MATH 108 REVIEW TOPIC 10 Quadratic Equations. B. Solving Quadratics by Completing the Square
|
|
- Lionel Walters
- 7 years ago
- Views:
Transcription
1 Math 108 T10-Review Topic 10 Page 1 MATH 108 REVIEW TOPIC 10 Quadratic Equations I. Finding Roots of a Quadratic Equation A. Factoring B. Quadratic Formula C. Taking Roots II. III. Guidelines for Finding Roots of a Quadratic Completing the Square A. Perfect Square Trinomials B. Solving Quadratics by Completing the Square Answers to Exercises
2 Math 108 T10-Review Topic 10 Page Introduction: Any equation that can be expressed in the form ax + bx + c =0, a 0 is called a quadratic equation. Illustration: x + x 6 = 0 quadratic in x 16t +80t = 0 quadratic in t. The values that satisfy a quadratic (or any polynomial equation) are called roots. I. Finding Roots of a Quadratic Equation There are 3 primary methods for finding roots to a quadratic. Here are examples and comments on each. A. Factoring Consider the equation x + x 6 = 0. When expressed as a polynomial, its roots are not easily apparent. Notice what happens if we rewrite this expression in factored form: x + x 6=0 (x 3)(x +)=0 The roots now become clear: x = 3 or x =. When solving a quadratic equation, factored form has a distinct advantage over polynomial form. Any value that turns a factor into 0 will automatically make the overall product into 0 (and is therefore a root). In algebreeze (that strange language used by math instructors), if ab =0, then a =0orb =0. Example: Solve x(x ) = 5. Warning: You can only make use of factors when their product is 0. If the problem read x(x ) = 0, you would have roots of 0 and. No such conclusions about roots can be drawn from x(x ) = 5. Our only recourse is to remove parentheses and put into ab = 0 form.
3 Math 108 T10-Review Topic 10 Page 3 Solution: x(x ) = 5 x x 5=0 (x 5)(x +1)=0 ab =0 x =5orx = 1 Example: Solve 9x = x. Solution: 9x = x 9x x =0 x(9x 1) = 0 ab =0 x =0orx = 1 9 What if you attempted the same problem using the following method? 9x = x 9x = 1 divide by x x = 1 9 Every quadratic equation has roots. Dividing by x removes the root x =0. However, dividing by a constant does not effect roots. Example: Solve 16t +80t =0. 16t +80t =0 t 5t = 0 t(t 5) = 0 t =0ort =5 divide by ( 16) Here s one last example of how factoring finds roots. Example: x 3 +3x x 1 = 0. Even though the example is not quadratic, any factorable polynomial can be solved using the same principles. Solution: x 3 +3x x 1 = 0 x (x +3) (x +3)=0 (x )(x +3)=0 ] Grouping Review Topic (x )(x + )(x +3)=0 x = ± or x = 3
4 Math 108 T10-Review Topic 10 Page Exercise 1: Solve for x. a) x(x +1)=15 b) 1x +60x +75=0 c) 5x x + 3 x += d) x 3 9x =0 6 x x Hint: Find LCD and clear fractions. e) x 3 5x 18x +5=0 Answers B. Quadratic Formula Another method for finding roots to a quadratic equation is the quadratic formula. For ax + bx + c =0,a 0, x = b ± b ac a Example: Solve x 6x =0. Solution: With a =1,b= 6 andc =,. 5 = 13 = 13 x = 6 ± ( 6) (1)( ) (1) = 6 ± 5 =3± 13 = 6 ± 13 Any help you need with simplifying radicals can be found in Review Topic 6. Comments: A quadratic has real roots when b ac 0. The discussion of non-real or complex roots (when b ac < 0) will be left for the course.
5 Math 108 T10-Review Topic 10 Page 5 C. Taking Roots For quadratics with no middle term (when b = 0), the simplest approach is to take square roots of both sides of the equation. Example: Solve 3x =0. Solution: 3x =0 x = isolate x and take roots. 3 x = ± = ± Common error: Don t forget the negative root. Example: Solve (x ) =17. x =5 x = 5 or 5. You could expand the binomial, put into quadratic form and solve. Instead, as in the previous example, square root both sides of the equation. Solution: (x ) =17 x =± 17 x =± 17 It will help if you keep this example in mind when looking at Section III of this topic, completing the square. Exercise : Find all real solutions. a) x x = 0 e) 3x + = x x 3 b) (x +3) = f) 5 3 x +3x +1=0 c) x x 3=0 g) k = 1 mv for v (Kinetic energy) d) x 3x +=0 Answers
6 Math 108 T10-Review Topic 10 Page 6 II. Guidelines for Finding Roots of a Quadratic You should now be able to solve quadratic equations using any of the three methods shown: factoring, quadratic formula, or taking roots. Here is a summary of what has been covered. 1) For ax + c = 0, isolate x and square root both sides. Don t forget the negative root. Otherwise... ) Put into the form ax + bx + c =0. This may require removing parentheses or clearing fractions. Dividing out a constant is helpful but not necessary. 3) Find roots by factoring or the *quadratic formula. If b ac < 0, the equation has no real roots. ) Check solutions, especially if original equation is fractional. *Don t overuse the quadratic formula. Factoring is an important skill to maintain so use it at every opportunity. III. Completing the Square Section Ic) demonstrated how quadratics in the form (x ± ( solved. )) = k are Illustration: (x ) =17 x =± 17 x =± 17. How is this related to completing the square? By expanding (x ) and setting equal to 0, (x ) =17 x x 13 = 0. This would seem to indicate that any quadratic can be changed into (x ± ( )) = k form (and then solved). Such a process is called completing the square. A. Perfect Square Trinomials Completing the square requires a thorough understanding of how trinomials of the form a ± ab + b always factor into (a ± b) (Review Topic ).
7 Math 108 T10-Review Topic 10 Page 7 Illustration: a +ab+b (a+b) { }} { { }} { x +6x +9= x + (3)x +3 = (x +3) x 10x +5=x (5)x +5 =(x 5) x 5x + 5 ( ) ( ) ( 5 5 = x x + = x 5 Trinomials such as these are referred to as Perfect Square Trinomials (PST). Exercise 3: Find the term needed to make a PST, then express in factored form. ( ) ( ) ( ) ( x 5x +? = x x +? = x x + = x 5 ) ) a) x 1x +? b) a +9a +? c) x 1 x +? Answers B. Solving Quadratics by Completing the Square Example: Solve x 10x =0. Solution. x 10x = x (5)x +5 =+5 (x 5) =9 x 5=± 9 x =5± 9 Move the constant so it won t interfere with completing the square Comp. Square, maintain equality by adding to both sides Factor and take roots
8 Math 108 T10-Review Topic 10 Page 8 Example: Solve x +7x 15 = 0 Solution. x + 7 x = 15 ( 7 x + ( x + 7 ) x + ) = ( 7 ) = 15 + x + 7 = ±13 x = 7 ± 13 = 3 or 5 ( ) 7 Why are these the roots of x +7x 15 = 0? Divide out coef. of x Check: if x = 3 ( ( ) 3 3 ), = =0 if x = 5, ( 5) +7( 5) 15 = = 0. Final Comment: Completing the square may not be your preferred method for solving quadratics. However, the process is important to learn. You will need to complete squares when working with equations of circles and parabolas. Exercise : Solve by completing the square. a) x +8x 6=0 b) x 11x =0 c) 5 + x x =0 Answers Beginning of Topic 108 Skills Assessment
9 Math 108 Exercise 1 Topic 10 Page 9 Solve for x. a) x(x + 1) = 15 (d) x 3 9x =0 b) 1x +60x + 75 = 0 (e) x 3 5x 18x +5=0 c) 5x x + 3 x += 6 x x Hint: Find LCD and clear fractions. Answers: a) x + x 15 = 0 (x 5)(x +3)=0 x = 5 or x = 3 b) 1x +60x +75=0 x +0x +5=0 (x +5) =0 x = 5 Since (x + 5) is a repeated factor, 5 is a repeated or double root. c) Multiply by x(x ) to clear fractions. 5x +3(x ) + x(x ) = 0 7x x =0 x(7x 1) = 0 Thus x = 0 or x = 1 7 appear to be roots. Since division by 0 is not permissible, the only root is 1. Always check roots when variables 7 appear in any denominator. d) x(x 9) = 0 x(x 3)(x +3)=0 x =0 or ± 3 e) x 3 5x 19x +5=0 x (x 5) 9(x 5) = 0 (x 9)(x 5) = 0 x = ±3 or 5 Return to Review Topic
10 Math 108 Exercise Topic 10 Page 10 Find all real solutions. a) x 10 = 0 e) x +1 3x + = x x 3 b) (x +3) = f) 5 3 x +3x +1=0 c) x x 3=0 g) k = 1 mu for v (Kinetic energy) d) x 3x +=0 Answers: a) x =5 x = 5or 5 b) x +3=± x = 3 ± x = 1 orx = 5 c) x = ± 7 = 1 ( ± 3 3) d) x = 3 ± 7 ; Since b ac < 0, this equation has no real roots. e) Solving as a proportion, (x 3)(x +1)=(3x + )(x ) x 3x 1=0 x = 3 ± 13 f) After clearing fractions, 5x +9x +3=0. x = 9 ± 1 = 1 ( 9 ± 1) g) k = 1 mv k = mv v = k m k v = m Since we solved for velocity, disregard the negative root. Return to Review Topic
11 Math 108 Exercise 3 Topic 10 Page 11 Find the term needed to make a PST, then express in factored form. a) x 1x +? b) a +9a +? c) x 1 x +? Answers: a) x (7)x +7 =(x 7) b) a + c) x ( ) 9 a + ( ) 1 x + ( ) ( 9 = a + 9 ) ( ) ( 1 = x 1 ) Return to Review Topic
12 Math 108 Exercise Topic 10 Page 1 Solve by completing the square. a) x +8x 6=0 b) x 11x =0 c) 5 + x x =0 Answers: a) (x +) =6+ x +=± x = ± b) x x 11 x =0 ( ) ( ) ( x + = ( x 11 ) ( 11 = ) ) x 11 = ±11 x = 11 ± 11 =0or 11 c) x x =5 (x ) =9 x =± 3=5or 1 Return to Review Topic
Tool 1. Greatest Common Factor (GCF)
Chapter 4: Factoring Review Tool 1 Greatest Common Factor (GCF) This is a very important tool. You must try to factor out the GCF first in every problem. Some problems do not have a GCF but many do. When
More information1.3 Algebraic Expressions
1.3 Algebraic Expressions A polynomial is an expression of the form: a n x n + a n 1 x n 1 +... + a 2 x 2 + a 1 x + a 0 The numbers a 1, a 2,..., a n are called coefficients. Each of the separate parts,
More informationFactoring Polynomials and Solving Quadratic Equations
Factoring Polynomials and Solving Quadratic Equations Math Tutorial Lab Special Topic Factoring Factoring Binomials Remember that a binomial is just a polynomial with two terms. Some examples include 2x+3
More informationAnswers to Basic Algebra Review
Answers to Basic Algebra Review 1. -1.1 Follow the sign rules when adding and subtracting: If the numbers have the same sign, add them together and keep the sign. If the numbers have different signs, subtract
More informationFACTORING QUADRATICS 8.1.1 and 8.1.2
FACTORING QUADRATICS 8.1.1 and 8.1.2 Chapter 8 introduces students to quadratic equations. These equations can be written in the form of y = ax 2 + bx + c and, when graphed, produce a curve called a parabola.
More informationALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form
ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form Goal Graph quadratic functions. VOCABULARY Quadratic function A function that can be written in the standard form y = ax 2 + bx+ c where a 0 Parabola
More informationFactoring and Applications
Factoring and Applications What is a factor? The Greatest Common Factor (GCF) To factor a number means to write it as a product (multiplication). Therefore, in the problem 48 3, 4 and 8 are called the
More informationAlgebra Practice Problems for Precalculus and Calculus
Algebra Practice Problems for Precalculus and Calculus Solve the following equations for the unknown x: 1. 5 = 7x 16 2. 2x 3 = 5 x 3. 4. 1 2 (x 3) + x = 17 + 3(4 x) 5 x = 2 x 3 Multiply the indicated polynomials
More informationSection 6.1 Factoring Expressions
Section 6.1 Factoring Expressions The first method we will discuss, in solving polynomial equations, is the method of FACTORING. Before we jump into this process, you need to have some concept of what
More informationSECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS
(Section 0.6: Polynomial, Rational, and Algebraic Expressions) 0.6.1 SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS LEARNING OBJECTIVES Be able to identify polynomial, rational, and algebraic
More informationNSM100 Introduction to Algebra Chapter 5 Notes Factoring
Section 5.1 Greatest Common Factor (GCF) and Factoring by Grouping Greatest Common Factor for a polynomial is the largest monomial that divides (is a factor of) each term of the polynomial. GCF is the
More informationDefinitions 1. A factor of integer is an integer that will divide the given integer evenly (with no remainder).
Math 50, Chapter 8 (Page 1 of 20) 8.1 Common Factors Definitions 1. A factor of integer is an integer that will divide the given integer evenly (with no remainder). Find all the factors of a. 44 b. 32
More informationexpression is written horizontally. The Last terms ((2)( 4)) because they are the last terms of the two polynomials. This is called the FOIL method.
A polynomial of degree n (in one variable, with real coefficients) is an expression of the form: a n x n + a n 1 x n 1 + a n 2 x n 2 + + a 2 x 2 + a 1 x + a 0 where a n, a n 1, a n 2, a 2, a 1, a 0 are
More informationSolving Quadratic Equations
9.3 Solving Quadratic Equations by Using the Quadratic Formula 9.3 OBJECTIVES 1. Solve a quadratic equation by using the quadratic formula 2. Determine the nature of the solutions of a quadratic equation
More informationThis is a square root. The number under the radical is 9. (An asterisk * means multiply.)
Page of Review of Radical Expressions and Equations Skills involving radicals can be divided into the following groups: Evaluate square roots or higher order roots. Simplify radical expressions. Rationalize
More informationAIP Factoring Practice/Help
The following pages include many problems to practice factoring skills. There are also several activities with examples to help you with factoring if you feel like you are not proficient with it. There
More informationGreatest Common Factor (GCF) Factoring
Section 4 4: Greatest Common Factor (GCF) Factoring The last chapter introduced the distributive process. The distributive process takes a product of a monomial and a polynomial and changes the multiplication
More information4.1. COMPLEX NUMBERS
4.1. COMPLEX NUMBERS What You Should Learn Use the imaginary unit i to write complex numbers. Add, subtract, and multiply complex numbers. Use complex conjugates to write the quotient of two complex numbers
More informationZero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P.
MATH 11011 FINDING REAL ZEROS KSU OF A POLYNOMIAL Definitions: Polynomial: is a function of the form P (x) = a n x n + a n 1 x n 1 + + a x + a 1 x + a 0. The numbers a n, a n 1,..., a 1, a 0 are called
More information0.8 Rational Expressions and Equations
96 Prerequisites 0.8 Rational Expressions and Equations We now turn our attention to rational expressions - that is, algebraic fractions - and equations which contain them. The reader is encouraged to
More informationMATH 10034 Fundamental Mathematics IV
MATH 0034 Fundamental Mathematics IV http://www.math.kent.edu/ebooks/0034/funmath4.pdf Department of Mathematical Sciences Kent State University January 2, 2009 ii Contents To the Instructor v Polynomials.
More informationName Intro to Algebra 2. Unit 1: Polynomials and Factoring
Name Intro to Algebra 2 Unit 1: Polynomials and Factoring Date Page Topic Homework 9/3 2 Polynomial Vocabulary No Homework 9/4 x In Class assignment None 9/5 3 Adding and Subtracting Polynomials Pg. 332
More informationFactoring Guidelines. Greatest Common Factor Two Terms Three Terms Four Terms. 2008 Shirley Radai
Factoring Guidelines Greatest Common Factor Two Terms Three Terms Four Terms 008 Shirley Radai Greatest Common Factor 008 Shirley Radai Factoring by Finding the Greatest Common Factor Always check for
More informationIn algebra, factor by rewriting a polynomial as a product of lower-degree polynomials
Algebra 2 Notes SOL AII.1 Factoring Polynomials Mrs. Grieser Name: Date: Block: Factoring Review Factor: rewrite a number or expression as a product of primes; e.g. 6 = 2 3 In algebra, factor by rewriting
More informationFlorida Math 0028. Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies - Upper
Florida Math 0028 Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies - Upper Exponents & Polynomials MDECU1: Applies the order of operations to evaluate algebraic
More information( ) FACTORING. x In this polynomial the only variable in common to all is x.
FACTORING Factoring is similar to breaking up a number into its multiples. For example, 10=5*. The multiples are 5 and. In a polynomial it is the same way, however, the procedure is somewhat more complicated
More informationMath 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.
Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used
More informationMATH 21. College Algebra 1 Lecture Notes
MATH 21 College Algebra 1 Lecture Notes MATH 21 3.6 Factoring Review College Algebra 1 Factoring and Foiling 1. (a + b) 2 = a 2 + 2ab + b 2. 2. (a b) 2 = a 2 2ab + b 2. 3. (a + b)(a b) = a 2 b 2. 4. (a
More informationThe Method of Partial Fractions Math 121 Calculus II Spring 2015
Rational functions. as The Method of Partial Fractions Math 11 Calculus II Spring 015 Recall that a rational function is a quotient of two polynomials such f(x) g(x) = 3x5 + x 3 + 16x x 60. The method
More informationFactoring Quadratic Expressions
Factoring the trinomial ax 2 + bx + c when a = 1 A trinomial in the form x 2 + bx + c can be factored to equal (x + m)(x + n) when the product of m x n equals c and the sum of m + n equals b. (Note: the
More informationChapter R.4 Factoring Polynomials
Chapter R.4 Factoring Polynomials Introduction to Factoring To factor an expression means to write the expression as a product of two or more factors. Sample Problem: Factor each expression. a. 15 b. x
More information3.2. Solving quadratic equations. Introduction. Prerequisites. Learning Outcomes. Learning Style
Solving quadratic equations 3.2 Introduction A quadratic equation is one which can be written in the form ax 2 + bx + c = 0 where a, b and c are numbers and x is the unknown whose value(s) we wish to find.
More informationAlgebra 2/Trig Unit 2 Notes Packet Period: Quadratic Equations
Algebra 2/Trig Unit 2 Notes Packet Name: Date: Period: # Quadratic Equations (1) Page 253 #4 6 **Check on Graphing Calculator (GC)** (2) Page 253 254 #20, 26, 32**Check on GC** (3) Page 253 254 #10 12,
More information1.3 Polynomials and Factoring
1.3 Polynomials and Factoring Polynomials Constant: a number, such as 5 or 27 Variable: a letter or symbol that represents a value. Term: a constant, variable, or the product or a constant and variable.
More informationEAP/GWL Rev. 1/2011 Page 1 of 5. Factoring a polynomial is the process of writing it as the product of two or more polynomial factors.
EAP/GWL Rev. 1/2011 Page 1 of 5 Factoring a polynomial is the process of writing it as the product of two or more polynomial factors. Example: Set the factors of a polynomial equation (as opposed to an
More informationPartial Fractions. (x 1)(x 2 + 1)
Partial Fractions Adding rational functions involves finding a common denominator, rewriting each fraction so that it has that denominator, then adding. For example, 3x x 1 3x(x 1) (x + 1)(x 1) + 1(x +
More informationCAHSEE on Target UC Davis, School and University Partnerships
UC Davis, School and University Partnerships CAHSEE on Target Mathematics Curriculum Published by The University of California, Davis, School/University Partnerships Program 006 Director Sarah R. Martinez,
More informationQuadratics - Build Quadratics From Roots
9.5 Quadratics - Build Quadratics From Roots Objective: Find a quadratic equation that has given roots using reverse factoring and reverse completing the square. Up to this point we have found the solutions
More informationZeros of a Polynomial Function
Zeros of a Polynomial Function An important consequence of the Factor Theorem is that finding the zeros of a polynomial is really the same thing as factoring it into linear factors. In this section we
More informationAlgebra 1 Course Title
Algebra 1 Course Title Course- wide 1. What patterns and methods are being used? Course- wide 1. Students will be adept at solving and graphing linear and quadratic equations 2. Students will be adept
More informationVeterans Upward Bound Algebra I Concepts - Honors
Veterans Upward Bound Algebra I Concepts - Honors Brenda Meery Kaitlyn Spong Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) www.ck12.org Chapter 6. Factoring CHAPTER
More informationChapter 7 - Roots, Radicals, and Complex Numbers
Math 233 - Spring 2009 Chapter 7 - Roots, Radicals, and Complex Numbers 7.1 Roots and Radicals 7.1.1 Notation and Terminology In the expression x the is called the radical sign. The expression under the
More informationARE YOU A RADICAL OR JUST A SQUARE ROOT? EXAMPLES
ARE YOU A RADICAL OR JUST A SQUARE ROOT? EXAMPLES 1. Squaring a number means using that number as a factor two times. 8 8(8) 64 (-8) (-8)(-8) 64 Make sure students realize that x means (x ), not (-x).
More informationFactoring Trinomials: The ac Method
6.7 Factoring Trinomials: The ac Method 6.7 OBJECTIVES 1. Use the ac test to determine whether a trinomial is factorable over the integers 2. Use the results of the ac test to factor a trinomial 3. For
More informationQUADRATIC EQUATIONS AND FUNCTIONS
Douglas College Learning Centre QUADRATIC EQUATIONS AND FUNCTIONS Quadratic equations and functions are very important in Business Math. Questions related to quadratic equations and functions cover a wide
More informationQUADRATIC EQUATIONS EXPECTED BACKGROUND KNOWLEDGE
MODULE - 1 Quadratic Equations 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
More informationFactoring Polynomials
Factoring Polynomials Factoring Factoring is the process of writing a polynomial as the product of two or more polynomials. The factors of 6x 2 x 2 are 2x + 1 and 3x 2. In this section, we will be factoring
More informationVocabulary Words and Definitions for Algebra
Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms
More informationCM2202: Scientific Computing and Multimedia Applications General Maths: 2. Algebra - Factorisation
CM2202: Scientific Computing and Multimedia Applications General Maths: 2. Algebra - Factorisation Prof. David Marshall School of Computer Science & Informatics Factorisation Factorisation is a way of
More informationSpringfield Technical Community College School of Mathematics, Sciences & Engineering Transfer
Springfield Technical Community College School of Mathematics, Sciences & Engineering Transfer Department: Mathematics Course Title: Algebra 2 Course Number: MAT-097 Semester: Fall 2015 Credits: 3 Non-Graduation
More information3.1. RATIONAL EXPRESSIONS
3.1. RATIONAL EXPRESSIONS RATIONAL NUMBERS In previous courses you have learned how to operate (do addition, subtraction, multiplication, and division) on rational numbers (fractions). Rational numbers
More information5.1 Radical Notation and Rational Exponents
Section 5.1 Radical Notation and Rational Exponents 1 5.1 Radical Notation and Rational Exponents We now review how exponents can be used to describe not only powers (such as 5 2 and 2 3 ), but also roots
More informationA. Factoring out the Greatest Common Factor.
DETAILED SOLUTIONS AND CONCEPTS - FACTORING POLYNOMIAL EXPRESSIONS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you!
More informationSection 5.0A Factoring Part 1
Section 5.0A Factoring Part 1 I. Work Together A. Multiply the following binomials into trinomials. (Write the final result in descending order, i.e., a + b + c ). ( 7)( + 5) ( + 7)( + ) ( + 7)( + 5) (
More informationAlgebra 1 If you are okay with that placement then you have no further action to take Algebra 1 Portion of the Math Placement Test
Dear Parents, Based on the results of the High School Placement Test (HSPT), your child should forecast to take Algebra 1 this fall. If you are okay with that placement then you have no further action
More informationBEST METHODS FOR SOLVING QUADRATIC INEQUALITIES.
BEST METHODS FOR SOLVING QUADRATIC INEQUALITIES. I. GENERALITIES There are 3 common methods to solve quadratic inequalities. Therefore, students sometimes are confused to select the fastest and the best
More informationA Systematic Approach to Factoring
A Systematic Approach to Factoring Step 1 Count the number of terms. (Remember****Knowing the number of terms will allow you to eliminate unnecessary tools.) Step 2 Is there a greatest common factor? Tool
More informationSOLVING QUADRATIC EQUATIONS - COMPARE THE FACTORING ac METHOD AND THE NEW DIAGONAL SUM METHOD By Nghi H. Nguyen
SOLVING QUADRATIC EQUATIONS - COMPARE THE FACTORING ac METHOD AND THE NEW DIAGONAL SUM METHOD By Nghi H. Nguyen A. GENERALITIES. When a given quadratic equation can be factored, there are 2 best methods
More informationAlgebra and Geometry Review (61 topics, no due date)
Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties
More informationSolving Rational Equations and Inequalities
8-5 Solving Rational Equations and Inequalities TEKS 2A.10.D Rational functions: determine the solutions of rational equations using graphs, tables, and algebraic methods. Objective Solve rational equations
More informationWentzville School District Algebra 1: Unit 8 Stage 1 Desired Results
Wentzville School District Algebra 1: Unit 8 Stage 1 Desired Results Unit Title: Quadratic Expressions & Equations Course: Algebra I Unit 8 - Quadratic Expressions & Equations Brief Summary of Unit: At
More informationMATH 143 Pre-calculus Algebra and Analytic Geometry
MATH 143 Pre-calculus Algebra and Analytic Geometry Course Guide Self-paced study. Anytime. Anywhere! Math 143 Pre-calculus Algebra and Analytic Geometry University of Idaho 3 Semester-Hour Credits Prepared
More informationHigher Education Math Placement
Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication
More informationSOLVING QUADRATIC EQUATIONS BY THE DIAGONAL SUM METHOD
SOLVING QUADRATIC EQUATIONS BY THE DIAGONAL SUM METHOD A quadratic equation in one variable has as standard form: ax^2 + bx + c = 0. Solving it means finding the values of x that make the equation true.
More informationSolving Rational Equations
Lesson M Lesson : Student Outcomes Students solve rational equations, monitoring for the creation of extraneous solutions. Lesson Notes In the preceding lessons, students learned to add, subtract, multiply,
More informationCONVERT QUADRATIC FUNCTIONS FROM ONE FORM TO ANOTHER (Standard Form <==> Intercept Form <==> Vertex Form) (By Nghi H Nguyen Dec 08, 2014)
CONVERT QUADRATIC FUNCTIONS FROM ONE FORM TO ANOTHER (Standard Form Intercept Form Vertex Form) (By Nghi H Nguyen Dec 08, 2014) 1. THE QUADRATIC FUNCTION IN INTERCEPT FORM The graph of the quadratic
More informationFactoring Trinomials of the Form
Section 4 6B: Factoring Trinomials of the Form A x 2 + Bx + C where A > 1 by The AC and Factor By Grouping Method Easy Trinomials: 1 x 2 + Bx + C The last section covered the topic of factoring second
More informationMath 25 Activity 6: Factoring Advanced
Instructor! Math 25 Activity 6: Factoring Advanced Last week we looked at greatest common factors and the basics of factoring out the GCF. In this second activity, we will discuss factoring more difficult
More informationAnswer Key for California State Standards: Algebra I
Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.
More informationAlum Rock Elementary Union School District Algebra I Study Guide for Benchmark III
Alum Rock Elementary Union School District Algebra I Study Guide for Benchmark III Name Date Adding and Subtracting Polynomials Algebra Standard 10.0 A polynomial is a sum of one ore more monomials. Polynomial
More informationSOLVING QUADRATIC EQUATIONS BY THE NEW TRANSFORMING METHOD (By Nghi H Nguyen Updated Oct 28, 2014))
SOLVING QUADRATIC EQUATIONS BY THE NEW TRANSFORMING METHOD (By Nghi H Nguyen Updated Oct 28, 2014)) There are so far 8 most common methods to solve quadratic equations in standard form ax² + bx + c = 0.
More informationLAKE ELSINORE UNIFIED SCHOOL DISTRICT
LAKE ELSINORE UNIFIED SCHOOL DISTRICT Title: PLATO Algebra 1-Semester 2 Grade Level: 10-12 Department: Mathematics Credit: 5 Prerequisite: Letter grade of F and/or N/C in Algebra 1, Semester 2 Course Description:
More informationALGEBRA REVIEW LEARNING SKILLS CENTER. Exponents & Radicals
ALGEBRA REVIEW LEARNING SKILLS CENTER The "Review Series in Algebra" is taught at the beginning of each quarter by the staff of the Learning Skills Center at UC Davis. This workshop is intended to be an
More informationReview of Intermediate Algebra Content
Review of Intermediate Algebra Content Table of Contents Page Factoring GCF and Trinomials of the Form + b + c... Factoring Trinomials of the Form a + b + c... Factoring Perfect Square Trinomials... 6
More informationMath 10C. Course: Polynomial Products and Factors. Unit of Study: Step 1: Identify the Outcomes to Address. Guiding Questions:
Course: Unit of Study: Math 10C Polynomial Products and Factors Step 1: Identify the Outcomes to Address Guiding Questions: What do I want my students to learn? What can they currently understand and do?
More informationFactoring Polynomials
Factoring Polynomials 4-1-2014 The opposite of multiplying polynomials is factoring. Why would you want to factor a polynomial? Let p(x) be a polynomial. p(c) = 0 is equivalent to x c dividing p(x). Recall
More informationMTH 092 College Algebra Essex County College Division of Mathematics Sample Review Questions 1 Created January 17, 2006
MTH 092 College Algebra Essex County College Division of Mathematics Sample Review Questions Created January 7, 2006 Math 092, Elementary Algebra, covers the mathematical content listed below. In order
More informationIntegrals of Rational Functions
Integrals of Rational Functions Scott R. Fulton Overview A rational function has the form where p and q are polynomials. For example, r(x) = p(x) q(x) f(x) = x2 3 x 4 + 3, g(t) = t6 + 4t 2 3, 7t 5 + 3t
More informationPartial Fractions Examples
Partial Fractions Examples Partial fractions is the name given to a technique of integration that may be used to integrate any ratio of polynomials. A ratio of polynomials is called a rational function.
More informationMATH 90 CHAPTER 6 Name:.
MATH 90 CHAPTER 6 Name:. 6.1 GCF and Factoring by Groups Need To Know Definitions How to factor by GCF How to factor by groups The Greatest Common Factor Factoring means to write a number as product. a
More informationLesson 9.1 Solving Quadratic Equations
Lesson 9.1 Solving Quadratic Equations 1. Sketch the graph of a quadratic equation with a. One -intercept and all nonnegative y-values. b. The verte in the third quadrant and no -intercepts. c. The verte
More informationMATH 0110 Developmental Math Skills Review, 1 Credit, 3 hours lab
MATH 0110 Developmental Math Skills Review, 1 Credit, 3 hours lab MATH 0110 is established to accommodate students desiring non-course based remediation in developmental mathematics. This structure will
More information1 Lecture: Integration of rational functions by decomposition
Lecture: Integration of rational functions by decomposition into partial fractions Recognize and integrate basic rational functions, except when the denominator is a power of an irreducible quadratic.
More informationCore Maths C1. Revision Notes
Core Maths C Revision Notes November 0 Core Maths C Algebra... Indices... Rules of indices... Surds... 4 Simplifying surds... 4 Rationalising the denominator... 4 Quadratic functions... 4 Completing the
More information7.1 Graphs of Quadratic Functions in Vertex Form
7.1 Graphs of Quadratic Functions in Vertex Form Quadratic Function in Vertex Form A quadratic function in vertex form is a function that can be written in the form f (x) = a(x! h) 2 + k where a is called
More informationFactoring Flow Chart
Factoring Flow Chart greatest common factor? YES NO factor out GCF leaving GCF(quotient) how many terms? 4+ factor by grouping 2 3 difference of squares? perfect square trinomial? YES YES NO NO a 2 -b
More informationCopy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.
Algebra 2 - Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers - {1,2,3,4,...}
More informationFactoring Methods. Example 1: 2x + 2 2 * x + 2 * 1 2(x + 1)
Factoring Methods When you are trying to factor a polynomial, there are three general steps you want to follow: 1. See if there is a Greatest Common Factor 2. See if you can Factor by Grouping 3. See if
More informationFACTORING ax 2 bx c. Factoring Trinomials with Leading Coefficient 1
5.7 Factoring ax 2 bx c (5-49) 305 5.7 FACTORING ax 2 bx c In this section In Section 5.5 you learned to factor certain special polynomials. In this section you will learn to factor general quadratic polynomials.
More informationWhat are the place values to the left of the decimal point and their associated powers of ten?
The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything
More informationAlgebra I Vocabulary Cards
Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression
More informationMathematics Online Instructional Materials Correlation to the 2009 Algebra I Standards of Learning and Curriculum Framework
Provider York County School Division Course Syllabus URL http://yorkcountyschools.org/virtuallearning/coursecatalog.aspx Course Title Algebra I AB Last Updated 2010 - A.1 The student will represent verbal
More informationSystems of Equations Involving Circles and Lines
Name: Systems of Equations Involving Circles and Lines Date: In this lesson, we will be solving two new types of Systems of Equations. Systems of Equations Involving a Circle and a Line Solving a system
More information15.1 Factoring Polynomials
LESSON 15.1 Factoring Polynomials Use the structure of an expression to identify ways to rewrite it. Also A.SSE.3? ESSENTIAL QUESTION How can you use the greatest common factor to factor polynomials? EXPLORE
More informationThis unit has primarily been about quadratics, and parabolas. Answer the following questions to aid yourselves in creating your own study guide.
COLLEGE ALGEBRA UNIT 2 WRITING ASSIGNMENT This unit has primarily been about quadratics, and parabolas. Answer the following questions to aid yourselves in creating your own study guide. 1) What is the
More information7-6. Choosing a Factoring Model. Extension: Factoring Polynomials with More Than One Variable IN T RO DUC E T EACH. Standards for Mathematical Content
7-6 Choosing a Factoring Model Extension: Factoring Polynomials with More Than One Variable Essential question: How can you factor polynomials with more than one variable? What is the connection between
More informationChapter 4 -- Decimals
Chapter 4 -- Decimals $34.99 decimal notation ex. The cost of an object. ex. The balance of your bank account ex The amount owed ex. The tax on a purchase. Just like Whole Numbers Place Value - 1.23456789
More informationSOLVING QUADRATIC EQUATIONS - COMPARE THE FACTORING AC METHOD AND THE NEW TRANSFORMING METHOD (By Nghi H. Nguyen - Jan 18, 2015)
SOLVING QUADRATIC EQUATIONS - COMPARE THE FACTORING AC METHOD AND THE NEW TRANSFORMING METHOD (By Nghi H. Nguyen - Jan 18, 2015) GENERALITIES. When a given quadratic equation can be factored, there are
More information1.7. Partial Fractions. 1.7.1. Rational Functions and Partial Fractions. A rational function is a quotient of two polynomials: R(x) = P (x) Q(x).
.7. PRTIL FRCTIONS 3.7. Partial Fractions.7.. Rational Functions and Partial Fractions. rational function is a quotient of two polynomials: R(x) = P (x) Q(x). Here we discuss how to integrate rational
More information