Table of C ntents - SAP Education

Synergy for Success in Mathematics 9 is designed for Grade 9 students based on the new K to 12 Curriculum released by the Department of Education. The textbook contains all the required learning competencies and is supplemented with some additional topics for enrichment. Lessons are presented using effective Singapore Math strategies that are intended for easy understanding and grasp of ideas for its target readers. Various exercises are also provided to help the learners acquire the necessary skills needed.

The book is organized with the following recurring features in every chapter:

Learning Goals Thisgivesthespecificobjectivesthatareintendedtobeachieved in the end.

Introduction The reader is given a bird's eye view of the contents.Historical Note A brief historical account of a related topic is included giving the

reader an awareness of some important contributions of some great mathematicians or even stories of great achievements related to mathematics.

Method/Exam Notes These additional tools help students recall important information, formulas, and shortcuts, needed in working out solutions.

Examples Step-by-stepanddetaileddemonstrationsofhowaspecificconcept or technique is applied in solving problems.

Enhancing Skills These are practice exercises found after every lesson, that will consolidate and reinforce what the students have learned.

Linking Together This visual tool can help the students realize the connection of all the ideas presented in the chapter.

Chapter Test This is a summative test given at the end in preparation for the expected actual classroom examination containing the topics included in the chapter. A challenging task is designed for the learner giving him an opportunity to use what he/she has learned in the chapter.

Chapter Project Thismaybeamanipulativetypeofactivitythatisspecificallychosen to enhance understanding of the concepts learned in the chapter.

Making Connection The students are exposed to facts and information that connect mathematics and culture. This is for the purpose of letting the learnersappreciatethesubjectbecauseoftangibleortrue-to-life stories that show how mathematics is useful and relevant.

Every effort has been made in order for all the discussions in this book to be clear, simple, and straightforward.ItscontentscatertotheneedsspecifiedintheKto12curriculumbutbeyondall these, this book also gives opportunities for the readers to see the beauty of mathematics as an essential tool in understanding the world we live in. With this in mind, appreciation of mathematics goes beyond seeing; realizing its critical application to decision making in life completes the purpose of knowing and understanding mathematics.

PREFACE

Table of C ntents

Introduction ............................................................................................................................1Historical Note .......................................................................................................................21.1 QuadraticEquations ..............................................................................................31.2 SolvingQuadraticEquationsbyExtractingSquareRoots .....................9

Imaginary Roots ............................................................................................... 121.3 SolvingQuadraticEquationsbyFactoring ............................................... 161.4 SolvingQuadraticEquationsbyCompletingtheSquare ..................... 241.5 TheQuadraticFormula ...................................................................................... 341.6 StudyingtheRootsofaQuadraticEquation ............................................. 43

The Nature of the Roots and the Discriminant of Quadratic Equations ....................................................................................... 43 Deriving the Quadratic Equation Given its Roots ............................... 45

1.7 SolvingProblemsInvolvingQuadraticEquations .................................. 56 Rational Equations Leading to Quadratic Equations ......................... 56 Equations Involving Integral Exponents Leading to Quadratic Equations ....................................................................................... 59 Application of Quadratic Equations ......................................................... 62

Linking Together ................................................................................................................ 70Chapter Test ........................................................................................................................ 71ChapterProject ................................................................................................................... 73Making Connection ........................................................................................................... 74

Introduction ......................................................................................................................... 75Historical Note .................................................................................................................... 762.1 QuadraticFunctionsandTheirGraphs ....................................................... 77

The Standard Form of Quadratic Function ............................................ 77 The Graph of a Quadratic Function ........................................................... 78

2.2 TheZerosofQuadraticFunctions ...............................................................109 The Zeros of a Quadratic Function ..........................................................109 The Graphical Method of Finding the Zeros of a Quadratic Function ........................................................................................110

CHAPTER 1 QUADRATIC EQUATIONS AND THEIR ROOTS

CHAPTER 2 QUADRATIC FUNCTIONS AND THEIR GRAPHS

The Algebraic Method of Finding the Zeros of a Quadratic Function .......................................................................................114 Discriminant and the Nature of Zeros ....................................................124

2.3 TheVertexFormandShiftingParabolas ..................................................139 The Vertex Form of a Quadratic Function .............................................139 The Domain and Range of a Quadratic Function ...............................142 Shifting Parabolas .........................................................................................144

2.4 DerivingQuadraticFunctions ......................................................................1572.5 ProblemSolvingInvolvingQuadraticFunctions ...................................1652.6 QuadraticInequalities ......................................................................................172Linking Together ..............................................................................................................183Chapter Test ......................................................................................................................184ChapterProject .................................................................................................................187Making Connection .........................................................................................................188

Introduction .......................................................................................................................189Historical Note ..................................................................................................................1903.1 RatioandProportion ........................................................................................191

Ratio and Percentage ...................................................................................191 The Fundamental Property of Proportion ...........................................194 Rate ......................................................................................................................199

3.2 DirectandInverseVariations ........................................................................204 Variation ............................................................................................................204 Direct Variation ...............................................................................................204 Inverse Variation ............................................................................................215

3.3 JointandCombinedVariations .....................................................................228 Joint Variation .................................................................................................228 Combined Variation .......................................................................................232

3.4 SolvingProblemsInvolvingVariations .....................................................239Linking Together ..............................................................................................................251Chapter Test ......................................................................................................................252ChapterProject .................................................................................................................255Making Connection .........................................................................................................258

CHAPTER 3 VARIATIONS

Introduction .......................................................................................................................259Historical Note ..................................................................................................................2604.1 RootsofNumbersandRadicalExpressions ...........................................261

Squares and Square Roots .........................................................................261 Radical Expressions .......................................................................................265 Cubes and Cube Roots ...................................................................................268 The Principal nth Root ................................................................................269

4.2 ExpressionswithRationalExponents ......................................................276 Rational Exponents ........................................................................................276 Positive Rational Exponents .......................................................................278 Negative Rational Exponents ....................................................................280

4.3 OperationsInvolvingRadicalExpressions ..............................................289 Simplifying Radical Expressions Using Rational Exponents .........289 Multiplying Radical Expressions ...............................................................291 Simplifying Radical Expressions Using Factoring ..............................292 Dividing Radical Expressions .....................................................................295 Rationalizing Denominators with One Term .......................................297

4.4 MoreabouttheOperationsInvolvingRadicalExpressions ............303 Adding and Subtracting Radical Expressions ......................................303 Multiplying Radical Expressions with Two or More Terms ............306 Multiplying and Dividing Radical Expressions with Different Indices ..............................................................................................307 Rationalizing the Denominator with Two Terms ...............................310

4.5 EquationsInvolvingRadicalExpressionsandApplications ............315 Radical Equations Leading to Linear Equations ................................315 Radical Equations that Lead to Quadratic Equations ......................318 Solving Problems Involving Expressions with Rational Exponents and Radical Expressions ......................................321

Linking Together ..............................................................................................................326Chapter Test ......................................................................................................................327ChapterProject .................................................................................................................330Making Connection .........................................................................................................332

CHAPTER 4 RATIONAL EXPONENTS AND RADICAL EXPRESSIONS

Introduction .......................................................................................................................333Historical Note ...................................................................................................................3345.1 ProportionalityandSimilarity ......................................................................335

Review on Ratio and Proportion ...............................................................335 Geometric Mean ..............................................................................................339 The Side-Splitter Theorem ..........................................................................341 Similarity of Polygons ...................................................................................345

5.2 TriangleSimilarityTheorems .......................................................................3535.3 AboutRightTriangles .......................................................................................372

Proportions in a Right Triangle ................................................................372 The Pythagorean Theorem .........................................................................376

5.4 SpecialRightTriangles .....................................................................................386Linking Together ..............................................................................................................395Chapter Test ......................................................................................................................396ChapterProject .................................................................................................................403Making Connection .........................................................................................................404

Introduction .......................................................................................................................405Historical Note ...................................................................................................................4066.1 ReviewonQuadrilaterals ................................................................................407

The Sum of the Measures of the Interior Angles of a Quadrilateral ...........................................................................................409 Types of Quadrilateral ..................................................................................411

6.2 Parallelograms .....................................................................................................419 Quadrilaterals That Are Parallelograms ...............................................424

6.3 OnSomeSpecialParallelograms ..................................................................432 Rectangles .........................................................................................................432 Rhombi ...............................................................................................................433 Squares ...............................................................................................................434

6.4 TrapezoidsandKites.........................................................................................438 Trapezoids .........................................................................................................438 Kites .....................................................................................................................442

Linking Together ..............................................................................................................451Chapter Test ......................................................................................................................452ChapterProject .................................................................................................................457Making Connection .........................................................................................................458

CHAPTER 5 SIMILARITY AND RIGHT TRIANGLES

CHAPTER 6 QUADRILATERALS

Introduction .......................................................................................................................459Historical Note ...................................................................................................................4607.1 AngleMeasure .....................................................................................................461

Angles ..................................................................................................................461 Degree Measure ..............................................................................................462 Coterminal Angles ..........................................................................................464 Radian Measure ..............................................................................................467 Relation of Degree Measure and Radian Measure .............................469 Reference Angles .............................................................................................471 Degree-Minute-Second Form of an Angle ............................................474

7.2 TheSixTrigonometricFunctions ................................................................481 Trigonometric Functions of 45º ................................................................483 Trigonometric Functions of 30º and 60º ...............................................484 Trigonometric Identities ..............................................................................485

7.3 TrigonometricFunctionsofanyAngle ......................................................4957.4 ApplicationsofTrigonometry .......................................................................506

Solving a Right Triangle ..............................................................................506 Angles of Elevation and Depression ........................................................507

7.5 LawsofSinesandCosines ..............................................................................515Linking Together ..............................................................................................................531Chapter Test ......................................................................................................................532ChapterProject .................................................................................................................535Making Connection .........................................................................................................536

Glossary .....................................................................................................................................537 Index ...........................................................................................................................................544 Bibliography ............................................................................................................................548 PhotoCredits ...........................................................................................................................550

CHAPTER 7 TRIGONOMETRIC FUNCTIONS

QUADRATIC EQUATIONS AND THEIR ROOTS1

Learning GoalsAt the end of the chapter, you should be able to:

1.1 Define quadratic equation and simplify quadratic equations in one variable into their standard form

1.2 Establish the square root property and solve quadratic equations by extracting square roots

1.3 Recall factoring techniques and apply these techniques to solve quadratic equations

1.4 Establish the method of completing the square and solve quadratic equations by completing the square

1.5 Derive the quadratic formula using completing the square and solve quadratic equations by applying the quadratic formula

1.6 Describe the nature of the roots using the discriminant of a quadratic equation, establish the relationship of the quadratic equation to the sum and product of its roots, and form quadratic equations using the roots

1.7 Solve problems involving quadratic equations using different methods

The famous and remarkable shoe industry in Marikina gives pride to the Philippines. Known as the shoe capital of the country, the city was once recognized by the Guinness World Records because of the world’s largest pair of shoes that the shoemakers in the city made. Also, the shoe industry helps alleviate the unemployment problem. The production of shoes depends on the number of employees and profit. To help the manufacturers and shoemakers maximize the production and profit, a quadratic equation can be used as a model.

In this chapter, we study the fundamental concepts revolving around quadratic equations in one variable. We find ways and methods to solve problems involving quadratic equations in one variable. We also look at the connection of the quadratic equation and its roots.

2

Historical Note

As early as 2000 BC, Babylonians used cuneiform, a system of writing on clay tablets, to write algebraic and geometric problems. There are clay tablets showing evidence that the Babylonians were familiar with today’s way of solving quadratic equations. Their solution is generally known today as "completing the square" method of solving quadratic equations.

Aside from the Babylonians, other ancient civilizations formulated and studied quadratic equations. The ancient Egyptians listed the areas of possible sides and shapes. Pythagoras (569–500 BC), a Greek philosopher and mathematician, formulated the Pythagorean theorem which states that the square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the lengths of the other two sides.

3

Synergy for Success in Mathematics • Chapter 1

1.1 Quadratic Equations

Certain problems in mathematics and science frequently lead to situations involving quadratic equations. Consider the following problem.

May and Paul want to put a fence around their rectangular garden. The width of the garden is 2 feet shorter than its length. Also, the area of the garden is 80 square feet. How long should the fence be?

Let x be the length of the garden. Thus, the width measures x −( )2 feet. Since the area of the rectangular garden is 80

square feet, x x −( )=2 80.

When the distributive property and the addition property of equality are applied, the simplest form of the equation is shown as follows:

x x

x xx x

−( )=

− =

− − =

2 80

2 80

2 80 0

2

2

Notice that the highest degree of the resulting equation is 2, which makes the equation not linear. The resulting equation is an example of a second-degree equation, also known as quadratic equation.

Exam Note

The area of a rectangle is equal to the product of its length l and width w; that is,

A lw= .

Exam Note

A linear equation is an algebraic equation whose highest degree among the terms is 1.

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A Quadratic Equation in One Variable

A quadratic equation in one variable is a second-degree equation written in the standard form

ax bx c20+ + =

where a, b, and c are real numbers and a ≠ 0.

Other examples of quadratic equations in one variable are as follows:

This signifies that the equation is quadratic.

This signifies that the equation is quadratic.

The quadratic equation has only one variable, m.

The quadratic equation has only one variable, x.

5 2 10 02m m− + =

3 2 5 02x x+ − =

Example 1

Identify whether the equation is a quadratic equation in one variable or not.(a) x x x2 2

5 7 2 0+ − + =

(b) x y2 21+ =

(c) 2 3 5 3 02y y y− +( )=

(d) a a a2 1 12−( )= −

(e) b b b+( ) −( )= −( )3 3 2 92

SOLUTION

Simplify each equation to determine whether the equation is a quadratic equation in one variable or not.

(a) x x xx x x

x x

2 2

2 2

2

5 7 2 0

2 5 7 0

3 5 7 0

+ − + =

+ + − =

+ − =

The resulting equation is a quadratic equation in one variable.

Exam Note

The term quadratic is derived from the Latin word quadratus, which means “squared.”

5


(b) The equation x y2 21+ = is a quadratic equation

since the highest degree among the terms is 2. However, it is not expressed in one variable.

(c) 2 3 5 3 0

6 10 6 0

2

3 2

y y y

y y y

− +( )=

− + =

The highest degree among the terms is 3. Hence, it is not a quadratic equation.

(d) a a a

a a aa a

2 1 1

2 1

1 0

2

2 2

2

−( )= −

− = −

− + =

The resulting equation is a quadratic equation in one variable.

(e) b b b

b bbb

+( ) −( )= −( )− = −

− + =

− =

3 3 2 9

9 2 18

9 0

9 0

2

2 2

2

2

The highest degree among the terms is 2. The equation is expressed in one variable. Hence, it is a quadratic equation in one variable.

When a quadratic equation in one variable is written in standard form, you can easily identify the numerical coefficients of each term. For example, the equation x x2

5 6 0+ − = is already in standard form. The numerical coefficients of each term of the equation are indicated below.

x x25 6 0+ − =

The numerical coefficient of x2 is 1.

The numerical coefficient of 5x. is 5.

The constant term is −6.

Exam Note

Not all quadratic equations are expressed in one variable. Some have two.

6

Based on the standard form of quadratic equation in one variable, you can represent the numerical coefficient of the first term (the term with the highest degree) as a, the numerical coefficient of the middle term (the term in linear degree) as b, and the constant term as c. Hence, the values of a, b, and c in the equation x x2

5 6 0+ − = are 1, 5, and −6, respectively

a b c= = =−1 5 6; ; . and

Example 2

Simplify each quadratic equation into its standard form. Identify the values of a, b, and c.

(a) 3 7 02x x− + =

(b) 4 3 2 02− − =x x

(c) 5 2 32+ =−x x

SOLUTION

(a) The quadratic equation 3 7 02x x− + = is already

in standard form. Thus, a b c= =− =3 1 7, . , and

(b) The terms in 4 3 2 02− − =x x need to be

rearranged. The commutative property of addition should be applied.

4 3 2 0

2 3 4 0

2

2

− − =

− − + =

x xx x .

Thus, a b c=− =− =2 3 4, , and .

(c) Based on the addition property of equality and the commutative property of addition, the standard form of the equation is indicated as follows:

5 2 3

5 2 3 0

2 3 5 0

2

2

2

+ =−

+ + =

+ + =

x xx x

x x

Hence, a b c= = =2 3 5, , . and

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ENHANCING SKILLS

A Write QUADRATIC if the equation is a quadratic equation in one variable. Otherwise, write EQUATION. Justify your answer.

(1) 4 7 02x x+ − = :

(2) x y2 2

4 91− = :

(3) x x2

4 68 0+ − = :

(4) x x+( ) = −5 2 72 2 :

(5) 4 4 02 2a ab b+ + = :

(6) x y x y+( ) −( )= 0 :

(7) 2 1 2 1 4x x+( ) −( )= :

(8) 5 8 1x x −( )= :

(9) 5 1 3 7a a−( ) +( )= :

(10) 5 5 6 02x x x+ −( )= :

B Simplify each quadratic equation into its standard form. Then, identify the values of the numerical coefficients a, b, and c. Express the leading coefficient a as a nonnegative real number.

(11) 5 92x x− =

Standard form: a = __________b = __________c = __________

(12) 2 1 8x x +( )=− Standard form: a = __________b = __________c = __________

(13) 8 2 13 02− + =x x


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(14) 2 1 3 0x x+( ) −( )= Standard form: a = __________b = __________c = __________

(15) 2 1 3 6 0x x−( ) +( )+ = Standard form: a = __________b = __________c = __________

(16) y y+( ) −( )=7 7 0 Standard form: a = __________b = __________c = __________

(17) 2 7 42y −( ) =


(18) y +( ) − =4 16 02


(19) 2 7 0m m−( )= Standard form: a = __________b = __________c = __________

(20) m m m m+( ) −( )= +( ) −( )2 2 2 1 2 1


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Table of C ntents - SAP Education