module 1( final 2) quadtraic equations and inequalities jq

Regional Training of MathematicsTeachers for

Grade 9 of The K to 12 Enhanced Basic Education

ProgramMay 15-19, 2014

Notre Dame of Marbel UniversityKoronadal City

AIRLINE

ASSESSMENT

QUADRATIC

SQUARE ROOT

OFACTORING

YINEQUALITY

SUGGESTIONS

RI SQUADRILATERALS

GPROVING

Grade 9 Mathematics

Quarter I – First Grading Period

Module 2 – Quadratic Functions

Module 1 – Quadratic Equations and Inequalities

Quarter II – Second Grading Period

Module 4 – Zero Exponents, Negative Integral Exponents,

Rational Exponents and Radicals

Module 3 – Variations

Grade 9 Mathematics

Quarter III – Third Grading Period

Module 6 – Similarity

Module 5 – Quadrilaterals

Grade 9 Mathematics

Quarter IV – Fourth Grading Period

Module 7 – Triangle Trigonometry

Grade 9 Mathematics

Curriculum Guide LegendSample: M9AL-Ic-d-1

M9 AL I c-d 1

Math 9 Algebra Quarter 1

Week 3-4

Competency 1

Curriculum Guide LegendSample: M9AL-IIg-2

M9 AL II g 2

Math 9 Algebra Quarter 2

Week 7

Competency 2

Domain/Component Code

Number Sense NS

Geometry GE

Patterns and Algebra AL

Measurement ME

Statistics and Probability SP

MODULE 1QUADRATIC EQUATIONS AND

INEQUALITIES

LIVE C. ANGGA

Module 1

QUADRATIC EQUATIONS AND INEQUALITIES LM -pages 1 – 118TG- pages 1 – 78

CG-pages - 11

Quadratic Equations, Quadratic

Inequalities, and Rational Algebraic Equations

Illustrations of Quadratic Equations

Solving Quadratic Equations

Nature of Roots of Quadratic Equations

Sum and Product of Roots of Quadratic Equations

Extracting Square Roots

MODULE MAP

Equations Transformable to Quadratic Equations

Applications of Quadratic Equations and Rational

Algebraic Equations

Quadratic Inequalities

Rational Algebraic Equations

Illustrations of Quadratic

Inequalities

Solving Quadratic Inequalities

Application of Quadratic

Inequalities

MODULE MAP

Group Number Module 1 ActivityGroup Lessons 1 Group Lesson 2aGroup Lesson 2b Group Lesson 2cGroup Lesson 2dGroup Lesson 3Group Lesson 4 Group Lesson 5Group Lesson 6Group Lesson 7

Group Assignments

Lessons Coverage and its Objective:

Lesson I. Illustrations of Quadratic Equations

Objective:* Illustrate Quadratic Equations

Lessons Coverage and Objective:

Lesson 2- Solving Quadratic Equations Extracting Square RootsFactoringCompleting the SquareQuadratic Formula

Objective:* Solve Quadratic Equations by:

a. Extracting square rootsb. factoringc. completing the squaresd. using quadratic formula

Lesson 3. Nature of roots of Quadratic Equations

Objective:* characterize the roots of a quadratic equation using the discriminant.

Lesson 4: Sum and Product or Roots of Quadratic Equations

Objective: describe the relationship between the coefficient and the roots of a quadratic equation

Lesson 5 : Equations Transformable to Quadratic

Equations ( Including Rational Algebraic Equations)

Objective:* solve equations transformable to quadratic

equations (Including rational algebraic equations)

Lesson 6 : Applications of Quadratic Equations and Rational Algebraic Equations

Objective:* solve problems involving quadratic equations

and rational algebraic equations.

Lesson7 : Quadratic Inequalities

Objective:* Illustrate quadratic inequalities* solve quadratic inequalities and* solve problems involving quadratic

inequalities

Pretest

• Group Activity:5 groups

Group 1 – answer item 1-7Group 2 – answer item 8-14Group 3 – answer item 15-21Group 4 – answer item nos. 22-28Group5 – answer Part II items 1-7

Lesson No.

Topic What to Know

What to Process

What to Reflect

What to Transfer

1Illustrations of Quadratic Equations

( 1,2,3) = 3

(4,5,6) = 3

( 7) = 1

( 8 ) = 1 8

Lesson No.

Topic What to Know

What to Process

What to Reflect

What to Transfer

Lesson 2A

Solving Quadratic Equations by Extracting the Square Roots

(1,2,3,4,5) = 5

( 6,7) = 2

( 8, 9, ) = 2

( 10) = 1

Lesson No.

Topic What to Know

What to Process

What to Reflect

What to Transfer

2BSolving Quadratic Equations by Factoring

( 1,2,3) = 3 (4,5) = 2 ( 6) = 1 (7) = 1 7

Lesson No.

Topic What to Know

What to Process

What to Reflect

What to Transfer

Solving Quadratic Equations

by Completing the Square

(1,2,3,4) = 4

( 5,6) = 2

( 7) = 1

( 8 ) = 1 8

Lesson No.

Topic What to Know

What to Process

What to Reflect

What to Transfer

Solving Quadratic Equations by Using Quadratic Formula

( 1,2,3,4) = 4

( 5,6 ) = 2

( 7 ) = 1

( 8 ) = 1 8

Lesson No.

Topic What to Know

What to Process

What to Reflect

What to Transfer

The Nature of the Roots of a Quadratic Equation

(1,2,3,4,5,6) = 6

( 7,8) = 2

( 9 ) = 1

( 10 ) = 1 10

Lesson No.

Topic What to Know

What to Process

What to Reflect

What to Transfer

The Sum and the Product of Roots of Quadratic Equations

(1,2,3,4) = 4

( 5,6) = 2

( 7 , 8) = 2 ( 9 ) = 1 9

Lesson No.

Topic What to Know

What to Process

What to Reflect

What to Transfer

Equations transformable to Quadratic Equations

(1,2,3) = 3

( 4,5,6,7) = 4 ( 8 ) = 1 ( 9 ) = 1 9

Lesson No.

Topic What to Know

What to Process

What to Reflect

What to Transfer

Solving Quadratic Equations by Using Quadratic Formula

(1,2,3) = 4 (4) = 3 (5) = 1 ( 6,7) =

Lesson No.

Topic What to Know

What to Process

What to Reflect

What to Transfer

7 Quadratic Inequalities

( 1,2,3) = 3

( 4,5,6, 7,8) = 5

( 9) = 1

( 10 ) = 1 10

Let’s do the Activity …

Norms to Follow During the Presentation of Outputs

A – accurate (exact)

B – Brief (short duration)

C – Concise (using only few words clearly stated)

D – direct (easy to understand or respond to)

Time Frame

20 minutes – simultaneous group preparation

10 minutes - group presentation

5 minutes - interaction

Lesson 1 –Illustrations of Quadratic Equations

• What to knowActivity I: Do you Remember these Products?

Answer the Questions Refer: LM. pp.11 TG. pp. 14

Activity 2 :Another Kind of EquationAnswer the Questions Refer: LM. pp12 TG. Pp.14

Activity 3: A Real Step to Quadratic Equations Refer: LM. pp12 TG. pp. 15

Continuation of Lesson 1

• What to Process Activity 4: Quadratic or Not Quadratic

Refer: LM. pp.14 TG. pp. 16Activity 5: Does it Illustrate Me?

Refer: LM. pp.14 TG. pp. 16Activity 6 : Set Me to Your Standard

Refer: LM. pp.15 TG. pp. 17

Continuation of Lesson 1

• What to Reflect or UnderstandActivity 7: Dig Deeper

Refer: LM. pp.16 TG. pp. 18• What to TransferActivity 8 Where in the Real World

Refer: LM. pp.18 TG. pp. 18 * Summary/Synthesis/Generalization

Refer: LM pp. 17 TG pp. 18

Abstraction

Quadratic Equations in one variable is a mathematical sentence of degree 2 that can be written in the general form:

ax2+bx+c =0

• A quadratic equation is an equation equivalent to one of the form

• Where a, b, and c are real numbers and a 0 a is the quadratic coefficient

b is a linear coefficient

c is the constant term or free term

Include in the standard form:

ax2 = 0

ax2+bx = 0

ax2+c = 0

Note: If a = 0 can’t be a quadratic equation

Application

• Journal Writing/ Self-Reflection:• I realize that I need to do the following in order to

improve the delivery of the lessons in ____________.

• ___________________________________________• ___________________________________________• ___________________________________________

Lesson 2A –Solving Quadratic Equations by Extracting Square Roots

• What to Know

Activity 1. Find My Roots

Refer: LM pp.18 TG pp. 19

Activity 2 . What Would Make A Statement True?

Activity 3. Air Out!!!

Activity 4. Learn to Solve Quadratic Equations!!!

Activity 5 . Anything Real or Nothing Real?

Continuation…. Lesson 2A• What to Process

Activity 6: Extract Me!!!

LM pp. 23 TG pp. 21

Activity 7 : What Does a Square Have?

LM pp. 24 TG pp. 22• What to Reflect and Further Understand

Activity 8: Extract Then Describe Me!

LM pp. 25 TG pp. 22

Activity 9: Intensify your Understanding

LM pp. 25 TG pp. 22

Continuation…. Lesson 2A

What to Transfer:

With activity 10.in the TG: What More can I do?

LM pp. 23 TG pp. 26

Summary/ Synthesis/Generalization

LM pp. 26 TG pp.23

Abstraction: a. Extracting Square Roots

• An alternate method of solving a quadratic equation is using the Principle of Taking the Square Root of Each Side of an Equation

If x2 = a, then

x = + a

Ex 1: Solve by taking square roots 5(x – 4)2 = 125

First, isolate the squared factor:5(x – 4)2 = 125

(x – 4)2 = 25Now take the square root of both sides:

25)4( 2 x

x – 4 = + 5 x = 4 + 5

x = 4 + 5 = 9 and x = 4 – 5 = – 1

Lesson 2B Solving Quadratic Equations by Factoring

• What to Know

Activity 1: What Made Me?

LM pp. 27 TG pp. 24

Activity 2: The Manhole

LM pp. 28 TG pp. 24

Activity 3: Why is the Product Zero?

LM pp. 28 TG pp. 25

Continuation…Lesson 2BSolving Quadratic Equations by Factoring

• What to Process

Activity 4 : Factor Then Solve!

LM pp. 31 TG pp. 25

Activity 5: What Must be My Length and Width?

Activity 6. How well Did I Understand?

LM pp. 33 TG pp. 26• What to Transfer

Activity 7. Meet My Demands!!! ( TG)

LM pp. 34 TG pp 27

* Summary/ synthesis/ Generalization

b. Factoring

• Ex 1: Solve x2 + 5x + 4 = 0Quadratic equation factor the left hand side (LHS)

x2 + 5x + 4 = (x + )(x + )1

x2 + 5x + 4 = (x + 4)(x + 1) = 0Now the equation as given is of the form ab = 0

set each factor equal to 0 and solvex + 4 = 0 x + 1 = 0

x = – 4 x = – 1

Solution: x = - 4 and –1 x = {-4, -1}

Ex 2: Solve x2 -10x = - 25

Quadratic equation but not of the form ax2 + bx + c = 0

x2 - 10x + 25 = (x - )(x - )5 5

x2 - 10x + 25 = (x - 5)(x - 5) = 0

Now the equation as given is of the form ab = 0 set each factor equal to 0 and solve

x - 5 = 0x = 5

x - 5 = 0

x = 5 Solution: x = 5 x = { 5} repeated root

Quadratic equation factor the left hand side (LHS) Add 25 x2 – 10x + 25 = 0

Ex 3: Solve 5x2 = 4x

Quadratic equation but not of the form ax2 + bx + c = 0

5x2 – 4x = x( )5x – 4

5x2 – 4x = x(5x – 4) = 0

Now the equation as given is of the form ab = 0 set each factor equal to 0 and solve

x = 05x – 4 = 05x = 4

Solution: x = 0 and 4/5 x = {0, 4/5}

Quadratic equation factor the left hand side (LHS) Subtract 6x 5x2 – 4x = 0

x = 4/5

Explain: Zero property and Factoring procedure:

Zero Property = If the product of two real numbers is zero, then either of the two is equal to zero or both numbers are equal to zero.

Procedure:1. Transform quadratic equation into standard form if necessary.2. Factor the quadratic expression3. Apply the zero property by setting ach factor of the quadratic

expression equal to zero4. Solve each resulting equation.5. Check the values of the variable obtained by substituting

each in the original equation.

Lesson 2C. Solving Quadratic Equations by Completing the Square

• What to Know

Activity 1: How Many Solutions Do I have?

LM pp. 35 TG pp. 28

Activity 2: Perfect Square Trinomial to Square of a Binomial

LM pp. 36 TG pp. 29

Activity 3: Make it Perfect

LM pp. 37 TG pp. 29

Activity 4. Finish the Contract

LM pp. 37 TG pp. 29

Continuation: Lesson 2C.

• What to Process

Activity 5. Complete Me!

LM pp. 42 TG . 30

Activity 6. Represent then Solve!

LM pp. 43 TG. Pp. 33

• What to Reflect and Further Understand

Activity 7 . What Solving Quadratic Equations by Completing the Square Means to Me…

LM pp. 44 TG pp. 31

Continuation: Lesson 2C.

• What to Transfer:

Activity 8. Design Packaging Boxes

LM pp 45 TG pp. 32• Summary/ Synthesis/Generalization

LM pp. 46 TG pp. 32

Completing the Square

• Recall from factoring that a Perfect-Square Trinomial is the square of a binomial:Perfect square Trinomial Binomial Square x2 + 8x + 16 (x + 4)2

x2 – 6x + 9 (x – 3)2

• The square of half of the coefficient of x equals the constant term: ( ½ * 8 )2 = 16 -----------------64/4 =16 [½ (-6)]2 = 9 ------------------36/4 = 9

• Write the equation in the form x2 + bx = c• Add to each side of the equation [½(b)]2

• Factor the perfect-square trinomial x2 + bx + [½(b)] 2 = c + [½(b)]2

• Take the square root of both sides of the equation

• Solve for x

Further explanation:

• Quadratic equation ax2+bx+c = 0 can be transformed into (x-h)2=k where k≥0.

• K should not be negative.. Why? • Explain how to transform general form to

standard or vertex form.

Steps in completing the square: LM page 38

1. Divide both sides of the equation by a then simplify.2. Write the equation such that the terms with variables are

on the left side of the equation and the constant term is on the right side.

3. Add the square of one-half of the coefficient of x on both sides of the resulting equation. The left side of the equation becomes a perfect square trinomial.

4. Express the perfect square trinomial on the left side of the equation as a square of the binomial.

5. Solve the resulting quadratic equation by extracting the square root.

6. Solve the resulting linear equation.7. Check the solutions obtained against the original equation.

Lesson 2D. Solving Quadratic Equations by Using Quadratic Formula

• What to Know

Activity 1: It’s Good to be Simple!

LM pp. 47 TG pp 33

Activity 2 Follow the Standard

LM pp. 48 TG pp. 34

Activity 3. Why do the Gardens Have to be Adjacent?

LM pp.48 TG pp. 35

Activity 4 Lead Me to the Formula

LM pp. 49 TG pp. 35

Continuation…Lesson 2D. • What to Process

Activity 5: Is the Formula Effective?

LM pp. 52 TG pp. 36

Activity 6. Cut to Fit!

Activity 7 : Make the Most Out of It!

LM pp. 53 TG pp. 37• What to Transfer:

Activity 8. Show Me the Best Floor Plan?

LM pp. 55 TG pp. 38• Summary/ Synthesis/Generalization

LM pp. 55 TG pp38

Abstraction: d. The Quadratic Formula

• Consider a quadratic equation of the form ax2 + bx + c = 0 for a nonzero

• Completing the square 2ax bx c

x xa a

b b c bx x

a 4a a 4a

The Quadratic Formula

Solutions to ax2 + bx + c = 0 for a nonzero are

b b 4acx2a 4a

2b b 4acx

b b 4ac bx x

a 4a 4a 4a

Ex: Use the Quadratic Formula to solve x2 + 7x + 6 = 0

Recall: For quadratic equation ax2 + bx + c = 0, the solutions to a quadratic equation are given by

a2ac4bb

Identify a, b, and c in ax2 + bx + c = 0:

a = b = c = 1 7 6

Now evaluate the quadratic formula at the identified values of a, b, and c

)1(2)6)(1(477

224497

x = ( - 7 + 5)/2 = - 1 and x = (-7 – 5)/2 = - 6

x = { - 1, - 6 }

Application

• ___________________________________________• ___________________________________________• ___________________________________________

Lesson 3. The Nature of the Roots of a Quadratic Equation

• What to Know

Activity 1. Which are Real? Which are Not?

LM pp. 56 TG pp.39

Activity 2: Math in A,B,C?

LM pp . 57 TG pp. 40

Activity 3: Math My Value?

LM pp. 57 TG 40

Activity 4: Find my Equation and Roots

LM pp. 58 TG pp. 40

Activity 5: Place Me on the Table

LM pp.58 TG pp. 41

Activity 6: Let’s Shoot that Ball!

LM pp. 59 TG pp. 41

Continuation….Lesson 3• What to Process

Activity 7: What is My Nature?

LM pp. 42 TG pp. 62

Activity 8: Lets Make a Table!

Activity 9: How Well Did I Understand the Lesson?

Activity 10 . Will It or Will It Not?

LM pp. 64 TG PP. 44• Summary/ Synthesis/Generalization

Lm PP. 65 TG PP. 44

Abstraction:

Explain Discriminant and its nature of roots.It is the value of the expression b2-4ac of the quadratic equation ax2+bx+c = 0;It describes the nature of the roots of the quadratic equation ; It can be:* zero* positive and perfect square* positive but not perfect square* negative

Nature of Roots LM: Page 59-61

Value of D Nature of Roots Roots

D=0 Real and equal Each root = to –b/2a

D˃0 and a perfect square

rational and are not equal

{-b+√D/2a}

D˃0 but not perfect square

Irrational and are not equal {-b+√D/2a}

D˂0 No real roots none

Application

• ___________________________________________• ___________________________________________• ___________________________________________

Lesson 4. The Sum and the Product of Roots of Quadratic Equations

• What to Know

Activity 1: Let’s Do Addition and Multiplication!

ML pp. 66 TG pp.45

Activity 2: Find My Roots!

LM pp. 67 TG pp. 45

Activity 3: Relate Me to My Roots

LM pp. 67 TG pp. 46

Activity 4 : What the Sum and Product Mean to Me..

LM pp. 68 TG pp. 46

Continuation…..Lesson 4.

• What to Process

Activity 5: This is My Sum and this is My Product. Who Am I?

LM pp. 71 TG pp. 47

Activity 6. Here Are the Roots. Where is the Trunk?

LM pp. 72 TG pp. 48

* What to Reflect and Further Understand

Activity 7. Fence My Lot!!

LM pp. 73 TG pp. 48

Activity 8. Think of These Further!

LM pp. 74 TG pp 49

Continuation…..Lesson 4.

• What to Transfer:• Activity 9: Lets Make a Scrap Book!

LM pp. 76 TG pp. 49

Abstract

• Solving quadratic equations by factoring,

Consider the general quadratic equation: where

Multiply to create a leading coefficient of 1:

Represent the roots of the equation as and :

Comparing the equations, it can be seen that:or and

Our investigation reveals that there is a definite relationship between the roots of a quadratic equation and the coefficient

of the second term and the constant term.The sum of the roots of a quadratic equation is equal to the negation of the coefficient of the second term divided by the leading coefficient. The product of the roots of a quadratic equation is

equal to the constant term divided by the leading coefficient.

Application

• ___________________________________________• ___________________________________________• ___________________________________________

Lesson 5. Equations Transformable to Quadratic Equations

• What to Know

Activity 1: Let’s Recall

LM pp. 77 TG pp. 50

Activity 2: Let’s Add and Subtract!

LM pp. 77 TG pp. 50

Activity 3: How Long Does It Take To Finish Your Job?

LM pp. 78 TG pp. 51

Lesson 5. Equations Transformable to Quadratic Equations ( Continuation)

• What to Process

Activity 4: View Me in Another Way!

LM pp. 83 TG pp. 51

Activity 5: What Must be The Right Value?

LM pp. 83 TG pp. 52

Activity 6: Let’s Be True!

LM pp. 84 TG pp. 52

Activity 7: Let’s Paint the House!

LM pp. 84 TG pp. 52

Lesson 5. Equations Transformable to Quadratic Equations ( Continuation)

What to Reflect and Further Understand

Activity 8: My Understanding of Equations Transformable to Quadratic

Activity 9: A Reality of Rational Algebraic Equation

LM pp. 87 TG pp. 53

Abstraction

An equation is said to be in a quadratic form if its original variable is in the highest degree of 2.

Example:

ax2+bx+c = 0 is said to be a quadratic form because the variable x has a highest degree of 2.

Example: By factoring

• Solve: x2-34x+ 225 = 0Solution:

(x-9) (x-25) = 0 (x-9) = 0 and (x-25) = 0

x = 9 x = 25

Solution set : {9, 25}

Application

• ___________________________________________• ___________________________________________• ___________________________________________

Lesson 6. Solving Quadratic Equations by Using Quadratic Formula

• What to Know

Activity 1: Find My Solution!

LM pp. 88 TG pp. 54

Activity 2: Translate into….

LM pp. 88 TG pp. 54

Activity 3: What are my Dimensions?

LM pp. 89 TG pp. 55• What to Process

Activity 4 : Let Me Try

LM pp. 92 TG56

Lesson 6. Solving Quadratic Equations by Using Quadratic Formula ( Continuation)

• What to Reflect and Further UnderstandActivity 5: Find Those Missing!

LM pp. 93 TG pp. 56• What to Transfer:Activity 6: Let’s Draw!

LM pp. 94 TG pp. 57 Activity 7: Play the Role of …

LM PP. 95 tg PP. 57

Abstraction

Quadratic Formula: For

The solutions of some quadratic equations, ( ), are not rational, and cannot be obtained by factoring.

Note: The quadratic formula can be used to solve ANY quadratic equation, even those that can be factored.

By factoring (this equation is factorable):

By Quadratic Formula: a = 1, b = 2, c = -8

Hints:Be careful with the signs of the values a, b and c. Don't drop the sign when substituting into the formula. Also remember your rules for multiplying and adding signed numbers as you solve the formula. MSJC ~ San Jacinto Campus

Math Center Workshop SeriesJanice Levasseur

Hints:Remember that a

negative value under the radical is creating an imaginary number (a number with an i).

Example 2: This equation cannot be solved by factoring. By Quadratic Formula: a = 1, b = 4, c = 5

MSJC ~ San Jacinto CampusMath Center Workshop Series

Janice Levasseur

Example 3. This equation cannot be solved by factoring. By Quadratic Formula: a = 3, b = -10, c = 5

Hints: Notice how the value for b was substituted into the formula using parentheses (-10). This helps you to remember to deal with the negative value of b. Also, notice how the (-10)2 is actually a positive value. When you square a value, the answer is always positive.If needed, these answers can be estimated as decimal values, such as (rounded to 3 decimal places):x = 2.721; x = 0.613The radical answers are the "exact" answers.The decimal answers are "approximate" answers.

MSJC ~ San Jacinto Campus

Math Center Workshop Series

Janice Levasseur

Application

• ___________________________________________• ___________________________________________• ___________________________________________

Lesson 7. Quadratic Inequalities

• What to Know

Activity 1 : What Makes Me True?

LM pp. 96 TG pp. 58

Activity 2 Which are Not Quadratic Equations?

LM pp. 97 TG pp. 59

Activity 3: Let’s Do Gardening

LM pp. 97 TG pp. 59

Lesson 7. Quadratic Inequalities ( Continuation)

• What to Process

Activity 4: Quadratic Inequalities or Not?

LM pp. 106 TG pp. 60

Activity 5. Describe My Solutions!

Activity 6: Am I a Solution or Not?

Activity 7: What Represents Me?

Activity 8: Make It Real!

Lesson 7. Quadratic Inequalities ( Continuation)

• What to Reflect and Further Understand

Activity 9: How Well I Understood…

LM pp. 111 TG pp. 63- 65• What to Transfer:

Activity 10: Investigate Me!

Activity 11. How Much Would It Cost to Tile a Floor?

• Summary/ Synthesis/Generalization

Abstraction

Quadratic inequalities can be solved graphically or algebraically.

The graph of an inequality is the collection of all solutions of the inequality.

The trick to solving a quadratic inequality is to replace the

inequality symbol with an equal sign and solve the resulting equation. The solutions to the equation will allow you to establish intervals that will let you solve the inequality.

Plot the solutions on a number line creating the intervals for investigation. Pick a number from each interval and test it in the original inequality. If the result is true, that interval

is a solution to the inequality.

Example 1 (one variable inequality):

Answer:x < -3 or x > 4

Example 2 (two variable inequality):

• Begin by graphing the corresponding equation .• (Use a dashed line for < or > and a solid line for < or >.) • Test a point above the parabola and a point below the

parabola into the original inequality. Shade the entire region where the test point yields a true result.

• The parabola graph was drawn using a solid line since the inequality was "greater than or equal to".

• The point (0,0) was tested into the inequality and found to be true.

• The point (0,-2) was tested into the inequality and found to be false.

• The graph was shaded in the region where the true test point was located. ANSWER: The shaded area (including the solid line of the parabola) contains all of the points that make this inequality true.

When you solved quadratic equations, you created factors whose product was zero, implying either one or both of the factors must be equal to zero.

When solving a quadratic inequality, you need to take more options into consideration. Consider these two different problems

Solving a quadratic inequality

From the graph we can see that in the intervals around the zeros, the graph is either above the x-axis (positive) or below the x-axis (negative). So we can see from the graph the interval or intervals where the inequality is positive. But how can we find this out without graphing the quadratic?

We can simply test the intervals around the zeros in the quadratic inequality and determine which make the inequality true.

Solving a quadratic inequality

For the quadratic inequality,we found zeros 3 and –2 by solving the equation

. Put these values on a number line and we can see three intervals that we will test in the inequality. We will test one value from each interval.

062 xx

Solving a quadratic inequalityInterval Test Point Evaluate in the inequality True/False

06639633 2

066416644 2

062 xx

(0)2- (0)-6= 0-0-6=-6˃0

Example 2:

Solve First find the zeros by solving the equation,

0132 2 xx

0112 xx

01or012 xx

Example 2:

Now consider the intervals around the zeros and test a value from each interval in the inequality.

The intervals can be seen by putting the zeros on a number line.

Forms of Quadratic Inequalitiesy<ax2+bx+c y>ax2+bx+cy≤ax2+bx+c y≥ax2+bx+c

• Graphs will look like a parabola with a solid or dotted line and a shaded section.

• The graph could be shaded inside the parabola or outside.

Steps for graphing1. Sketch the parabola y=ax2+bx+c(dotted line for < or >, solid line for ≤ or ≥)** remember to use 5 points for the graph!2. Choose a test point and see whether it is a

solution of the inequality.3. Shade the appropriate region.

(if the point is a solution, shade where the point is, if it’s not a solution, shade the other region)

Example:Graph y ≤ x2+6x- 4

* Vertex: (-3,-13)

* Opens up, solid line

134189

4)3(6)3( 2

y 9- 5-

12- 4-

13- 3-

12- 2-

•Test Point: (0,0)

0≤02+6(0)-4

0≤-4 So, shade where the point is NOT!

Test point

Graph: y>-x2+4x-3

* Opens down, dotted line.* Vertex: (2,1)

3)2(4)2(1 2

* Test point (0,0)

0>-02+4(0)-3

Test Point

Last Example! Sketch the intersection of the given inequalities.

1 y≥x2 and 2 y≤-x2+2x+4

• Graph both on the same coordinate plane. The place where the shadings overlap is the solution.

• Vertex of #1: (0,0)Other points: (-2,4), (-1,1), (1,1),

• Vertex of #2: (1,5)Other points: (-1,1), (0,4), (2,4),

* Test point (1,0): doesn’t work in #1, works in #2.

SOLUTION!

Application

• ___________________________________________• ___________________________________________• ___________________________________________

Thoughts to Remember• Speak 6 lines to yourself everyday:1. I am blessed2. I can do it3. I am a winner4. Today is my day5. God is always with me and6. I am a child of God

Be a blessing with others committed in sharing knowledge, skills and abilities ,nurturing learners, promoting better education.

God bless us all

module 1( final 2) quadtraic equations and inequalities jq

solving quadratic equations

quadratic equations

equations transformable

quadratic functions

d group lesson

c group lesson

b group lesson

group number module

Documents

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