Concept Category 4. Quadratic Equations

Size: px
Start display at page:

Download "Concept Category 4. Quadratic Equations"

Transcription

1 Concept Category 4 Quadratic Equations

2 1 Solving Quadratic Equations by the Square Root Property

3 Square Root Property We previously have used factoring to solve quadratic equations. This chapter will introduce additional methods for solving quadratic equations. Square Root Property If b is a real number and a = b, then a b Martin-Gay, Developmental Mathematics 3

4 Square Root Property Example Solve x = 49 x 49 7 Solve x = 4 x = x Solve (y 3) = 4 y 3 4 y = 3 y = 1 or 5 Martin-Gay, Developmental Mathematics 4

5 Square Root Property Example Solve x + 4 = 0 x = 4 There is no real solution because the square root of 4 is not a real number. Martin-Gay, Developmental Mathematics 5

6 Square Root Property Example Solve (x + ) = 5 x 5 5 x = ± 5 x = + 5 or x = 5 x = 3 or x = 7 Martin-Gay, Developmental Mathematics 6

7 Square Root Property Example Solve (3x 17) = 8 3x 17 = x 17 7 x Martin-Gay, Developmental Mathematics 7

8 Solving Quadratic Equations by the Quadratic Formula

9 The Quadratic Formula Another technique for solving quadratic equations is to use the quadratic formula. The formula is derived from completing the square of a general quadratic equation. Martin-Gay, Developmental Mathematics 9

10 The Quadratic Formula A quadratic equation written in standard form, ax + bx + c = 0, has the solutions. x b b 4ac a Martin-Gay, Developmental Mathematics 10

11 The Quadratic Formula Example Solve 11n 9n = 1 by the quadratic formula. 11n 9n 1 = 0 set one side = 0 a = 11, b = -9, c = -1 n 9 ( 9) 4(11)( 1) (11) Martin-Gay, Developmental Mathematics 11

12 Two kinds of answers: Decimal Answers (to graph): and 0.9 and 0.1 Simplified Radical Answers (SAT, ACT, and other college placement exams): Martin-Gay, Developmental Mathematics 1

13 Practice: Solve x using QFormula Present your answers in Decimals and Simplified Radicals: a] f ( x) 4x 1x 63 b] y x 1x 46 Martin-Gay, Developmental Mathematics 13

14 Martin-Gay, Developmental Mathematics 14

15 1/18 Practice Now: a f x x x b g x x ] ( ) 10 1 ] ( ) ( 3) 5 Vertex point? x-intercept points? y-intercept point? *Vertex point? *x-intercept points? *y-intercept point? Martin-Gay, Developmental Mathematics 15

16 Quadratic Formula SAT example Example Solve x + x = 0 by the quadratic formula. x + 8x 0 = 0 multiply both sides by 8 a = 1, b = 8, c = 0 x 8 (8) 4(1)( 0) (1) or, 10 or Martin-Gay, Developmental Mathematics 16

17 Concept Category 4 Quadratics Standard Form of Quadratic Equation b Vertex : x int : Factoring or Quadratic Formula a Vertex Form of Quadratic Equation Vertex: transformation first Radical Operations Solving Radical Equations Nth Roots Complex Numbers Martin-Gay, Developmental Mathematics 17

18 Radicals = = 4 = 5 = 10 = 1 Martin-Gay, Developmental Mathematics 18

19 Perfect Squares Martin-Gay, Developmental Mathematics 19

20 LEAVE IN RADICAL FORM Perfect Square Factor * Other Factor 8 = 4* = 0 = 4*5 = 5 3 = 16* = 4 75 = 5*3 = = 4*10 = 10 Martin-Gay, Developmental Mathematics 0

21 Simplify each expression A] B] Martin-Gay, Developmental Mathematics 1

22 + To combine radicals: combine the coefficients of like radicals Martin-Gay, Developmental Mathematics

23 Simplify each radical first, then combine * 3 16 * *5 3* Martin-Gay, Developmental Mathematics 3

24 Practice NOW * * 3 3* 3 3 5* Martin-Gay, Developmental Mathematics 4

25 * Multiply the coefficients and then multiply the radicands and then simplify Martin-Gay, Developmental Mathematics 5

26 Multiply and then simplify 5 * * * *14 6* * *10 80 Martin-Gay, Developmental Mathematics 6

27 X = X Y 6 = Y 3 P 4 X 6 Y = P X 3 Y 4X 4 Y = X Y 8 5C D 10 = 5C 4 D 5 Martin-Gay, Developmental Mathematics 7

28 X 3 = X * X = X X Y 5 = Y 4 Y = Y Y Martin-Gay, Developmental Mathematics 8

29 PX 3 Y 3 = X Y * PXY = XY PXY 1X 7 Y = Y 5 8 5C D 9 = 5 Y Martin-Gay, Developmental Mathematics 9

30 Happy Wed. 1/5 Did everyone have a chance to work on yesterday s SMC practice test? Placement Exams for English and Math are required for ALL CA colleges ( or 4 yr) This practice test have other parts: nd part of Algebra, Geometry, and Precalculus The less you score and more courses ($$$$ + Time) you will need to make up for Martin-Gay, Developmental Mathematics 30

31 CA College Placement Exams v.s. CAASP Why they all have similar questions? Because your HIGH SCHOOL standards/state Exams are mostly set by public colleges Martin-Gay, Developmental Mathematics 31

32 5 5 * *3 7 *3 7 *3 7 81* w 8 w 4 * 8w w 4 8w 3x 3 * 3 x x 4 9x 1x Martin-Gay, Developmental Mathematics 3

33 Practice NOW x 16x 4 Martin-Gay, Developmental Mathematics 33

34 SOLUTIONS x Martin-Gay, Developmental Mathematics 34

35 A] Divide the coefficients, divide the radicands if possible, B] rationalize the denominator so that the denominator is always an integer! Martin-Gay, Developmental Mathematics 35

36 Fractional Radicands Simplify exponents/radicals first. Then reduce Martin-Gay, Developmental Mathematics 36

37 * Martin-Gay, Developmental Mathematics 37

38 You Try: Martin-Gay, Developmental Mathematics 38

39 How about BIG NO NO! When simplifying radicals, there can never be a radical in the denominator of the final answer. Rationalization: the steps to change a radical denominator to a whole number. Martin-Gay, Developmental Mathematics 39

40 In the world of SAT, ACT, math placement exams: Method: 4 4 * * 7 7 Multiply the Denominator by the same radical 6 * Fraction expansion Martin-Gay, Developmental Mathematics 40

41 This can be divided which leaves the radical in the denominator. 5 1 * 10 We do not leave radicals in the denominator. So we need to rationalize. Martin-Gay, Developmental Mathematics 41

42 3 3 1 * Rationalize Reduce the fraction. 3 Martin-Gay, Developmental Mathematics 4

43 Try it yourself. Simplify the following. 1) How about with variables 6 x ) x3 3 y y y x 3 y y y y x y 3 y y 3 x yy y x 3 y y Martin-Gay, Developmental Mathematics 43

44 1 3 4 * x x 5 4 x x x x x x x x 3x 18x 4x x *3x x 6 x x 9 4 x 5 3 3xy 8x y 1x 6 4 3xy 4 *7x xy y 4 *3x Martin-Gay, Developmental Mathematics 44 6???

45 Happy MONDAY! Quick Check on Wedn. This week: Will be on gradebook *Standard Form: vertex point, x and y intercepts, sketch *Vertex Form: vertex point, x and y intercepts, sketch *Radicals: Simplify, add, subtr, multi, div (rationalization) *Solving Radical equations Martin-Gay, Developmental Mathematics 45

46 Quick Check Tomorrow Martin-Gay, Developmental Mathematics 46

47 Self-Evaluation, Correction, Practice 1) Self Evaluation: Mark right or wrong with. CO Conceptual Errors i.e. wrong formulas, wrong solving steps CA computational Errors ) Write the actual correction work on separate paper Turn in (1) and () today 3) Practice: use the other version Martin-Gay, Developmental Mathematics 47

48 Simplify When radicals have binomial radicals in the denominator, multiply the numerator and denominator by the conjugate of the denominator to eliminate the radical in the denominator. Simplify by distributing or using FOIL. then continue to completely simplify Be sure to simplify all numbers outside the radical if they all have common factors. Martin-Gay, Developmental Mathematics 48

49 Rationalizing Binomial Radicals Martin-Gay, Developmental Mathematics 49

50 The Quadratic Formula Example Solve x(x + 6) = 30 by the quadratic formula. x + 6x + 30 = 0 a = 1, b = 6, c = 30 x 6 (6) (1) 4(1)(30) This is Complex Number!!!! Martin-Gay, Developmental Mathematics 50

51 The Discriminant The expression under the radical sign in the formula (b 4ac) is called the discriminant. The discriminant will take on a value that is positive, 0, or negative. The value of the discriminant indicates two distinct real solutions, one real solution, or no real solutions, respectively. Martin-Gay, Developmental Mathematics 51

52 The Discriminant Example Use the discriminant to determine the number and type of solutions for the following equation. 5 4x + 1x = 0 a = 1, b = 4, and c = 5 b 4ac = ( 4) 4(1)(5) = = 4 There are no real solutions. Martin-Gay, Developmental Mathematics 5

53 Solving Quadratic Equations Steps in Solving Quadratic Equations 1) If the equation is in the form (ax+b) = c, use the square root property to solve. ) If not solved in step 1, write the equation in standard form. 3) Try to solve by factoring. 4) If you haven t solved it yet, use the quadratic formula. Martin-Gay, Developmental Mathematics 53

54 Solving Equations Example Solve 1x = 4x = 4x 1x = 4(x 3x + 1) Let a = 1, b = -3, c = 1 x 3 ( 3) (1) 4(1)(1) Martin-Gay, Developmental Mathematics 54

55 Example Solving Equations Solve the following quadratic equation. 5 1 m m 8 5m 8m ( 5m )( m ) 0 5m 0 or m m or m 5 0 Martin-Gay, Developmental Mathematics 55

56 /7/17 Complex Number Martin-Gay, Developmental Mathematics 56

57 What imaginary roots actually look like on a graph (which is why we don t usually graph them) Martin-Gay, Developmental Mathematics 57

58 Before, negative But now a special symbol is no solution. i is assigned so we can carry on the computation. 1 i Martin-Gay, Developmental Mathematics 58

59 Examples i i 9 i 4i 9i Martin-Gay, Developmental Mathematics 59

60 i 1 Always! i 1 1 i i 1 ALWAYS! i 3 i i i i i 1i i 4 i iiii i i i i 5 6 iiiii i i i i iiiiii i i i 1 the pattern repeats... Martin-Gay, Developmental Mathematics 60

61 Simplify i 6 Method Divide the exponent by 4 (or ) and look at the remainder =6 with remainder Method i 6 i 1 i =13 ( 1) 1 Martin-Gay, Developmental Mathematics 61

62 Simplify each expression. a] 8i 3i 4i b] 5 0 i 5 i 0 Remember that 1 i i Martin-Gay, Developmental Mathematics 6

63 When adding or subtracting complex numbers, combine like terms. Ex: 8 3i 5i 8 3i 5i 10 i Martin-Gay, Developmental Mathematics 63

64 Multiplication: 8 5i 3i F O I L 16 4i 10i 15i 16 14i i Martin-Gay, Developmental Mathematics 64

65 Complex Numbers Warm-Ups 8 minutes Multiply and Simplify : a} i i b} (3 4 i) c} (3 4 i)(3 4 i) d} (4 i)(4 i) Martin-Gay, Developmental Mathematics 65

66 Division & Rationalizing the Denominator Ex] Denominator can only be an integer! 1 ex] 36i 1 3i i i i 3i i i 3 3 Try] 18 3i 7i 6 i 6 i 9i 9i i i i 9 i i i Martin-Gay, Developmental Mathematics 66

67 Warm-up 8 minutes Simplify : a] i i 0 55 b] (4 5 i)(4 5 i) c] ( 6 )( 6 ) Solve using Quadratic Formula, what is the discriminant? d y x x ] Martin-Gay, Developmental Mathematics 67

68 Radical expression and Exponents Exponents v.s. Radicals (Roots) therefore Martin-Gay, Developmental Mathematics 68

69 Other examples Martin-Gay, Developmental Mathematics 69

70 More Examples: a] 64 3 b] 64 c d e f 4 ] 64 5 ] 64 6 ] 64 7 ] only 6 of them, not enough Martin-Gay, Developmental Mathematics 70

71 More Examples: a] 81 3 b] 81 c d 4 ] 81 5 ] Not enough numbers still 3 81 Martin-Gay, Developmental Mathematics 71

72 Try 6 79 on calculator Rational Exponents did you get 9? Rational Exponent is actually for calculators 16 ^ ( 1 4 ) Martin-Gay, Developmental Mathematics 7 x y

73 Practice NOW! DOK1 Simplify : a] ( 4 i )( 7 i ) b] ( 3 i )( 3 i ) c] ( 3 )( 3 ) d] i i e] ( 8i ) DOK: a] f ( x) x 8x 3 b] g( x) x 6x 10 vertex d ] Solve? x int? y int? sketch c] Solve x 3 x ( x 4) 6 8 Martin-Gay, Developmental Mathematics 73

74 Answers a] 4 i b]13 c]1 d] i1 e] 60 3i vertex(, 5) 8 40 x int 4 0.4, 3.58 y int (0,3) vertex(3,1) 6 4 x int 6 i 3 y int (0,10) i Martin-Gay, Developmental Mathematics 74

75 Rationalization for Binomials When you have a Binomial for Denominator: a. 5i 3 i 3i 3i 64i15i 10i 96i6i4i 1611i i b Martin-Gay, Developmental Mathematics 75

76 Ex: Solve x + 6x +10 = 0 x a = b = c = b b 4ac a nd 1 st i 6 i 6 i and 3 i and 3i Martin-Gay, Developmental Mathematics 76

Concept Category 4. Quadratic Equations

Concept 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 information

P.1 Prerequisite skills Basic Algebra Skills

P.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 information

Concept Category 4. Polynomial Functions

Concept Category 4. Polynomial Functions Concept Category 4 Polynomial Functions (CC1) A Piecewise Equation 2 ( x 4) x 2 f ( x) ( x 3) 2 x 1 The graph for the piecewise Polynomial Graph (preview) Still the same transformations CC4 Learning Targets

More information

Unit 2-1: Factoring and Solving Quadratics. 0. I can add, subtract and multiply polynomial expressions

Unit 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 information

CHAPTER 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 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 information

HONORS GEOMETRY Summer Skills Set

HONORS 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 information

Solving Quadratic Equations Review

Solving Quadratic Equations Review Math III Unit 2: Polynomials Notes 2-1 Quadratic Equations Solving Quadratic Equations Review Name: Date: Period: Some quadratic equations can be solved by. Others can be solved just by using. ANY quadratic

More information

Final Exam C Name i D) 2. Solve the equation by factoring. 4) x2 = x + 72 A) {1, 72} B) {-8, 9} C) {-8, -9} D) {8, 9} 9 ± i

Final Exam C Name i D) 2. Solve the equation by factoring. 4) x2 = x + 72 A) {1, 72} B) {-8, 9} C) {-8, -9} D) {8, 9} 9 ± i Final Exam C Name First, write the value(s) that make the denominator(s) zero. Then solve the equation. 7 ) x + + 3 x - = 6 (x + )(x - ) ) A) No restrictions; {} B) x -, ; C) x -; {} D) x -, ; {2} Add

More information

Reference Material /Formulas for Pre-Calculus CP/ H Summer Packet

Reference Material /Formulas for Pre-Calculus CP/ H Summer Packet Reference Material /Formulas for Pre-Calculus CP/ H Summer Packet Week # 1 Order of Operations Step 1 Evaluate expressions inside grouping symbols. Order of Step 2 Evaluate all powers. Operations Step

More information

SOLUTIONS FOR PROBLEMS 1-30

SOLUTIONS FOR PROBLEMS 1-30 . Answer: 5 Evaluate x x + 9 for x SOLUTIONS FOR PROBLEMS - 0 When substituting x in x be sure to do the exponent before the multiplication by to get (). + 9 5 + When multiplying ( ) so that ( 7) ( ).

More information

1 Quadratic Functions

1 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 information

SUMMER REVIEW PACKET. Name:

SUMMER REVIEW PACKET. Name: Wylie East HIGH SCHOOL SUMMER REVIEW PACKET For students entering Regular PRECALCULUS Name: Welcome to Pre-Calculus. The following packet needs to be finished and ready to be turned the first week of the

More information

Algebra 2 Honors: Final Exam Review

Algebra 2 Honors: Final Exam Review Name: Class: Date: Algebra 2 Honors: Final Exam Review Directions: You may write on this review packet. Remember that this packet is similar to the questions that you will have on your final exam. Attempt

More information

Chapter 4: Radicals and Complex Numbers

Chapter 4: Radicals and Complex Numbers Chapter : Radicals and Complex Numbers Section.1: A Review of the Properties of Exponents #1-: Simplify the expression. 1) x x ) z z ) a a ) b b ) 6) 7) x x x 8) y y y 9) x x y 10) y 8 b 11) b 7 y 1) y

More information

UNIT 4: RATIONAL AND RADICAL EXPRESSIONS. 4.1 Product Rule. Objective. Vocabulary. o Scientific Notation. o Base

UNIT 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 information

Study Guide for Math 095

Study Guide for Math 095 Study Guide for Math 095 David G. Radcliffe November 7, 1994 1 The Real Number System Writing a fraction in lowest terms. 1. Find the largest number that will divide into both the numerator and the denominator.

More information

30 Wyner Math Academy I Fall 2015

30 Wyner Math Academy I Fall 2015 30 Wyner Math Academy I Fall 2015 CHAPTER FOUR: QUADRATICS AND FACTORING Review November 9 Test November 16 The most common functions in math at this level are quadratic functions, whose graphs are parabolas.

More information

Lesson 7.1 Polynomial Degree and Finite Differences

Lesson 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 information

Herndon High School Geometry Honors Summer Assignment

Herndon 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 information

HOSTOS COMMUNITY COLLEGE DEPARTMENT OF MATHEMATICS. MAT 010 or placement on the COMPASS/CMAT

HOSTOS COMMUNITY COLLEGE DEPARTMENT OF MATHEMATICS. MAT 010 or placement on the COMPASS/CMAT 1 HOSTOS COMMUNITY COLLEGE DEPARTMENT OF MATHEMATICS MAT 00 ELEMENTARY ALGEBRA CREDIT HOURS: 0.0 EQUATED HOURS: 4.5 CLASS HOURS: 4.5 + PREREQUISITE: REQUIRED TEXTS: DESCRIPTION: EXAMINATIONS: GRADES: MAT

More information

Chapter 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 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 information

Solving Quadratic Equations by Formula

Solving 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 information

Algebra Summer Review Packet

Algebra Summer Review Packet Name: Algebra Summer Review Packet About Algebra 1: Algebra 1 teaches students to think, reason, and communicate mathematically. Students use variables to determine solutions to real world problems. Skills

More information

HOSTOS COMMUNITY COLLEGE DEPARTMENT OF MATHEMATICS

HOSTOS COMMUNITY COLLEGE DEPARTMENT OF MATHEMATICS HOSTOS COMMUNITY COLLEGE DEPARTMENT OF MATHEMATICS MAT 020 ELEMENTARY ALGEBRA CREDIT HOURS: 0.0 EQUATED HOURS: 4.5 CLASS HOURS: 4.5 PREREQUISITE: REQUIRED TEXTS: MAT 010 or placement on ACCUPLACER Martin-Gay,

More information

CHAPTER 3: Quadratic Functions and Equations; Inequalities

CHAPTER 3: Quadratic Functions and Equations; Inequalities MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 3: Quadratic Functions and Equations; Inequalities 3.1 The Complex Numbers 3.2 Quadratic Equations, Functions, Zeros, and

More information

Final Exam A Name. 20 i C) Solve the equation by factoring. 4) x2 = x + 30 A) {-5, 6} B) {5, 6} C) {1, 30} D) {-5, -6} -9 ± i 3 14

Final Exam A Name. 20 i C) Solve the equation by factoring. 4) x2 = x + 30 A) {-5, 6} B) {5, 6} C) {1, 30} D) {-5, -6} -9 ± i 3 14 Final Exam A Name First, write the value(s) that make the denominator(s) zero. Then solve the equation. 1 1) x + 3 + 5 x - 3 = 30 (x + 3)(x - 3) 1) A) x -3, 3; B) x -3, 3; {4} C) No restrictions; {3} D)

More information

LESSON 9.1 ROOTS AND RADICALS

LESSON 9.1 ROOTS AND RADICALS LESSON 9.1 ROOTS AND RADICALS LESSON 9.1 ROOTS AND RADICALS 67 OVERVIEW Here s what you ll learn in this lesson: Square Roots and Cube Roots a. Definition of square root and cube root b. Radicand, radical

More information

Equations and Inequalities. College Algebra

Equations and Inequalities. College Algebra Equations and Inequalities College Algebra Radical Equations Radical Equations: are equations that contain variables in the radicand How to Solve a Radical Equation: 1. Isolate the radical expression on

More information

Beginning Algebra. 1. Review of Pre-Algebra 1.1 Review of Integers 1.2 Review of Fractions

Beginning Algebra. 1. Review of Pre-Algebra 1.1 Review of Integers 1.2 Review of Fractions 1. Review of Pre-Algebra 1.1 Review of Integers 1.2 Review of Fractions Beginning Algebra 1.3 Review of Decimal Numbers and Square Roots 1.4 Review of Percents 1.5 Real Number System 1.6 Translations:

More information

Chapter 4: Radicals and Complex Numbers

Chapter 4: Radicals and Complex Numbers Section 4.1: A Review of the Properties of Exponents #1-42: Simplify the expression. 1) x 2 x 3 2) z 4 z 2 3) a 3 a 4) b 2 b 5) 2 3 2 2 6) 3 2 3 7) x 2 x 3 x 8) y 4 y 2 y 9) 10) 11) 12) 13) 14) 15) 16)

More information

Never leave a NEGATIVE EXPONENT or a ZERO EXPONENT in an answer in simplest form!!!!!

Never leave a NEGATIVE EXPONENT or a ZERO EXPONENT in an answer in simplest form!!!!! 1 ICM Unit 0 Algebra Rules Lesson 1 Rules of Exponents RULE EXAMPLE EXPLANANTION a m a n = a m+n A) x x 6 = B) x 4 y 8 x 3 yz = When multiplying with like bases, keep the base and add the exponents. a

More information

Are you ready for Algebra 3? Summer Packet *Required for all Algebra 3/Trigonometry Students*

Are you ready for Algebra 3? Summer Packet *Required for all Algebra 3/Trigonometry Students* Name: Date: Period: Are you ready for Algebra? Summer Packet *Required for all Students* The course prepares students for Pre Calculus and college math courses. In order to accomplish this, the course

More information

Math 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. 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 information

Secondary Math 2H Unit 3 Notes: Factoring and Solving Quadratics

Secondary Math 2H Unit 3 Notes: Factoring and Solving Quadratics Secondary Math H Unit 3 Notes: Factoring and Solving Quadratics 3.1 Factoring out the Greatest Common Factor (GCF) Factoring: The reverse of multiplying. It means figuring out what you would multiply together

More information

ACP ALGEBRA II MIDTERM REVIEW PACKET

ACP ALGEBRA II MIDTERM REVIEW PACKET ACP ALGEBRA II MIDTERM REVIEW PACKET 0-8 Name Per Date This review packet includes new problems and a list of problems from the textbook. The answers to the problems from the textbook can be found in the

More information

Evaluate the expression if x = 2 and y = 5 6x 2y Original problem Substitute the values given into the expression and multiply

Evaluate the expression if x = 2 and y = 5 6x 2y Original problem Substitute the values given into the expression and multiply Name EVALUATING ALGEBRAIC EXPRESSIONS Objective: To evaluate an algebraic expression Example Evaluate the expression if and y = 5 6x y Original problem 6() ( 5) Substitute the values given into the expression

More information

Module 1: Whole Numbers Module 2: Fractions Module 3: Decimals and Percent Module 4: Real Numbers and Introduction to Algebra

Module 1: Whole Numbers Module 2: Fractions Module 3: Decimals and Percent Module 4: Real Numbers and Introduction to Algebra Course Title: College Preparatory Mathematics I Prerequisite: Placement with a score below 20 on ACT, below 450 on SAT, or assessing into Basic Applied Mathematics or Basic Algebra using Accuplacer, ASSET

More information

Pre-Calculus Summer Packet

Pre-Calculus Summer Packet 2013-2014 Pre-Calculus Summer Packet 1. Complete the attached summer packet, which is due on Friday, September 6, 2013. 2. The material will be reviewed in class on Friday, September 6 and Monday, September

More information

Chapter 1.6. Perform Operations with Complex Numbers

Chapter 1.6. Perform Operations with Complex Numbers Chapter 1.6 Perform Operations with Complex Numbers EXAMPLE Warm-Up 1 Exercises Solve a quadratic equation Solve 2x 2 + 11 = 37. 2x 2 + 11 = 37 2x 2 = 48 Write original equation. Subtract 11 from each

More information

Math 0320 Final Exam Review

Math 0320 Final Exam Review Math 0320 Final Exam Review SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Factor out the GCF using the Distributive Property. 1) 6x 3 + 9x 1) Objective:

More information

Summer Work for students entering PreCalculus

Summer 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 information

Unit 2 Day 7. Quadratic Formula & the Discriminant

Unit 2 Day 7. Quadratic Formula & the Discriminant Unit Day 7 Quadratic Formula & the Discriminant 1 Warm Up Day 7 1. Solve each of the quadratic functions by graphing and algebraic reasoning: a. x 3 = 0 b. x + 5x 8 = 0 c. Explain why having alternative

More information

Algebra I Unit Report Summary

Algebra I Unit Report Summary Algebra I Unit Report Summary No. Objective Code NCTM Standards Objective Title Real Numbers and Variables Unit - ( Ascend Default unit) 1. A01_01_01 H-A-B.1 Word Phrases As Algebraic Expressions 2. A01_01_02

More information

Summer Mathematics Packet Say Hello to Algebra 2. For Students Entering Algebra 2

Summer Mathematics Packet Say Hello to Algebra 2. For Students Entering Algebra 2 Summer Math Packet Student Name: Say Hello to Algebra 2 For Students Entering Algebra 2 This summer math booklet was developed to provide students in middle school an opportunity to review grade level

More information

B. Complex number have a Real part and an Imaginary part. 1. written as a + bi some Examples: 2+3i; 7+0i; 0+5i

B. Complex number have a Real part and an Imaginary part. 1. written as a + bi some Examples: 2+3i; 7+0i; 0+5i Section 11.8 Complex Numbers I. The Complex Number system A. The number i = -1 1. 9 and 24 B. Complex number have a Real part and an Imaginary part II. Powers of i 1. written as a + bi some Examples: 2+3i;

More information

Intermediate Algebra with Applications

Intermediate Algebra with Applications Lakeshore Technical College 10-804-118 Intermediate Algebra with Applications Course Outcome Summary Course Information Alternate Title Description Total Credits 4 Total Hours 72 Pre/Corequisites Prerequisite

More information

( ) c. m = 0, 1 2, 3 4

( ) c. m = 0, 1 2, 3 4 G Linear Functions Probably the most important concept from precalculus that is required for differential calculus is that of linear functions The formulas you need to know backwards and forwards are:

More information

Chapter 1A -- Real Numbers. iff. Math Symbols: Sets of Numbers

Chapter 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 information

Section 3.6 Complex Zeros

Section 3.6 Complex Zeros 04 Chapter Section 6 Complex Zeros When finding the zeros of polynomials, at some point you're faced with the problem x = While there are clearly no real numbers that are solutions to this equation, leaving

More information

Natural Numbers Positive Integers. Rational Numbers

Natural Numbers Positive Integers. Rational Numbers Chapter A - - Real Numbers Types of Real Numbers, 2,, 4, Name(s) for the set Natural Numbers Positive Integers Symbol(s) for the set, -, - 2, - Negative integers 0,, 2,, 4, Non- negative integers, -, -

More information

ALGEBRA 2 Summer Review Assignments Graphing

ALGEBRA 2 Summer Review Assignments Graphing ALGEBRA 2 Summer Review Assignments Graphing To be prepared for algebra two, and all subsequent math courses, you need to be able to accurately and efficiently find the slope of any line, be able to write

More information

Homework. Basic properties of real numbers. Adding, subtracting, multiplying and dividing real numbers. Solve one step inequalities with integers.

Homework. Basic properties of real numbers. Adding, subtracting, multiplying and dividing real numbers. Solve one step inequalities with integers. Morgan County School District Re-3 A.P. Calculus August What is the language of algebra? Graphing real numbers. Comparing and ordering real numbers. Finding absolute value. September How do you solve one

More information

To solve a radical equation, you must take both sides of an equation to a power.

To solve a radical equation, you must take both sides of an equation to a power. Topic 5 1 Radical Equations A radical equation is an equation with at least one radical expression. There are four types we will cover: x 35 3 4x x 1x 7 3 3 3 x 5 x 1 To solve a radical equation, you must

More information

EXPONENT REVIEW!!! Concept Byte (Review): Properties of Exponents. Property of Exponents: Product of Powers. x m x n = x m + n

EXPONENT REVIEW!!! Concept Byte (Review): Properties of Exponents. Property of Exponents: Product of Powers. x m x n = x m + n Algebra B: Chapter 6 Notes 1 EXPONENT REVIEW!!! Concept Byte (Review): Properties of Eponents Recall from Algebra 1, the Properties (Rules) of Eponents. Property of Eponents: Product of Powers m n = m

More information

Honors Algebra II Final Exam Order - Fall 2018

Honors Algebra II Final Exam Order - Fall 2018 Honors Algebra II Final Exam Order - Fall 2018 For the Final Exam for Algebra II, students will be given the opportunity to re-take any of their Fall 2018 Assessments. To do so they will need to place

More information

5.1 Monomials. Algebra 2

5.1 Monomials. Algebra 2 . Monomials Algebra Goal : A..: Add, subtract, multiply, and simplify polynomials and rational expressions (e.g., multiply (x ) ( x + ); simplify 9x x. x Goal : Write numbers in scientific notation. Scientific

More information

Radicals: To simplify means that 1) no radicand has a perfect square factor and 2) there is no radical in the denominator (rationalize).

Radicals: To simplify means that 1) no radicand has a perfect square factor and 2) there is no radical in the denominator (rationalize). Summer Review Packet for Students Entering Prealculus Radicals: To simplify means that 1) no radicand has a perfect square factor and ) there is no radical in the denominator (rationalize). Recall the

More information

Math-2 Lesson 2-4. Radicals

Math-2 Lesson 2-4. Radicals Math- Lesson - Radicals = What number is equivalent to the square root of? Square both sides of the equation ( ) ( ) = = = is an equivalent statement to = 1.7 1.71 1.70 1.701 1.7008... There is no equivalent

More information

due date: third day of class estimated time: 10 hours (for planning purposes only; work until you finish)

due date: third day of class estimated time: 10 hours (for planning purposes only; work until you finish) Honors PreCalculus Summer Work 016 due date: third day of class estimated time: 10 hours (for planning purposes only; work until you finish) Dear Honors PreCalculus Students, This assignment is designed

More information

Algebra II Notes Quadratic Functions Unit Complex Numbers. Math Background

Algebra II Notes Quadratic Functions Unit Complex Numbers. Math Background Complex Numbers Math Background Previously, you Studied the real number system and its sets of numbers Applied the commutative, associative and distributive properties to real numbers Used the order of

More information

Polynomials: Adding, Subtracting, & Multiplying (5.1 & 5.2)

Polynomials: 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 information

Section 1.3 Review of Complex Numbers

Section 1.3 Review of Complex Numbers 1 Section 1. Review of Complex Numbers Objective 1: Imaginary and Complex Numbers In Science and Engineering, such quantities like the 5 occur all the time. So, we need to develop a number system that

More information

Summer Work for students entering PreCalculus

Summer 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 information

Simplifying Radical Expressions

Simplifying Radical Expressions Simplifying Radical Expressions Product Property of Radicals For any real numbers a and b, and any integer n, n>1, 1. If n is even, then When a and b are both nonnegative. n ab n a n b 2. If n is odd,

More information

REAL WORLD SCENARIOS: PART IV {mostly for those wanting 114 or higher} 1. If 4x + y = 110 where 10 < x < 20, what is the least possible value of y?

REAL WORLD SCENARIOS: PART IV {mostly for those wanting 114 or higher} 1. If 4x + y = 110 where 10 < x < 20, what is the least possible value of y? REAL WORLD SCENARIOS: PART IV {mostly for those wanting 114 or higher} REAL WORLD SCENARIOS 1. If 4x + y = 110 where 10 < x < 0, what is the least possible value of y? WORK AND ANSWER SECTION. Evaluate

More information

3.1 Solving Quadratic Equations by Factoring

3.1 Solving Quadratic Equations by Factoring 3.1 Solving Quadratic Equations by Factoring A function of degree (meaning the highest exponent on the variable is ) is called a Quadratic Function. Quadratic functions are written as, for example, f(x)

More information

Solving Quadratic & Higher Degree Equations

Solving 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 information

SECTION Types of Real Numbers The natural numbers, positive integers, or counting numbers, are

SECTION Types of Real Numbers The natural numbers, positive integers, or counting numbers, are SECTION.-.3. Types of Real Numbers The natural numbers, positive integers, or counting numbers, are The negative integers are N = {, 2, 3,...}. {..., 4, 3, 2, } The integers are the positive integers,

More information

Foundations of Math II Unit 5: Solving Equations

Foundations 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 information

Scope and Sequence Mathematics Algebra 2 400

Scope and Sequence Mathematics Algebra 2 400 Scope and Sequence Mathematics Algebra 2 400 Description : Students will study real numbers, complex numbers, functions, exponents, logarithms, graphs, variation, systems of equations and inequalities,

More information

Complex Numbers: Definition: A complex number is a number of the form: z = a + bi where a, b are real numbers and i is a symbol with the property: i

Complex Numbers: Definition: A complex number is a number of the form: z = a + bi where a, b are real numbers and i is a symbol with the property: i Complex Numbers: Definition: A complex number is a number of the form: z = a + bi where a, b are real numbers and i is a symbol with the property: i 2 = 1 Sometimes we like to think of i = 1 We can treat

More information

2.1 Quadratic Functions

2.1 Quadratic Functions Date:.1 Quadratic Functions Precalculus Notes: Unit Polynomial Functions Objective: The student will sketch the graph of a quadratic equation. The student will write the equation of a quadratic function.

More information

A quadratic expression is a mathematical expression that can be written in the form 2

A quadratic expression is a mathematical expression that can be written in the form 2 118 CHAPTER Algebra.6 FACTORING AND THE QUADRATIC EQUATION Textbook Reference Section 5. CLAST OBJECTIVES Factor a quadratic expression Find the roots of a quadratic equation A quadratic expression is

More information

Chapter R - Basic Algebra Operations (94 topics, no due date)

Chapter R - Basic Algebra Operations (94 topics, no due date) Course Name: Math 00024 Course Code: N/A ALEKS Course: College Algebra Instructor: Master Templates Course Dates: Begin: 08/15/2014 End: 08/15/2015 Course Content: 207 topics Textbook: Barnett/Ziegler/Byleen/Sobecki:

More information

Answers to Sample Exam Problems

Answers to Sample Exam Problems Math Answers to Sample Exam Problems () Find the absolute value, reciprocal, opposite of a if a = 9; a = ; Absolute value: 9 = 9; = ; Reciprocal: 9 ; ; Opposite: 9; () Commutative law; Associative law;

More information

Day 6: 6.4 Solving Polynomial Equations Warm Up: Factor. 1. x 2-2x x 2-9x x 2 + 6x + 5

Day 6: 6.4 Solving Polynomial Equations Warm Up: Factor. 1. x 2-2x x 2-9x x 2 + 6x + 5 Day 6: 6.4 Solving Polynomial Equations Warm Up: Factor. 1. x 2-2x - 15 2. x 2-9x + 14 3. x 2 + 6x + 5 Solving Equations by Factoring Recall the factoring pattern: Difference of Squares:...... Note: There

More information

Math 101 Study Session Spring 2016 Test 4 Chapter 10, Chapter 11 Chapter 12 Section 1, and Chapter 12 Section 2

Math 101 Study Session Spring 2016 Test 4 Chapter 10, Chapter 11 Chapter 12 Section 1, and Chapter 12 Section 2 Math 101 Study Session Spring 2016 Test 4 Chapter 10, Chapter 11 Chapter 12 Section 1, and Chapter 12 Section 2 April 11, 2016 Chapter 10 Section 1: Addition and Subtraction of Polynomials A monomial is

More information

Algebra 2 Segment 1 Lesson Summary Notes

Algebra 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 information

P.1: Algebraic Expressions, Mathematical Models, and Real Numbers

P.1: Algebraic Expressions, Mathematical Models, and Real Numbers Chapter P Prerequisites: Fundamental Concepts of Algebra Pre-calculus notes Date: P.1: Algebraic Expressions, Mathematical Models, and Real Numbers Algebraic expression: a combination of variables and

More information

Math Academy I Fall Study Guide. CHAPTER ONE: FUNDAMENTALS Due Thursday, December 8

Math Academy I Fall Study Guide. CHAPTER ONE: FUNDAMENTALS Due Thursday, December 8 Name: Math Academy I Fall Study Guide CHAPTER ONE: FUNDAMENTALS Due Thursday, December 8 1-A Terminology natural integer rational real complex irrational imaginary term expression argument monomial degree

More information

Roots 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

Roots 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 information

MATH Spring 2010 Topics per Section

MATH Spring 2010 Topics per Section MATH 101 - Spring 2010 Topics per Section Chapter 1 : These are the topics in ALEKS covered by each Section of the book. Section 1.1 : Section 1.2 : Ordering integers Plotting integers on a number line

More information

Algebra 1. Math Review Packet. Equations, Inequalities, Linear Functions, Linear Systems, Exponents, Polynomials, Factoring, Quadratics, Radicals

Algebra 1. Math Review Packet. Equations, Inequalities, Linear Functions, Linear Systems, Exponents, Polynomials, Factoring, Quadratics, Radicals Algebra 1 Math Review Packet Equations, Inequalities, Linear Functions, Linear Systems, Exponents, Polynomials, Factoring, Quadratics, Radicals 2017 Math in the Middle 1. Clear parentheses using the distributive

More information

7.5 Rationalizing Denominators and Numerators of Radical Expressions

7.5 Rationalizing Denominators and Numerators of Radical Expressions 440 CHAPTER Rational Exponents, Radicals, and Complex Numbers 86. Find the area and perimeter of the trapezoid. (Hint: The area of a trapezoid is the product of half the height 6 meters and the sum of

More information

Algebra 2 Honors Final Exam StudyGuide

Algebra 2 Honors Final Exam StudyGuide Name: Score: 0 / 80 points (0%) Algebra 2 Honors Final Exam StudyGuide Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Simplify. 2. D Multiply the numerator

More information

Chapter R - Review of Basic Algebraic Concepts (26 topics, no due date)

Chapter R - Review of Basic Algebraic Concepts (26 topics, no due date) Course Name: Math 00023 Course Code: N/A ALEKS Course: Intermediate Algebra Instructor: Master Templates Course Dates: Begin: 08/15/2014 End: 08/15/2015 Course Content: 245 topics Textbook: Miller/O'Neill/Hyde:

More information

COWLEY COLLEGE & Area Vocational Technical School

COWLEY COLLEGE & Area Vocational Technical School COWLEY COLLEGE & Area Vocational Technical School COURSE PROCEDURE FOR Student Level: This course is open to students on the college level in their freshman or sophomore year. Catalog Description: INTERMEDIATE

More information

MAT116 Final Review Session Chapter 3: Polynomial and Rational Functions

MAT116 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 information

CP Algebra 2. Unit 2-1 Factoring and Solving Quadratics

CP 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 information

Solving Multi-Step Equations

Solving Multi-Step Equations 1. Clear parentheses using the distributive property. 2. Combine like terms within each side of the equal sign. Solving Multi-Step Equations 3. Add/subtract terms to both sides of the equation to get the

More information

Multiplication of Polynomials

Multiplication of Polynomials Summary 391 Chapter 5 SUMMARY Section 5.1 A polynomial in x is defined by a finite sum of terms of the form ax n, where a is a real number and n is a whole number. a is the coefficient of the term. n is

More information

Algebra II Unit #2 4.6 NOTES: Solving Quadratic Equations (More Methods) Block:

Algebra 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 information

Subtract 6 to both sides Divide by 2 on both sides. Cross Multiply. Answer: x = -9

Subtract 6 to both sides Divide by 2 on both sides. Cross Multiply. Answer: x = -9 Subtract 6 to both sides Divide by 2 on both sides Answer: x = -9 Cross Multiply. = 3 Distribute 2 to parenthesis Combine like terms Subtract 4x to both sides Subtract 10 from both sides x = -20 Subtract

More information

Algebra 2 Chapter 3 Part 1 Practice Test 2018

Algebra 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 information

Math 95 Practice Final Exam

Math 95 Practice Final Exam Part 1: No Calculator Evaluating Functions and Determining their Domain and Range The domain of a function is the set of all possible inputs, which are typically x-values. The range of a function is the

More information

One box per group ( star group of 6)

One box per group ( star group of 6) 4 markers 2 erasers One box per group ( star group of 6) 1 pencil (just in case) Some small post-it notes 1 glue stick One person from each group collect all items and place them back into the box. Concept

More information

Polynomial Functions

Polynomial Functions Polynomial Functions Polynomials A Polynomial in one variable, x, is an expression of the form a n x 0 a 1 x n 1... a n 2 x 2 a n 1 x a n The coefficients represent complex numbers (real or imaginary),

More information

MA094 Part 2 - Beginning Algebra Summary

MA094 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 information

Quadratic Functions. Key Terms. Slide 1 / 200. Slide 2 / 200. Slide 3 / 200. Table of Contents

Quadratic Functions. Key Terms. Slide 1 / 200. Slide 2 / 200. Slide 3 / 200. Table of Contents Slide 1 / 200 Quadratic Functions Table of Contents Key Terms Identify Quadratic Functions Explain Characteristics of Quadratic Functions Solve Quadratic Equations by Graphing Solve Quadratic Equations

More information