UNIT 5 QUADRATIC FUNCTIONS Lesson 2: Creating and Solving Quadratic Equations in One Variable Instruction
|
|
- Helena Ward
- 5 years ago
- Views:
Transcription
1 Lesson : Creating and Solving Quadratic Equations in One Variale Prerequisite Skills This lesson requires the use of the following skills: understanding real numers and complex numers understanding rational numers and irrational numers applying the order of operations Introduction Completing the square can e a long process, and not all quadratic expressions can e factored. Rather than completing the square or factoring, we can use a formula that can e derived from the process of completing the square. This formula, called the quadratic formula, can e used to solve any quadratic equation in standard form, ax + x + c = 0. Key Concepts A quadratic equation in standard form, ax + x + c = 0, can e solved for x y using the ac quadratic formula: x = ±. a Solutions of quadratic equations are also called roots. The expression under the radical, ac, is called the discriminant. The discriminant tells us the numer and type of solutions for the equation. Discriminant Numer and type of solutions Negative Two complex solutions 0 One real, rational solution Positive and a perfect square Two real, rational solutions Positive and not a perfect square Two real, irrational solutions Common Errors/Misconceptions not setting the quadratic equation equal to 0 efore determining the values of a,, and c forgetting to use ± for prolems with two solutions forgetting to change the sign of dividing y a or y instead of y a not correctly following the order of operations not fully simplifying solutions U-78
2 Lesson : Creating and Solving Quadratic Equations in One Variale Guided Practice.. Example 1 Given the standard form of a quadratic equation, ax + x + c, derive the quadratic formula y completing the square. 1. Begin with a quadratic equation in standard form. ax + x + c = 0. Sutract c from oth sides. ax + x = c 3. Divide oth sides y a. x a x c + = a. Complete the square. Add the square of half of the coefficient of the x-term to oth sides to complete the square. Add a to oth sides to complete the square. x + a x + c a = + a a U-79
3 Lesson : Creating and Solving Quadratic Equations in One Variale. Write the left side as a inomial squared and simplify the right side. c a x + a = + a ac x + a = + a a ac x + a = a 6. Take the square root of oth sides and simplify the right side. x + =± a x + =± a 7. Sutract a x = ± a ac a ac a from oth sides to solve for x. ac a 8. The quadratic formula can e written as two fractions, as in step 7. However, the fractions are often comined. ac a U-80
4 Lesson : Creating and Solving Quadratic Equations in One Variale Example Use the discriminant of 3x x + 1 = 0 to identify the numer and type of solutions. 1. Determine a,, and c. a = 3, =, and c = 1. Sustitute the values for a,, and c into the formula for the discriminant, ac. ac = ( ) (3)(1) = 1 = 13 The discriminant of 3x x + 1 = 0 is Use what you know aout the discriminant to determine the numer and type of solutions for the quadratic equation. The discriminant, 13, is positive, ut it is not a perfect square. Therefore, there will e two real, irrational solutions. U-81
5 Lesson : Creating and Solving Quadratic Equations in One Variale Example 3 Solve x x = 1 using the quadratic formula. 1. Write the quadratic in standard form. x x = 1 Original equation x x 1 = 0 Sutract 1 from oth sides.. Determine the values of a,, and c. a =, =, and c = 1 3. Sustitute the values of a,, and c into the quadratic formula. ac a x = ( ) ± ( ) ( )( 1 ) ( ) x = ± + 96 x = ± 11 x = ± 11 Quadratic formula Sustitute values for a,, and c. Simplify.. Determine the solution(s). Since the discriminant, 11, is positive and a perfect square, there are two real, rational solutions. Write the fraction from step 3 as two fractions and simplify. x = = = x = 11 = 6 = 3 x = x = 3 The solutions to the equation x x = 1 are x = or x = 3. U-8
6 Lesson : Creating and Solving Quadratic Equations in One Variale Example Solve x = x 1 using the quadratic formula. 1. Put the quadratic in standard form. x = x 1 x x = 1 x x + 1 = 0 Original equation Sutract x from oth sides. Add 1 to oth sides.. Determine the values of a,, and c. a = 1, =, and c = 1 3. Sustitute the values of a,, and c into the quadratic formula. ac a x = ( ) ± ( ) 1 ()() 1 1 () x = ± x = ± 0 x = ± 0 Quadratic formula Sustitute values for a,, and c. Simplify.. Determine the solution(s). Since the discriminant is 0, there is one real, rational solution. x = ± 0 = = 1 x = 1 The solution to the equation x = x 1 is x = 1. U-83
7 Lesson : Creating and Solving Quadratic Equations in One Variale Example Solve x + x + 3 = 0 using the quadratic formula. 1. The quadratic is already in standard form. Determine the values of a,, and c. a =, =, and c = 3. Sustitute these values into the quadratic formula. ac a x = ( ) ± ( ) ( )() 3 ( ) x = ± 60 x = ± 6 Quadratic formula Sustitute values for a,, and c. Simplify. 3. Determine the solution(s). Since the discriminant is negative, there are two complex solutions. Simplify to determine the solutions. x = ± 6 x = ± 1 1 i x = ± 1 x = 1±i 1 The solutions to the equation x + x + 3 = 0 are x = 1±i 1. U-8
UNIT 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 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 informationFinding Complex Solutions of Quadratic Equations
y - y - - - x x Locker LESSON.3 Finding Complex Solutions of Quadratic Equations Texas Math Standards The student is expected to: A..F Solve quadratic and square root equations. Mathematical Processes
More informationLesson 3.5 Exercises, pages
Lesson 3.5 Exercises, pages 232 238 A 4. Calculate the value of the discriminant for each quadratic equation. a) 5x 2-9x + 4 = 0 b) 3x 2 + 7x - 2 = 0 In b 2 4ac, substitute: In b 2 4ac, substitute: a 5,
More informationUNIT 4 EXTENDING THE NUMBER SYSTEM Lesson 3: Operating with Complex Numbers Instruction
Prerequisite Skills This lesson requires the use of the following skills: simplifying expressions using properties of exponents finding quotients that include remainders understanding the real number system
More informationQUADRATIC EQUATIONS. + 6 = 0 This is a quadratic equation written in standard form. x x = 0 (standard form with c=0). 2 = 9
QUADRATIC EQUATIONS A quadratic equation is always written in the form of: a + b + c = where a The form a + b + c = is called the standard form of a quadratic equation. Eamples: 5 + 6 = This is a quadratic
More informationUNIT 5 QUADRATIC FUNCTIONS Lesson 1: Interpreting Structure in Expressions Instruction
Prerequisite Skills This lesson requires the use of the following skills: evaluating expressions using the order of operations evaluating expressions for a given value identifying parts of an expression
More informationUNIT 3 REASONING WITH EQUATIONS Lesson 2: Solving Systems of Equations Instruction
Prerequisite Skills This lesson requires the use of the following skills: graphing equations of lines using properties of equality to solve equations Introduction Two equations that are solved together
More informationGUIDED NOTES 2.5 QUADRATIC EQUATIONS
GUIDED NOTES 5 QUADRATIC EQUATIONS LEARNING OBJECTIVES In this section, you will: Solve quadratic equations by factoring. Solve quadratic equations by the square root property. Solve quadratic equations
More informationQUADRATIC EQUATIONS EXPECTED BACKGROUND KNOWLEDGE
6 QUADRATIC EQUATIONS In this lesson, you will study aout quadratic equations. You will learn to identify quadratic equations from a collection of given equations and write them in standard form. You will
More informationSolving Systems of Linear Equations Symbolically
" Solving Systems of Linear Equations Symolically Every day of the year, thousands of airline flights crisscross the United States to connect large and small cities. Each flight follows a plan filed with
More informationSection 2.1: Reduce Rational Expressions
CHAPTER Section.: Reduce Rational Expressions Section.: Reduce Rational Expressions Ojective: Reduce rational expressions y dividing out common factors. A rational expression is a quotient of polynomials.
More informationPolynomial Degree and Finite Differences
CONDENSED LESSON 7.1 Polynomial Degree and Finite Differences In this lesson, you Learn the terminology associated with polynomials Use the finite differences method to determine the degree of a polynomial
More informationa b a b ab b b b Math 154B Elementary Algebra Spring 2012
Math 154B Elementar Algera Spring 01 Stud Guide for Eam 4 Eam 4 is scheduled for Thursda, Ma rd. You ma use a " 5" note card (oth sides) and a scientific calculator. You are epected to know (or have written
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 informationObjectives To solve equations by completing the square To rewrite functions by completing the square
4-6 Completing the Square Content Standard Reviews A.REI.4. Solve quadratic equations y... completing the square... Ojectives To solve equations y completing the square To rewrite functions y completing
More informationCompleting the Square
3.5 Completing the Square Essential Question How can you complete the square for a quadratic epression? Using Algera Tiles to Complete the Square Work with a partner. Use algera tiles to complete the square
More informationStudy Guide and Intervention. The Quadratic Formula and the Discriminant. Quadratic Formula. Replace a with 1, b with -5, and c with -14.
4-6 Study Guide and Intervention Quadratic Formula The Quadratic Formula can be used to solve any quadratic equation once it is written in the form ax 2 + bx + c = 0. Quadratic Formula The solutions of
More informationInstructor Quick Check: Question Block 11
Instructor Quick Check: Question Block 11 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 informationSummer MA Lesson 11 Section 1.5 (part 1)
Summer MA 500 Lesson Section.5 (part ) The general form of a quadratic equation is a + b + c = 0, where a, b, and c are real numbers and a 0. This is a second degree equation. There are four ways to possibly
More informationLEARNING OBJECTIVES. guided practice Teacher: anticipates, monitors, selects, sequences, and connects student work. - developing essential skills
H Quadratics, Lesson, Using the Discriminant (r. 018) QUADRATICS Using the Discriminant Common Core Standard A-REI.4 Solve quadratic equations y inspection (e.g., for x =49), taking square roots, completing
More informationNAME DATE PERIOD. A negative exponent is the result of repeated division. Extending the pattern below shows that 4 1 = 1 4 or 1. Example: 6 4 = 1 6 4
Lesson 4.1 Reteach Powers and Exponents A number that is expressed using an exponent is called a power. The base is the number that is multiplied. The exponent tells how many times the base is used as
More information1Number ONLINE PAGE PROOFS. systems: real and complex. 1.1 Kick off with CAS
1Numer systems: real and complex 1.1 Kick off with CAS 1. Review of set notation 1.3 Properties of surds 1. The set of complex numers 1.5 Multiplication and division of complex numers 1.6 Representing
More informationUnit 5 Solving Quadratic Equations
SM Name: Period: Unit 5 Solving Quadratic Equations 5.1 Solving Quadratic Equations by Factoring Quadratic Equation: Any equation that can be written in the form a b c + + = 0, where a 0. Zero Product
More informationERASMUS UNIVERSITY ROTTERDAM Information concerning the Entrance examination Mathematics level 2 for International Business Administration (IBA)
ERASMUS UNIVERSITY ROTTERDAM Information concerning the Entrance examination Mathematics level 2 for International Business Administration (IBA) General information Availale time: 2.5 hours (150 minutes).
More informationMTH 65 WS 3 ( ) Radical Expressions
MTH 65 WS 3 (9.1-9.4) Radical Expressions Name: The next thing we need to develop is some new ways of talking aout the expression 3 2 = 9 or, more generally, 2 = a. We understand that 9 is 3 squared and
More informationMATH 225: Foundations of Higher Matheamatics. Dr. Morton. 3.4: Proof by Cases
MATH 225: Foundations of Higher Matheamatics Dr. Morton 3.4: Proof y Cases Chapter 3 handout page 12 prolem 21: Prove that for all real values of y, the following inequality holds: 7 2y + 2 2y 5 7. You
More informationSolving Quadratic Equations Using the Quadratic Formula
Section 9 : Solving Quadratic Equations Using the Quadratic Fmula Quadratic Equations are equations that have an x term as the highest powered term. They are also called Second Degree Equations. The Standard
More informationMATHEMATICS National Qualifications - Intermediate 2 Maths Unit 3 + Units 1/2 Revision
Mini-Prelim MATHEMATICS National Qualifications - Intermediate Maths Unit + Units / Revision Time allowed - minutes Read carefully. You may use a calculator.. Full credit will e given only where the solution
More informationSimplifying a Rational Expression. Factor the numerator and the denominator. = 1(x 2 6)(x 2 1) Divide out the common factor x 6. Simplify.
- Plan Lesson Preview Check Skills You ll Need Factoring ± ± c Lesson -5: Eamples Eercises Etra Practice, p 70 Lesson Preview What You ll Learn BJECTIVE - To simplify rational epressions And Why To find
More informationOnline Courses for High School Students
Online Courses for High School Students 1-888-972-6237 Algebra I Course Description: Students explore the tools of algebra and learn to identify the structure and properties of the real number system;
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 informationThe greatest common factor, or GCF, is the largest factor that two or more terms share.
Unit, Lesson Factoring Recall that a factor is one of two or more numbers or expressions that when multiplied produce a given product You can factor certain expressions by writing them as the product of
More informationERASMUS UNIVERSITY ROTTERDAM
Information concerning Colloquium doctum Mathematics level 2 for International Business Administration (IBA) and International Bachelor Economics & Business Economics (IBEB) General information ERASMUS
More information1.1. Linear Equations
1.1 Linear Equations Basic Terminology of Equations Solving Linear Equations Identities, Conditional Equations, and Contradictions Solving for a Specified Variable (Literal Equations) 1.1 Example 1 Solving
More informationGraphs and polynomials
1 1A The inomial theorem 1B Polnomials 1C Division of polnomials 1D Linear graphs 1E Quadratic graphs 1F Cuic graphs 1G Quartic graphs Graphs and polnomials AreAS of STud Graphs of polnomial functions
More informationGraphs and polynomials
5_6_56_MQVMM - _t Page Frida, Novemer 8, 5 :5 AM MQ Maths Methods / Final Pages / 8//5 Graphs and polnomials VCEcoverage Areas of stud Units & Functions and graphs Algera In this chapter A The inomial
More informationQuadratic Equations 6 QUESTIONS. Relatively Easy: Questions 1 to 2 Moderately Difficult: Questions 3 to 4 Difficult: Questions 5 to 6
Quadratic Equations 6 QUESTIONS Relatively Easy: Questions 1 to 2 Moderately Difficult: Questions 3 to 4 Difficult: Questions 5 to 6 Questions www.tutornext.com Page 2 of 11 Q1. The factors of 2x² - 7x
More informationChapter 6 Complex Numbers
Chapter 6 Complex Numbers Lesson 1: Imaginary Numbers Lesson 2: Complex Numbers Lesson 3: Quadratic Formula Lesson 4: Discriminant This assignment is a teacher-modified version of Algebra 2 Common Core
More informationDON ROBERT B. ESTRELLA SR. NATIONAL HIGH SCHOOL Nagsaag, San Manuel, Pangasinan. (Effective Alternative Secondary Education) MATHEMATICS II
DON ROBERT B. ESTRELLA SR. NATIONAL HIGH SCHOOL Nagsaag, San Manuel, Pangasinan. (Effective Alternative Secondary Education) MATHEMATICS II Y X MODULE 1 Quadratic Equations BUREAU OF SECONDARY EDUCATION
More informationUNIT 4 EXTENDING THE NUMBER SYSTEM Lesson 3: Operating with Complex Numbers Instruction
Prerequisite Skills This lesson requires the use of the following skills: simplifying sums and differences of numeric and algebraic quantities using the Commutative Property to reorder sums and differences
More informationTo 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 informationThe Final Exam is comprehensive but identifying the topics covered by it should be simple.
Math 10 Final Eam Study Guide The Final Eam is comprehensive ut identifying the topics covered y it should e simple. Use the previous eams as your primary reviewing tool! This document is to help provide
More informationMay 16, Aim: To review for Quadratic Function Exam #2 Homework: Study Review Materials. Warm Up - Solve using factoring: 5x 2 + 7x + 2 = 0
Aim: To review for Quadratic Function Exam #2 Homework: Study Review Materials Warm Up - Solve using factoring: 5x 2 + 7x + 2 = 0 Review Topic Index 1. Consecutive Integer Word Problems 2. Pythagorean
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 informationSolving Quadratic Equations
Student Probe Probe 1 Solve 9. Solving Quadratic Equations Answer: 1or 5 (Refer to Part 1 of the lesson.) Probe Solve 6 5 0. Answer: 1or 5 (Refer to Part of the lesson.) Probe 3 Solve 3 1 0. Answer: 3
More informationHigher. Polynomials and Quadratics. Polynomials and Quadratics 1
Higher Mathematics Polnomials and Quadratics Contents Polnomials and Quadratics 1 1 Quadratics EF 1 The Discriminant EF Completing the Square EF Sketching Paraolas EF 7 5 Determining the Equation of a
More informationThere are two types of solutions
There are two types of solutions 1) Real solutions which are also x intercept(s) on the graph of the parabola b 2 4ac > 0 b 2 4ac = 0 2) Non real solutions which are not x intercept(s) on the graph of
More informationMath 110 (S & E) Textbook: Calculus Early Transcendentals by James Stewart, 7 th Edition
Math 110 (S & E) Textbook: Calculus Early Transcendentals by James Stewart, 7 th Edition 1 Appendix A : Numbers, Inequalities, and Absolute Values Sets A set is a collection of objects with an important
More informationWhen two letters name a vector, the first indicates the and the second indicates the of the vector.
8-8 Chapter 8 Applications of Trigonometry 8.3 Vectors, Operations, and the Dot Product Basic Terminology Algeraic Interpretation of Vectors Operations with Vectors Dot Product and the Angle etween Vectors
More informationcorrelated to the Utah 2007 Secondary Math Core Curriculum Algebra 1
correlated to the Utah 2007 Secondary Math Core Curriculum Algebra 1 McDougal Littell Algebra 1 2007 correlated to the Utah 2007 Secondary Math Core Curriculum Algebra 1 The main goal of Algebra is to
More informationRadical Zeros. Lesson #5 of Unit 1: Quadratic Functions and Factoring Methods (Textbook Ch1.5)
Radical Zeros Lesson #5 of Unit 1: Quadratic Functions and Factoring Methods (Textbook Ch1.5) Learner Goals 1. Evaluate and approximate square roots 2. Solve quadratic equations by finding square roots.
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 informationSkills Practice Skills Practice for Lesson 4.1
Skills Practice Skills Practice for Lesson.1 Name Date Thinking About Numbers Counting Numbers, Whole Numbers, Integers, Rational and Irrational Numbers Vocabulary Define each term in your own words. 1.
More informationEquations 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 informationStudy Guide and Intervention. The Quadratic Formula and the Discriminant. Quadratic Formula. Replace a with 1, b with -5, and c with -14.
Study Guide and Intervention Quadratic Formula The Quadratic Formula can be used to solve any quadratic equation once it is written in the form a 2 + b + c = 0. Quadratic Formula The solutions of a 2 +
More informationAt first numbers were used only for counting, and 1, 2, 3,... were all that was needed. These are called positive integers.
1 Numers One thread in the history of mathematics has een the extension of what is meant y a numer. This has led to the invention of new symols and techniques of calculation. When you have completed this
More informationUNIT 4 EXTENDING THE NUMBER SYSTEM Lesson 3: Operating with Complex Numbers Instruction
Prerequisite Skills This lesson requires the use of the following skills: finding the product of two binomials simplifying powers of i adding two fractions with different denominators (for application
More informationSchool of Business. Blank Page
Equations 5 The aim of this unit is to equip the learners with the concept of equations. The principal foci of this unit are degree of an equation, inequalities, quadratic equations, simultaneous linear
More informationPolynomial Form. Factored Form. Perfect Squares
We ve seen how to solve quadratic equations (ax 2 + bx + c = 0) by factoring and by extracting square roots, but what if neither of those methods are an option? What do we do with a quadratic equation
More informationSTEP Support Programme. Hints and Partial Solutions for Assignment 2
STEP Support Programme Hints and Partial Solutions for Assignment 2 Warm-up 1 (i) You could either expand the rackets and then factorise, or use the difference of two squares to get [(2x 3) + (x 1)][(2x
More informationA. Graph the parabola. B. Where are the solutions to the equation, 0= x + 1? C. What does the Fundamental Theorem of Algebra say?
Hart Interactive Honors Algebra 1 Lesson 6 M4+ Opening Exercises 1. Watch the YouTube video Imaginary Numbers Are Real [Part1: Introduction] by Welch Labs (https://www.youtube.com/watch?v=t647cgsuovu).
More information( ) ( ) or ( ) ( ) Review Exercise 1. 3 a 80 Use. 1 a. bc = b c 8 = 2 = 4. b 8. Use = 16 = First find 8 = 1+ = 21 8 = =
Review Eercise a Use m m a a, so a a a Use c c 6 5 ( a ) 5 a First find Use a 5 m n m n m a m ( a ) or ( a) 5 5 65 m n m a n m a m a a n m or m n (Use a a a ) cancelling y 6 ecause n n ( 5) ( 5)( 5) (
More informationx y x y ax bx c x Algebra I Course Standards Gap 1 Gap 2 Comments a. Set up and solve problems following the correct order of operations (including proportions, percent, and absolute value) with rational
More informationSolve Quadratic Equations by Using the Quadratic Formula. Return to Table of Contents
Solve Quadratic Equations by Using the Quadratic Formula Return to Table of Contents 128 Solving Quadratics At this point you have learned how to solve quadratic equations by: graphing factoring using
More information5 Section 9.1 Prop of Radicals. 7 Section Section 9. 1b Properties of Radicals. 8 Quick Quiz Section 9.4 Completing the Square
Unit 10B Quadratics - Chapter 9 Name: The calendar and all assignments are subject to change. Students will be notified of any changes during class, so it is their responsibility to pay attention and make
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 informationModule 9: Further Numbers and Equations. Numbers and Indices. The aim of this lesson is to enable you to: work with rational and irrational numbers
Module 9: Further Numers and Equations Lesson Aims The aim of this lesson is to enale you to: wor with rational and irrational numers wor with surds to rationalise the denominator when calculating interest,
More informationUnit 3: HW3.5 Sum and Product
Unit 3: HW3.5 Sum and Product Without solving, find the sum and product of the roots of each equation. 1. x 2 8x + 7 = 0 2. 2x + 5 = x 2 3. -7x + 4 = -3x 2 4. -10x 2 = 5x - 2 5. 5x 2 2x 3 4 6. 1 3 x2 3x
More informationLesson 9-5 Completing the Square Name
Lesson 9-5 Completing the Square Name In the next two lessons, you will be solving quadratic equations by a couple of new methods that often produce solutions involving square roots. In Lesson 9-5A, you
More information1-2 Study Guide and Intervention
1- Study Guide and Intervention Real Numbers All real numbers can be classified as either rational or irrational. The set of rational numbers includes several subsets: natural numbers, whole numbers, and
More informationCreating Good Equations for the Classroom: Rational, Logarithmic and Square Root aversion ""Î# 9Î#!"'
Creating Good Equations for the Classroom: Rational, Logarithmic and Square Root aversion ""Î 9Î!"' Alan Roeuck Chaffey College alan.roeuck@chaffey.edu We're math teachers. We don't just solve equations,
More informationEquations in Quadratic Form
Equations in Quadratic Form MATH 101 College Algebra J. Robert Buchanan Department of Mathematics Summer 2012 Objectives In this lesson we will learn to: make substitutions that allow equations to be written
More informationSection 3.2 Quadratic Functions & Their Graphs
Week 2 Handout MAC 1140 Professor Niraj Wagh J Section 3.2 Quadratic Functions & Their Graphs Quadratic Function: Standard Form A quadratic function is a function that can be written in the form: f (x)
More informationMCS 115 Exam 2 Solutions Apr 26, 2018
MCS 11 Exam Solutions Apr 6, 018 1 (10 pts) Suppose you have an infinitely large arrel and a pile of infinitely many ping-pong alls, laeled with the positive integers 1,,3,, much like in the ping-pong
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 information1-1. Expressions and Formulas. Lesson 1-1. What You ll Learn. Active Vocabulary
1-1 Expressions and Formulas What You ll Learn Skim the lesson. Write two things you already know about expressions and formulas. 1. Active Vocabulary 2. Review Vocabulary Identify the four grouping symbols
More information9-8 Completing the Square
In the previous lesson, you solved quadratic equations by isolating x 2 and then using square roots. This method works if the quadratic equation, when written in standard form, is a perfect square. When
More informationAlg1 Texas TEKS/STAAR/EOC (Second Semester)
Alg 1_sem2; Texas-TEKS/STAAR/EOC page 1 Alg1 Texas TEKS/STAAR/EOC (Second Semester) Legend: Example 3[R]-A.5(B) 3, The reporting category [R], Either Readiness or Supporting A.5, The TEKS (B) Expectation
More informationLinear equations are equations involving only polynomials of degree one.
Chapter 2A Solving Equations Solving Linear Equations Linear equations are equations involving only polynomials of degree one. Examples include 2t +1 = 7 and 25x +16 = 9x 4 A solution is a value or a set
More informationPolynomial Form. Factored Form. Perfect Squares
We ve seen how to solve quadratic equations (ax 2 + bx + c = 0) by factoring and by extracting square roots, but what if neither of those methods are an option? What do we do with a quadratic equation
More informationOctober 28, Complex Numbers.notebook. Discriminant
OBJECTIVE Students will be able to utilize complex numbers to simplify roots of negative numbers. Students will be able to plot complex numbers on a complex coordinate plane. Students will be able to add
More informationStudy Guide and Intervention
Study Guide and Intervention An algeraic expression is a comination of variales, numers, and at least one operation. To evaluate an algeraic expression, replace the variale(s) with numers and follow the
More information7.8 Improper Integrals
CHAPTER 7. TECHNIQUES OF INTEGRATION 67 7.8 Improper Integrals Eample. Find Solution. Z e d. Z e d = = e e!! e = (e ) = Z Eample. Find d. Solution. We do this prolem twice: once the WRONG way, and once
More informationTopic 4: Matrices Reading: Jacques: Chapter 7, Section
Topic 4: Matrices Reading: Jacques: Chapter 7, Section 7.1-7.3 1. dding, sutracting and multiplying matrices 2. Matrix inversion 3. pplication: National Income Determination What is a matrix? Matrix is
More informationFactor each expression. Remember, always find the GCF first. Then if applicable use the x-box method and also look for difference of squares.
NOTES 11: RATIONAL EXPRESSIONS AND EQUATIONS Name: Date: Period: Mrs. Nguyen s Initial: LESSON 11.1 SIMPLIFYING RATIONAL EXPRESSIONS Lesson Preview Review Factoring Skills and Simplifying Fractions Factor
More information11.2 The Quadratic Formula
11.2 The Quadratic Formula Solving Quadratic Equations Using the Quadratic Formula. By solving the general quadratic equation ax 2 + bx + c = 0 using the method of completing the square, one can derive
More informationMath-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 informationALGEBRA 1B GOALS. 1. The student should be able to use mathematical properties to simplify algebraic expressions.
GOALS 1. The student should be able to use mathematical properties to simplify algebraic expressions. 2. The student should be able to add, subtract, multiply, divide, and compare real numbers. 3. The
More informationUNIT #9 ROOTS AND IRRATIONAL NUMBERS REVIEW QUESTIONS
Answer Key Name: Date: UNIT #9 ROOTS AND IRRATIONAL NUMBERS REVIEW QUESTIONS Part I Questions. Which of the following is the value of 6? () 6 () 4 () (4). The epression is equivalent to 6 6 6 6 () () 6
More informationAlgebra 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 informationRADNOR TOWNSHIP SCHOOL DISTRICT Course Overview Seminar Algebra 2 ( )
RADNOR TOWNSHIP SCHOOL DISTRICT Course Overview Seminar Algebra 2 (05040430) General Information Prerequisite: Seminar Geometry Honors with a grade of C or teacher recommendation. Length: Full Year Format:
More informationExpansion formula using properties of dot product (analogous to FOIL in algebra): u v 2 u v u v u u 2u v v v u 2 2u v v 2
Least squares: Mathematical theory Below we provide the "vector space" formulation, and solution, of the least squares prolem. While not strictly necessary until we ring in the machinery of matrix algera,
More informationLesson 23: The Defining Equation of a Line
Student Outomes Students know that two equations in the form of ax + y = and a x + y = graph as the same line when a = = and at least one of a or is nonzero. a Students know that the graph of a linear
More informationLadies and Gentlemen: Please Welcome the Quadratic Formula!
Lesson.1 Skills Practice Name Date Ladies and Gentlemen: Please Welcome the Quadratic Formula! The Quadratic Formula Vocabulary Complete the Quadratic Formula. Then, identify the discriminant and explain
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 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 informationSecondary 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 informationLesson 21 Not So Dramatic Quadratics
STUDENT MANUAL ALGEBRA II / LESSON 21 Lesson 21 Not So Dramatic Quadratics Quadratic equations are probably one of the most popular types of equations that you ll see in algebra. A quadratic equation has
More informationNAME DATE PERIOD. Operations with Polynomials. Review Vocabulary Evaluate each expression. (Lesson 1-1) 3a 2 b 4, given a = 3, b = 2
5-1 Operations with Polynomials What You ll Learn Skim the lesson. Predict two things that you expect to learn based on the headings and the Key Concept box. 1. Active Vocabulary 2. Review Vocabulary Evaluate
More information