# SECTION 1-6 Quadratic Equations and Applications

Size: px
Start display at page:

Transcription

1 58 Equations and Inequalities Supply the reasons in the proofs for the theorems stated in Problems 65 and Theorem: The complex numbers are commutative under addition. Proof: Let a bi and c di be two arbitrary complex numbers; then: Statement. (a bi) (c di) (a c) (b d)i. (c a) (d b)i. (c di) (a bi) Reason Theorem: The complex numbers are commutative under multiplication. Proof: Let a bi and c di be two arbitrary complex numbers; then: Statement. (a bi) (c di) (ac bd) (ad bc)i. (ca db) (da cb)i. (c di)(a bi) Reason... Letters z and w are often used as complex variables, where z x yi, w u vi, and x, y, u, v are real numbers. The conjugates of z and w, denoted by z and w, respectively, are given by z x yi and w u vi. In Problems 67 7, express each property of conjugates verbally and then prove the property. 67. zz is a real number. 68. z z is a real number. 69. z z if and only if z is real. 70. z z 7. z w z w 7. z w z w 7. zw z w 7. z/w z/ w SECTION -6 Quadratic Equations and Applications Solution by Factoring Solution by Square Root Solution by Completing the Square Solution by Quadratic Formula Applications The next class of equations we consider are the second-degree polynomial equations in one variable, called quadratic equations. DEFINITION Quadratic Equation A quadratic equation in one variable is any equation that can be written in the form ax bx c 0 a 0 Standard Form where x is a variable and a, b, and c are constants.

2 -6 Quadratic Equations and Applications 59 Now that we have discussed the complex number system, we will use complex numbers when solving equations. Recall that a solution of an equation is also called a root of the equation. A real number solution of an equation is called a real root, and an imaginary number solution is called an imaginary root. In this section we develop methods for finding all real and imaginary roots of a quadratic equation. Solution by Factoring If ax bx c can be written as the product of two first-degree factors, then the quadratic equation can be quickly and easily solved. The method of solution by factoring rests on the zero property of complex numbers, which is a generalization of the zero property of real numbers reviewed in Section A-. Zero Property If m and n are complex numbers, then m n 0 if and only if m 0 or n 0 (or both) EXAMPLE Solving Quadratic Equations by Factoring Solve by factoring: (A) 6x 9x 7 0 (B) x 6x 5 (C) x x Solutions (A) (B) (C) 6x 9x 7 0 (x 7)(x ) 0 Factor left side. x 7 0 or x 0 x 7 x The solution set is {, 7 }. x 6x 5 x 6x 9 0 Write in standard form. (x ) 0 Factor left side. x The solution set is {}. The equation has one root,. But since it came from two factors, we call a double root. x x 0 x(x ) 0 x 0 x x or Solution set: {0, } x 0 x

3 60 Equations and Inequalities Matched Problem Solve by factoring: (A) x 7x 0 0 (B) x x 9 0 (C) x 5x CAUTION. One side of an equation must be 0 before the zero property can be applied. Thus x 6x 5 (x )(x 5) does not imply that x or x 5. See Example B for the correct solution of this equation.. The equations x x and x are not equivalent. The first has solution set {0, }, while the second has solution set { }. The root x 0 is lost when each member of the first equation is divided by the variable x. See Example C for the correct solution of this equation. Do not divide both members of an equation by an expression containing the variable for which you are solving. You may be dividing by 0. Solution by Square Root We now turn our attention to quadratic equations that do not have the first-degree term that is, equations of the special form ax c 0 a 0 The method of solution of this special form makes direct use of the square root property: Square Root Property If A C, then A C.

4 -6 Quadratic Equations and Applications 6 EXPLORE-DISCUSS Determine if each of the following pairs of equations is equivalent or not. Explain your answer. (A) x and x (B) x and x (C) x and x (D) x and x The use of the square root property is illustrated in the next example. Note: It is common practice to represent solutions of quadratic equations informally by the last equation rather than by writing a solution set using set notation. From now on, we will follow this practice unless a particular emphasis is desired. EXAMPLE Using the Square Root Property Solve using the square root property: (A) x 0 (B) x 7 0 (C) (x ) 5 Solutions (A) x 0 x x or 6 Solution set: 6, 6 (B) x 7 0 x 9 x 9 or i Solution set: { i, i} (C) (x ) 5 x 5 x 5 5 Matched Problem Solve using the square root property: (A) x 5 0 (B) x 8 0 (C) (x ) 9

5 6 Equations and Inequalities EXPLORE-DISCUSS Replace? in each of the following with a number that makes the equation valid. (A) (x ) x x? (B) (x ) x x? (C) (x ) x 6x? (D) (x ) x 8x? Replace? in each of the following with a number that makes the trinomial a perfect square. (E) x 0x? (F) x x? (G) x bx? Solution by Completing the Square The methods of square root and factoring are generally fast when they apply; however, there are equations, such as x 6x 0 (see Example A), that cannot be solved directly by these methods. A more general procedure must be developed to take care of this type of equation for example, the method of completing the square. This method is based on the process of transforming the standard quadratic equation ax bx c 0 into the form (x A) B where A and B are constants. The last equation can easily be solved by using the square root property. But how do we transform the first equation into the second? The following brief discussion provides the key to the process. What number must be added to x bx so that the result is the square of a firstdegree polynomial? There is a simple mechanical rule for finding this number, based on the square of the following binomials: (x m) x mx m (x m) x mx m In either case, we see that the third term on the right is the square of one-half of the coefficient of x in the second term on the right. This observation leads directly to the rule for completing the square. Completing the Square To complete the square of a quadratic of the form x bx, add the square of one-half the coefficient of x; that is, add (b/). Thus, x bx b x b x 5x x bx x 5x 5 x 5

6 -6 Quadratic Equations and Applications 6 EXAMPLE Completing the Square Complete the square for each of the following: (A) x x (B) x bx Solutions (A) x x x x 9 (x ) (B) x bx x bx b x b 9 Add ; that is,. b Add b ; that is,. Matched Problem Complete the square for each of the following: (A) x 5x (B) x mx It is important to note that the rule for completing the square applies only to quadratic forms in which the coefficient of the second-degree term is. This causes little trouble, however, as you will see. We now solve two equations by the method of completing the square. EXAMPLE Solution by Completing the Square Solve by completing the square: (A) x 6x 0 (B) x x 0 Solutions (A) (B) x 6x 0 x 6x x 6x 9 9 (x ) x x x x 0 x x 0 x x x x Complete the square on the left side, and add the same number to the right side. Make the leading coefficient by dividing by. Complete the square on the left side and add the same number to the right side.

7 6 Equations and Inequalities (x ) Factor the left side. x x i i Answer in a bi form. Matched Problem Solve by completing the square: (A) x 8x 0 (B) x x 0 Solution by Quadratic Formula Now consider the general quadratic equation with unspecified coefficients: ax bx c 0 a 0 We can solve it by completing the square exactly as we did in Example B. To make the leading coefficient, we must multiply both sides of the equation by /a. Thus, x b a x c a 0 Adding c/a to both sides of the equation and then completing the square of the left side, we have We now factor the left side and solve using the square root property: x b a b ac a x b a b ac a x b a x b a b a c a x b a b ac a See Problem 75 in Exercise -6. b b ac a We have thus derived the well-known and widely used quadratic formula:

9 66 Equations and Inequalities Solution 5.7x 6.0x.7 0 x 6.0 ( 6.0) (5.7)(.7) (5.7) 0.5, 0.87 Matched Problem 6 Solve.79x 5.07x to two decimal places using a calculator. We conclude this part of the discussion by noting that b ac in the quadratic formula is called the discriminant and gives us useful information about the corresponding roots as shown in Table. TABLE Discriminant and Roots Discriminant Roots of ax bx c 0 b ac a, b, and c real numbers, a 0 Positive Two distinct real roots 0 One real root (a double root) Negative Two imaginary roots, one the conjugate of the other For example: (A) x x 0 has two real roots, since b ac ( ) ()( ) 0 (B) x x 0 has one real (double) root, since b ac ( ) ()() 0 (C) x x 0 has two imaginary roots, since b ac ( ) ()() 0 Applications We now consider several applications that make use of quadratic equations. First, the strategy for solving word problems, presented earlier in Section -, is repeated below. Strategy for Solving Word Problems. Read the problem carefully several times if necessary that is, until you understand the problem, know what is to be found, and know what is given.

10 -6 Quadratic Equations and Applications 67. Let one of the unknown quantities be represented by a variable, say x, and try to represent all other unknown quantities in terms of x. This is an important step and must be done carefully.. If appropriate, draw figures or diagrams and label known and unknown parts.. Look for formulas connecting the known quantities to the unknown quantities. 5. Form an equation relating the unknown quantities to the known quantities. 6. Solve the equation and write answers to all questions asked in the problem. 7. Check and interpret all solutions in terms of the original problem not just the equation found in step 5 since a mistake may have been made in setting up the equation in step 5. EXAMPLE 7 Setting Up and Solving a Word Problem The sum of a number and its reciprocal is. Find all such numbers. 6 Solution Let x the number; then: x x 6 (6x)x (6x) x (6x) 6 Multiply both sides by 6x. [Note: x 0.] 6x 6 x 6x x 6 0 (x )(x ) 0 x 0 x or A quadratic equation x 0 x Thus, two such numbers are and. Check 6 6 Matched Problem 7 The sum of two numbers is and their product is. Find the two numbers. [Hint: If one number is x, then the other number is x.]

11 68 Equations and Inequalities EXAMPLE 8 A Distance Rate Time Problem An excursion boat takes.6 hours longer to go 6 miles up a river than to return. If the rate of the current is miles per hour, what is the rate of the boat in still water? Solution Let T D, x, x x x R 6(x ) 6(x ).6(x )(x ) 6x 6x.6x 5.6.6x.6 x 96 x 96 The rate in still water is miles per hour. x Rate of boat in still water x Rate downstream x Rate upstream Time Time.6 upstream downstream [Note: 96 must be discarded, since it doesn t make sense in the problem to have a negative rate.] Check Time upstream D R 6.6 Time downstream D R 6.6 Difference of times Matched Problem 8 Two boats travel at right angles to each other after leaving a dock at the same time. One hour later they are 5 miles apart. If one boat travels 5 miles per hour faster than the other, what is the rate of each? [Hint: Use the Pythagorean theorem,* remembering that distance equals rate times time.] a c b *Pythagorean theorem: A triangle is a right triangle if and only if the square of the length of the longest side is equal to the sum of the squares of the lengths of the two shorter sides: c a b.

12 -6 Quadratic Equations and Applications 69 EXAMPLE 9 A Quantity Rate Time Problem A payroll can be completed in hours by two computers working simultaneously. How many hours are required for each computer to complete the payroll alone if the older model requires hours longer than the newer model? Compute answers to two decimal places. Solution Let x Time for new model to complete the payroll alone x Time for old model to complete the payroll alone Time for both computers to complete the payroll together Then, Part of job completed by new model in hours x () x Rate for new model x Rate for old model x Part of job completed by old model in hours Completes Completes whole job x 0, x x () of the payroll per hour of the payroll per hour x (x ) x x(x ) Multiply both sides by x(x ). x x x x x 5x 0 x x x 9.77 x is discarded since x cannot be negative. The new model would complete the payroll in 6.77 hours working alone, and the old model would complete the payroll in 9.77 hours working alone. x

13 70 Equations and Inequalities Check 6.77 () 9.77 () ± Note: We do not expect the check to be exact, since we rounded the answers to two decimal places. An exact check would be produced by using x (5 7)/. The latter is left to the reader. Matched Problem 9 Two technicians can complete a mailing in hours when working together. Alone, one can complete the mailing hours faster than the other. How long will it take each person to complete the mailing alone? Compute the answers to two decimal places. Answers to Matched Problems 5 5. (A) x, (B) x (a double root) (C) x 0,. (A) x 5 or 5/ (B) x i (C) x ( )/. (A) x 5 5 5x (x ) (B) x mx (m /) [x (m/)]. (A) x 9 (B) x (6 i )/ or ( /)i 5. x ( 9)/ 6. x.80, and 8. 5 and 0 miles per hour and 7.6 hours EXERCISE -6 A Leave all answers involving radicals in simplified radical form unless otherwise stated. B In Problems 6, solve by factoring.. x 8x. y 5y. t 9 t. s 6s 5. w w x 9 x In Problems 7 8, solve by using the square root property. 7. m n c d 6 0. y x z 0. 6w (s ) 5 6. (t ) 7. (n ) 8. (m ) In Problems 9 6, solve using the quadratic formula. 9. x x 0 0. y y 7 0. x x 0. y y 0. t 8 6t. 9s s 5. t 6t 6. 9s 7 s In Problems 7, solve by completing the square. 7. x x 0 8. y y 0 9. r 0r 0 0. s 6s 7 0. u 8u 5 0. v 6v 0. w w 0. z 8z 0 In Problems 5 5, solve by any method. 5. x 7x x 9x 7. (y ) 5 8. (m ) 9. x x 0. x x. 7n n. 8u u x x m 0 m x x x u u. y. y x x x

14 -6 Quadratic Equations and Applications x x x x 9 x x x x x x x 5. u u 5. 7x x In Problems 5 56, solve for the indicated variable in terms of the other variables. Use positive square roots only. 5. s gt for t 5. a b c for a 55. P EI RI for I 56. A P( r) for r Solve Problems to two decimal places using a calculator x.79x x.8x x.0x x 7.7x Consider the quadratic equation x x c 0 where c is a real number. Discuss the relationship between the values of c and the three types of roots listed in Table. 6. Consider the quadratic equation x x c 0 where c is a real number. Discuss the relationship between the values of c and the three types of roots listed in Table. Use the discriminant to determine whether the equations in Problems 6 66 have real solutions x 0.0x x 0.8x x 0.0x x 0.8x C Solve Problems and leave answers in simplified radical form (i is the imaginary unit). 67. x 8 x 68. x x 69. x ix 70. x ix In Problems 7 and 7, find all solutions. 7. x 0 7. x 0 7. Can a quadratic equation with rational coefficients have one rational root and one irrational root? Explain. 7. Can a quadratic equation with real coefficients have one real root and one imaginary root? Explain. 75. Show that if r and r are the two roots of ax bx c 0, then r r c/a. 76. For r and r in Problem 75, show that r r b/a. 77. In one stage of the derivation of the quadratic formula, we replaced the expression with What justifies using a in place of a? 78. Find the error in the following proof that two arbitrary numbers are equal to each other: Let a and b be arbitrary numbers such that a b. Then APPLICATIONS (a b) (b a) a b b a a b a b (b ac)/a b ac/a (a b) a ab b b ab a 79. Numbers. Find two numbers such that their sum is and their product is Numbers. Find all numbers with the property that when the number is added to itself the sum is the same as when the number is multiplied by itself. 8. Numbers. Find two consecutive positive even integers whose product is Numbers. The sum of a number and its reciprocal is. Find the number. 8. Geometry. If the length and width of a - by -inch rectangle are each increased by the same amount, the area of the new rectangle will be twice that of the original. What are the dimensions of the new rectangle (to two decimal places)? 8. Geometry. Find the base b and height h of a triangle with an area of square feet if its base is feet longer than its height and the formula for area is A bh. 85. Business. If \$P are invested at an interest rate r compounded annually, at the end of years the amount will be A P( r). At what interest rate will \$,000 increase to \$,0 in years? [Note: A \$,0 and P \$,000.]

15 7 Equations and Inequalities 86. Economics. In a certain city, the price demand and price supply equations for CDs are p 75,000 q p q.5 Demand equation Supply equation where q represents quantity and p represents the price in dollars. Find the equilibrium price. 87. Puzzle. Two planes travel at right angles to each other after leaving the same airport at the same time. One hour later they are 60 miles apart. If one travels 0 miles per hour faster than the other, what is the rate of each? 88. Navigation. A speedboat takes hour longer to go miles up a river than to return. If the boat cruises at 0 miles per hour in still water, what is the rate of the current? 89. Engineering. One pipe can fill a tank in 5 hours less than another. Together they can fill the tank in 5 hours. How long would it take each alone to fill the tank? Compute the answer to two decimal places. 90. Engineering. Two gears rotate so that one completes more revolution per minute than the other. If it takes the smaller gear second less than the larger gear to complete 5 revolution, how many revolutions does each gear make in minute? 9. Physics Engineering. For a car traveling at a speed of v miles per hour, under the best possible conditions the shortest distance d necessary to stop it (including reaction time) is given by the empirical formula d 0.0v.v, where d is measured in feet. Estimate the speed of a car that requires 65 feet to stop in an emergency. 9. Physics Engineering. If a projectile is shot vertically into the air (from the ground) with an initial velocity of 76 feet per second, its distance y (in feet) above the ground t seconds after it is shot is given by y 76t 6t (neglecting air resistance). (A) Find the times when y is 0, and interpret the results physically. (B) Find the times when the projectile is 6 feet off the ground. Compute answers to two decimal places. 9. Construction. A developer wants to erect a rectangular building on a triangular-shaped piece of property that is 00 feet wide and 00 feet long (see the figure). Find the dimensions of the building if its cross-sectional area is 5,000 square feet. [Hint: Use Euclid s theorem* to find a relationship between the length and width of the building.] 9. Architecture. An architect is designing a small A-frame cottage for a resort area. A cross section of the cottage is an isosceles triangle with an area of 98 square feet. The front wall of the cottage must accommodate a sliding door that is 6 feet wide and 8 feet high (see the figure). Find the width and height of the cross section of the cottage. [Recall: The area of a triangle with base b and altitude h is bh/.] 95. Transportation. A delivery truck leaves a warehouse and travels north to factory A. From factory A the truck travels east to factory B and then returns directly to the warehouse (see the figure). The driver recorded the truck s odometer reading at the warehouse at both the beginning and the end of the trip and also at factory B, but forgot to record it at factory A (see the table). The driver does recall that it was further from the warehouse to factory A than it was from factory A to factory B. Since delivery charges are based on distance from the warehouse, the driver needs to know how far factory A is from the warehouse. Find this distance. Factory A 6 feet 8 feet Factory B Warehouse 00 feet 5,000 square feet 00 feet *Euclid s theorem: If two triangles are similar, their corresponding sides are proportional: a b c a b c a a b b c c

16 -7 Equations Reducible to Quadratic Form 7 Warehouse Factory A Factory B Warehouse Odometer Readings ??? enclose an area of 00,000 square feet. Find the length of the straightaways and the diameter of the semicircles to the nearest foot. [Recall: The area A and circumference C of a circle of diameter d are given by A d / and C d.] 96. Construction. A -mile track for racing stock cars consists of two semicircles connected by parallel straightaways (see the figure). In order to provide sufficient room for pit crews, emergency vehicles, and spectator parking, the track must 00,000 square feet SECTION -7 Equations Reducible to Quadratic Form Equations Involving Radicals Equations Involving Rational Exponents Equations Involving Radicals In solving an equation involving a radical like x x it appears that we can remove the radical by squaring each side and then proceed to solve the resulting quadratic equation. Thus, x x 0 (x )(x ) 0 x ( x ) x x x, Now we check these results in the original equation. Check: x x x Check: x x x Thus, is a solution, but is not. These results are a special case of Theorem.

### Factoring 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

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

### MATH 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

476 CHAPTER 7 Graphs, Equations, and Inequalities 7. Quadratic Equations Now Work the Are You Prepared? problems on page 48. OBJECTIVES 1 Solve Quadratic Equations by Factoring (p. 476) Solve Quadratic

### Math 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

### Answer 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.

### Review 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

### 1.3 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,

### Sect 6.7 - Solving Equations Using the Zero Product Rule

Sect 6.7 - Solving Equations Using the Zero Product Rule 116 Concept #1: Definition of a Quadratic Equation A quadratic equation is an equation that can be written in the form ax 2 + bx + c = 0 (referred

### MATH 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

### Higher 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

### How To Solve Factoring Problems

05-W4801-AM1.qxd 8/19/08 8:45 PM Page 241 Factoring, Solving Equations, and Problem Solving 5 5.1 Factoring by Using the Distributive Property 5.2 Factoring the Difference of Two Squares 5.3 Factoring

### Mathematics Placement

Mathematics Placement The ACT COMPASS math test is a self-adaptive test, which potentially tests students within four different levels of math including pre-algebra, algebra, college algebra, and trigonometry.

### Factoring Polynomials

UNIT 11 Factoring Polynomials You can use polynomials to describe framing for art. 396 Unit 11 factoring polynomials A polynomial is an expression that has variables that represent numbers. A number can

### 4.1. 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

### A.3. Polynomials and Factoring. Polynomials. What you should learn. Definition of a Polynomial in x. Why you should learn it

Appendi A.3 Polynomials and Factoring A23 A.3 Polynomials and Factoring What you should learn Write polynomials in standard form. Add,subtract,and multiply polynomials. Use special products to multiply

### What 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

### How do you compare numbers? On a number line, larger numbers are to the right and smaller numbers are to the left.

The verbal answers to all of the following questions should be memorized before completion of pre-algebra. Answers that are not memorized will hinder your ability to succeed in algebra 1. Number Basics

### MATH 095, College Prep Mathematics: Unit Coverage Pre-algebra topics (arithmetic skills) offered through BSE (Basic Skills Education)

MATH 095, College Prep Mathematics: Unit Coverage Pre-algebra topics (arithmetic skills) offered through BSE (Basic Skills Education) Accurately add, subtract, multiply, and divide whole numbers, integers,

### 12) 13) 14) (5x)2/3. 16) x5/8 x3/8. 19) (r1/7 s1/7) 2

DMA 080 WORKSHEET # (8.-8.2) Name Find the square root. Assume that all variables represent positive real numbers. ) 6 2) 8 / 2) 9x8 ) -00 ) 8 27 2/ Use a calculator to approximate the square root to decimal

### MATH 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

### Algebra 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

### ALGEBRA 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

### 1.1 Practice Worksheet

Math 1 MPS Instructor: Cheryl Jaeger Balm 1 1.1 Practice Worksheet 1. Write each English phrase as a mathematical expression. (a) Three less than twice a number (b) Four more than half of a number (c)

### Algebra 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

### Vocabulary 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

### MATH 60 NOTEBOOK CERTIFICATIONS

MATH 60 NOTEBOOK CERTIFICATIONS Chapter #1: Integers and Real Numbers 1.1a 1.1b 1.2 1.3 1.4 1.8 Chapter #2: Algebraic Expressions, Linear Equations, and Applications 2.1a 2.1b 2.1c 2.2 2.3a 2.3b 2.4 2.5

### 43 Perimeter and Area

43 Perimeter and Area Perimeters of figures are encountered in real life situations. For example, one might want to know what length of fence will enclose a rectangular field. In this section we will study

### MATH 100 PRACTICE FINAL EXAM

MATH 100 PRACTICE FINAL EXAM Lecture Version Name: ID Number: Instructor: Section: Do not open this booklet until told to do so! On the separate answer sheet, fill in your name and identification number

### Veterans 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

### Definitions 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

### Polynomials. Key Terms. quadratic equation parabola conjugates trinomial. polynomial coefficient degree monomial binomial GCF

Polynomials 5 5.1 Addition and Subtraction of Polynomials and Polynomial Functions 5.2 Multiplication of Polynomials 5.3 Division of Polynomials Problem Recognition Exercises Operations on Polynomials

### SECTION 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

### a 1 x + a 0 =0. (3) ax 2 + bx + c =0. (4)

ROOTS OF POLYNOMIAL EQUATIONS In this unit we discuss polynomial equations. A polynomial in x of degree n, where n 0 is an integer, is an expression of the form P n (x) =a n x n + a n 1 x n 1 + + a 1 x

### Algebra 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

### Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks

Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson

### Quick Reference ebook

This file is distributed FREE OF CHARGE by the publisher Quick Reference Handbooks and the author. Quick Reference ebook Click on Contents or Index in the left panel to locate a topic. The math facts listed

### Florida 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

### Expression. Variable Equation Polynomial Monomial Add. Area. Volume Surface Space Length Width. Probability. Chance Random Likely Possibility Odds

Isosceles Triangle Congruent Leg Side Expression Equation Polynomial Monomial Radical Square Root Check Times Itself Function Relation One Domain Range Area Volume Surface Space Length Width Quantitative

### SPECIAL PRODUCTS AND FACTORS

CHAPTER 442 11 CHAPTER TABLE OF CONTENTS 11-1 Factors and Factoring 11-2 Common Monomial Factors 11-3 The Square of a Monomial 11-4 Multiplying the Sum and the Difference of Two Terms 11-5 Factoring the

### Biggar High School Mathematics Department. National 5 Learning Intentions & Success Criteria: Assessing My Progress

Biggar High School Mathematics Department National 5 Learning Intentions & Success Criteria: Assessing My Progress Expressions & Formulae Topic Learning Intention Success Criteria I understand this Approximation

### Math Review. for the Quantitative Reasoning Measure of the GRE revised General Test

Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important

### POLYNOMIAL FUNCTIONS

POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a

### A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions

A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions Marcel B. Finan Arkansas Tech University c All Rights Reserved First Draft February 8, 2006 1 Contents 25

### MATH 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.

### ( ) ( ) Math 0310 Final Exam Review. # Problem Section Answer. 1. Factor completely: 2. 2. Factor completely: 3. Factor completely:

Math 00 Final Eam Review # Problem Section Answer. Factor completely: 6y+. ( y+ ). Factor completely: y+ + y+ ( ) ( ). ( + )( y+ ). Factor completely: a b 6ay + by. ( a b)( y). Factor completely: 6. (

### Summer Math Exercises. For students who are entering. Pre-Calculus

Summer Math Eercises For students who are entering Pre-Calculus It has been discovered that idle students lose learning over the summer months. To help you succeed net fall and perhaps to help you learn

### CAMI Education linked to CAPS: Mathematics

- 1 - TOPIC 1.1 Whole numbers _CAPS curriculum TERM 1 CONTENT Mental calculations Revise: Multiplication of whole numbers to at least 12 12 Ordering and comparing whole numbers Revise prime numbers to

### The program also provides supplemental modules on topics in geometry and probability and statistics.

Algebra 1 Course Overview Students develop algebraic fluency by learning the skills needed to solve equations and perform important manipulations with numbers, variables, equations, and inequalities. Students

### Geometry: Classifying, Identifying, and Constructing Triangles

Geometry: Classifying, Identifying, and Constructing Triangles Lesson Objectives Teacher's Notes Lesson Notes 1) Identify acute, right, and obtuse triangles. 2) Identify scalene, isosceles, equilateral

### of surface, 569-571, 576-577, 578-581 of triangle, 548 Associative Property of addition, 12, 331 of multiplication, 18, 433

Absolute Value and arithmetic, 730-733 defined, 730 Acute angle, 477 Acute triangle, 497 Addend, 12 Addition associative property of, (see Commutative Property) carrying in, 11, 92 commutative property

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

### Algebra 2 Chapter 1 Vocabulary. identity - A statement that equates two equivalent expressions.

Chapter 1 Vocabulary identity - A statement that equates two equivalent expressions. verbal model- A word equation that represents a real-life problem. algebraic expression - An expression with variables.

### NSM100 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

### POLYNOMIALS and FACTORING

POLYNOMIALS and FACTORING Exponents ( days); 1. Evaluate exponential expressions. Use the product rule for exponents, 1. How do you remember the rules for exponents?. How do you decide which rule to use

### 6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives

6 EXTENDING ALGEBRA Chapter 6 Extending Algebra Objectives After studying this chapter you should understand techniques whereby equations of cubic degree and higher can be solved; be able to factorise

### Students will be able to simplify and evaluate numerical and variable expressions using appropriate properties and order of operations.

Outcome 1: (Introduction to Algebra) Skills/Content 1. Simplify numerical expressions: a). Use order of operations b). Use exponents Students will be able to simplify and evaluate numerical and variable

### Florida Algebra 1 End-of-Course Assessment Item Bank, Polk County School District

Benchmark: MA.912.A.2.3; Describe the concept of a function, use function notation, determine whether a given relation is a function, and link equations to functions. Also assesses MA.912.A.2.13; Solve

### Characteristics of the Four Main Geometrical Figures

Math 40 9.7 & 9.8: The Big Four Square, Rectangle, Triangle, Circle Pre Algebra We will be focusing our attention on the formulas for the area and perimeter of a square, rectangle, triangle, and a circle.

### SOLVING EQUATIONS WITH RADICALS AND EXPONENTS 9.5. section ( 3 5 3 2 )( 3 25 3 10 3 4 ). The Odd-Root Property

498 (9 3) Chapter 9 Radicals and Rational Exponents Replace the question mark by an expression that makes the equation correct. Equations involving variables are to be identities. 75. 6 76. 3?? 1 77. 1

### Polynomial Degree and Finite Differences

CONDENSED LESSON 7.1 Polynomial Degree and Finite Differences In this lesson you will learn the terminology associated with polynomials use the finite differences method to determine the degree of a polynomial

### 13. Write the decimal approximation of 9,000,001 9,000,000, rounded to three significant

æ If 3 + 4 = x, then x = 2 gold bar is a rectangular solid measuring 2 3 4 It is melted down, and three equal cubes are constructed from this gold What is the length of a side of each cube? 3 What is the

### Anchorage School District/Alaska Sr. High Math Performance Standards Algebra

Anchorage School District/Alaska Sr. High Math Performance Standards Algebra Algebra 1 2008 STANDARDS PERFORMANCE STANDARDS A1:1 Number Sense.1 Classify numbers as Real, Irrational, Rational, Integer,

### Warm-Up Oct. 22. Daily Agenda:

Evaluate y = 2x 3x + 5 when x = 1, 0, and 2. Daily Agenda: Grade Assignment Go over Ch 3 Test; Retakes must be done by next Tuesday 5.1 notes / assignment Graphing Quadratic Functions 5.2 notes / assignment

### Copy 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,...}

### Section 1.1. Introduction to R n

The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to

### Applications of the Pythagorean Theorem

9.5 Applications of the Pythagorean Theorem 9.5 OBJECTIVE 1. Apply the Pythagorean theorem in solving problems Perhaps the most famous theorem in all of mathematics is the Pythagorean theorem. The theorem

### ModuMath Basic Math Basic Math 1.1 - Naming Whole Numbers Basic Math 1.2 - The Number Line Basic Math 1.3 - Addition of Whole Numbers, Part I

ModuMath Basic Math Basic Math 1.1 - Naming Whole Numbers 1) Read whole numbers. 2) Write whole numbers in words. 3) Change whole numbers stated in words into decimal numeral form. 4) Write numerals in

### Zero: 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

9.4 Multiplying and Dividing Radicals 9.4 OBJECTIVES 1. Multiply and divide expressions involving numeric radicals 2. Multiply and divide expressions involving algebraic radicals In Section 9.2 we stated

### FACTORING 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.

### Algebra I. In this technological age, mathematics is more important than ever. When students

In this technological age, mathematics is more important than ever. When students leave school, they are more and more likely to use mathematics in their work and everyday lives operating computer equipment,

### Solving Quadratic Equations by Factoring

4.7 Solving Quadratic Equations by Factoring 4.7 OBJECTIVE 1. Solve quadratic equations by factoring The factoring techniques you have learned provide us with tools for solving equations that can be written

### PERIMETER AND AREA. In this unit, we will develop and apply the formulas for the perimeter and area of various two-dimensional figures.

PERIMETER AND AREA In this unit, we will develop and apply the formulas for the perimeter and area of various two-dimensional figures. Perimeter Perimeter The perimeter of a polygon, denoted by P, is the

### QUADRATIC 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

### COMPETENCY TEST SAMPLE TEST. A scientific, non-graphing calculator is required for this test. C = pd or. A = pr 2. A = 1 2 bh

BASIC MATHEMATICS COMPETENCY TEST SAMPLE TEST 2004 A scientific, non-graphing calculator is required for this test. The following formulas may be used on this test: Circumference of a circle: C = pd or

### Polynomial Operations and Factoring

Algebra 1, Quarter 4, Unit 4.1 Polynomial Operations and Factoring Overview Number of instructional days: 15 (1 day = 45 60 minutes) Content to be learned Identify terms, coefficients, and degree of polynomials.

### MTH 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

### Lesson 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

### 0.4 FACTORING POLYNOMIALS

36_.qxd /3/5 :9 AM Page -9 SECTION. Factoring Polynomials -9. FACTORING POLYNOMIALS Use special products and factorization techniques to factor polynomials. Find the domains of radical expressions. Use

### 6.1 Add & Subtract Polynomial Expression & Functions

6.1 Add & Subtract Polynomial Expression & Functions Objectives 1. Know the meaning of the words term, monomial, binomial, trinomial, polynomial, degree, coefficient, like terms, polynomial funciton, quardrtic

### Indiana State Core Curriculum Standards updated 2009 Algebra I

Indiana State Core Curriculum Standards updated 2009 Algebra I Strand Description Boardworks High School Algebra presentations Operations With Real Numbers Linear Equations and A1.1 Students simplify and

### CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA

We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical

### Law of Cosines. If the included angle is a right angle then the Law of Cosines is the same as the Pythagorean Theorem.

Law of Cosines In the previous section, we learned how the Law of Sines could be used to solve oblique triangles in three different situations () where a side and two angles (SAA) were known, () where

### Factor and Solve Polynomial Equations. In Chapter 4, you learned how to factor the following types of quadratic expressions.

5.4 Factor and Solve Polynomial Equations Before You factored and solved quadratic equations. Now You will factor and solve other polynomial equations. Why? So you can find dimensions of archaeological

### McDougal Littell California:

McDougal Littell California: Pre-Algebra Algebra 1 correlated to the California Math Content s Grades 7 8 McDougal Littell California Pre-Algebra Components: Pupil Edition (PE), Teacher s Edition (TE),

### Algebra 1 Course Information

Course Information Course Description: Students will study patterns, relations, and functions, and focus on the use of mathematical models to understand and analyze quantitative relationships. Through

### ALGEBRA 2 CRA 2 REVIEW - Chapters 1-6 Answer Section

ALGEBRA 2 CRA 2 REVIEW - Chapters 1-6 Answer Section MULTIPLE CHOICE 1. ANS: C 2. ANS: A 3. ANS: A OBJ: 5-3.1 Using Vertex Form SHORT ANSWER 4. ANS: (x + 6)(x 2 6x + 36) OBJ: 6-4.2 Solving Equations by

### LAKE 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:

### Principles of Mathematics MPM1D

Principles of Mathematics MPM1D Grade 9 Academic Mathematics Version A MPM1D Principles of Mathematics Introduction Grade 9 Mathematics (Academic) Welcome to the Grade 9 Principals of Mathematics, MPM

### Algebra I Credit Recovery

Algebra I Credit Recovery COURSE DESCRIPTION: The purpose of this course is to allow the student to gain mastery in working with and evaluating mathematical expressions, equations, graphs, and other topics,

### Common Core Unit Summary Grades 6 to 8

Common Core Unit Summary Grades 6 to 8 Grade 8: Unit 1: Congruence and Similarity- 8G1-8G5 rotations reflections and translations,( RRT=congruence) understand congruence of 2 d figures after RRT Dilations

### MA107 Precalculus Algebra Exam 2 Review Solutions

MA107 Precalculus Algebra Exam 2 Review Solutions February 24, 2008 1. The following demand equation models the number of units sold, x, of a product as a function of price, p. x = 4p + 200 a. Please write

### Algebra 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

### Factoring 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

### PowerScore Test Preparation (800) 545-1750

Question 1 Test 1, Second QR Section (version 2) Two triangles QA: x QB: y Geometry: Triangles Answer: Quantity A is greater 1. The astute student might recognize the 0:60:90 and 45:45:90 triangle right

### Factor Polynomials Completely

9.8 Factor Polynomials Completely Before You factored polynomials. Now You will factor polynomials completely. Why? So you can model the height of a projectile, as in Ex. 71. Key Vocabulary factor by grouping

### Chapter 111. Texas Essential Knowledge and Skills for Mathematics. Subchapter B. Middle School

Middle School 111.B. Chapter 111. Texas Essential Knowledge and Skills for Mathematics Subchapter B. Middle School Statutory Authority: The provisions of this Subchapter B issued under the Texas Education