116 Final Review Session Chapter 3: Polynomial and Rational Functions
MAT116 Final Review Session Chapter 3: Polynomial and Rational Functions Quadratic Function A quadratic function is defined by a quadratic or second-degree polynomial. Standard Form f x = ax 2 + bx + c,
More informationExponential Functions Dr. Laura J. Pyzdrowski
1 Names: (4 communication points) About this Laboratory An exponential function is an example of a function that is not an algebraic combination of polynomials. Such functions are called trancendental
More informationSolving Equations Quick Reference
Solving Equations Quick Reference Integer Rules Addition: If the signs are the same, add the numbers and keep the sign. If the signs are different, subtract the numbers and keep the sign of the number
More informationPractice Test - Chapter 2
Graph and analyze each function. Describe the domain, range, intercepts, end behavior, continuity, and where the function is increasing or decreasing. 1. f (x) = 0.25x 3 Evaluate the function for several
More informationSystems of Nonlinear Equations and Inequalities: Two Variables
Systems of Nonlinear Equations and Inequalities: Two Variables By: OpenStaxCollege Halley s Comet ([link]) orbits the sun about once every 75 years. Its path can be considered to be a very elongated ellipse.
More informationAlgebra 2 Segment 1 Lesson Summary Notes
Algebra 2 Segment 1 Lesson Summary Notes For each lesson: Read through the LESSON SUMMARY which is located. Read and work through every page in the LESSON. Try each PRACTICE problem and write down the
More informationAlgebra I (2016) Wright City R-II. Mathematics Grades 8-9, Duration 1 Year, 1 Credit Required Course Course Description
Algebra I (2016) Course Algebra 1 introduces basic algebraic skills in a logical order, including relations, functions, graphing, systems of equations, radicals, factoring polynomials, rational equations,
More informationAlgebra Review. Finding Zeros (Roots) of Quadratics, Cubics, and Quartics. Kasten, Algebra 2. Algebra Review
Kasten, Algebra 2 Finding Zeros (Roots) of Quadratics, Cubics, and Quartics A zero of a polynomial equation is the value of the independent variable (typically x) that, when plugged-in to the equation,
More information3.4. ZEROS OF POLYNOMIAL FUNCTIONS
3.4. ZEROS OF POLYNOMIAL FUNCTIONS What You Should Learn Use the Fundamental Theorem of Algebra to determine the number of zeros of polynomial functions. Find rational zeros of polynomial functions. Find
More information(a) Write down the value of q and of r. (2) Write down the equation of the axis of symmetry. (1) (c) Find the value of p. (3) (Total 6 marks)
1. Let f(x) = p(x q)(x r). Part of the graph of f is shown below. The graph passes through the points ( 2, 0), (0, 4) and (4, 0). (a) Write down the value of q and of r. (b) Write down the equation of
More informationSummer Work for students entering PreCalculus
Summer Work for students entering PreCalculus Name Directions: The following packet represent a review of topics you learned in Algebra 1, Geometry, and Algebra 2. Complete your summer packet on separate
More informationMA094 Part 2 - Beginning Algebra Summary
MA094 Part - Beginning Algebra Summary Page of 8/8/0 Big Picture Algebra is Solving Equations with Variables* Variable Variables Linear Equations x 0 MA090 Solution: Point 0 Linear Inequalities x < 0 page
More informationPart 2 - Beginning Algebra Summary
Part - Beginning Algebra Summary Page 1 of 4 1/1/01 1. Numbers... 1.1. Number Lines... 1.. Interval Notation.... Inequalities... 4.1. Linear with 1 Variable... 4. Linear Equations... 5.1. The Cartesian
More informationReading Mathematical Expressions & Arithmetic Operations Expression Reads Note
Math 001 - Term 171 Reading Mathematical Expressions & Arithmetic Operations Expression Reads Note x A x belongs to A,x is in A Between an element and a set. A B A is a subset of B Between two sets. φ
More informationModule 2, Section 2 Solving Equations
Principles of Mathematics Section, Introduction 03 Introduction Module, Section Solving Equations In this section, you will learn to solve quadratic equations graphically, by factoring, and by applying
More information6.4. The Quadratic Formula. LEARN ABOUT the Math. Selecting a strategy to solve a quadratic equation. 2x 2 + 4x - 10 = 0
6.4 The Quadratic Formula YOU WILL NEED graphing calculator GOAL Understand the development of the quadratic formula, and use the quadratic formula to solve quadratic equations. LEARN ABOUT the Math Devlin
More informationWhat students need to know for... ALGEBRA II
What students need to know for... ALGEBRA II 2017-2018 NAME This is a MANDATORY assignment that will be GRADED. It is due the first day of the course. Your teacher will determine how it will be counted
More informationDefinition: Quadratic equation: A quadratic equation is an equation that could be written in the form ax 2 + bx + c = 0 where a is not zero.
We will see many ways to solve these familiar equations. College algebra Class notes Solving Quadratic Equations: Factoring, Square Root Method, Completing the Square, and the Quadratic Formula (section
More informationGeometry 21 Summer Work Packet Review and Study Guide
Geometry Summer Work Packet Review and Study Guide This study guide is designed to accompany the Geometry Summer Work Packet. Its purpose is to offer a review of the ten specific concepts covered in the
More informationLesson 7.1 Polynomial Degree and Finite Differences
Lesson 7.1 Polynomial Degree and Finite Differences 1. Identify the degree of each polynomial. a. 1 b. 0.2 1. 2 3.2 3 c. 20 16 2 20 2. Determine which of the epressions are polynomials. For each polynomial,
More informationA repeated root is a root that occurs more than once in a polynomial function.
Unit 2A, Lesson 3.3 Finding Zeros Synthetic division, along with your knowledge of end behavior and turning points, can be used to identify the x-intercepts of a polynomial function. This information allows
More information( ) 0. Section 3.3 Graphs of Polynomial Functions. Chapter 3
76 Chapter 3 Section 3.3 Graphs of Polynomial Functions In the previous section we explored the short run behavior of quadratics, a special case of polynomials. In this section we will explore the short
More informationChapter P. Prerequisites. Slide P- 1. Copyright 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide P- 1 Chapter P Prerequisites 1 P.1 Real Numbers Quick Review 1. List the positive integers between -4 and 4.. List all negative integers greater than -4. 3. Use a calculator to evaluate the expression
More informationA2 HW Imaginary Numbers
Name: A2 HW Imaginary Numbers Rewrite the following in terms of i and in simplest form: 1) 100 2) 289 3) 15 4) 4 81 5) 5 12 6) -8 72 Rewrite the following as a radical: 7) 12i 8) 20i Solve for x in simplest
More informationAlgebra 2 Chapter 3 Part 1 Practice Test 2018
Synthetic divisions in this worksheet were performed using the Algebra App for PCs that is available at www.mathguy.us/pcapps.php. 1) Given the polynomial f x x 5x 2x 24 and factor x 2, factor completely.
More informationSection 3.1 Quadratic Functions
Chapter 3 Lecture Notes Page 1 of 72 Section 3.1 Quadratic Functions Objectives: Compare two different forms of writing a quadratic function Find the equation of a quadratic function (given points) Application
More informationSolving Quadratic Equations by Formula
Algebra Unit: 05 Lesson: 0 Complex Numbers All the quadratic equations solved to this point have had two real solutions or roots. In some cases, solutions involved a double root, but there were always
More informationWest Windsor-Plainsboro Regional School District Math A&E Grade 7
West Windsor-Plainsboro Regional School District Math A&E Grade 7 Page 1 of 24 Unit 1: Introduction to Algebra Content Area: Mathematics Course & Grade Level: A&E Mathematics, Grade 7 Summary and Rationale
More informationSummer Packet A Math Refresher For Students Entering IB Mathematics SL
Summer Packet A Math Refresher For Students Entering IB Mathematics SL Name: PRECALCULUS SUMMER PACKET Directions: This packet is required if you are registered for Precalculus for the upcoming school
More informationAlgebra 1: Hutschenreuter Chapter 10 Notes Adding and Subtracting Polynomials
Algebra 1: Hutschenreuter Chapter 10 Notes Name 10.1 Adding and Subtracting Polynomials Polynomial- an expression where terms are being either added and/or subtracted together Ex: 6x 4 + 3x 3 + 5x 2 +
More information6.4 6.notebook December 03, 2018
6.4 Opening Activity: 1. Expand and Simplify 3. Expand and Simplify (x 5) 2 y = (x 5) 2 3 2. Expand and Simplify 4. Expand and Simplify (x 5) 2 3 y + 3 = (x 5) 2 5. What is the vertex of the following
More informationCourse Name: MAT 135 Spring 2017 Master Course Code: N/A. ALEKS Course: Intermediate Algebra Instructor: Master Templates
Course Name: MAT 135 Spring 2017 Master Course Code: N/A ALEKS Course: Intermediate Algebra Instructor: Master Templates Course Dates: Begin: 01/15/2017 End: 05/31/2017 Course Content: 279 Topics (207
More informationLesson 24: Modeling with Quadratic Functions
Student Outcomes Students create a quadratic function from a data set based on a contextual situation, sketch its graph, and interpret both the function and the graph in context. They answer questions
More informationWhat students need to know for PRE-CALCULUS Students expecting to take Pre-Calculus should demonstrate the ability to:
What students need to know for PRE-CALCULUS 2014-2015 Students expecting to take Pre-Calculus should demonstrate the ability to: General: keep an organized notebook take good notes complete homework every
More information