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This is a math module forr 9th graders.
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This instructional material was collaboratively developed and reviewed by educators from public and private schools, colleges, and/or universities. We encourage teachers and other education stakeholders to email their feedback, comments, and recommendations to the Department of Education at [email protected].
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Mathematics – Grade 9 Learner’s Material First Edition, 2014
Republic Act 8293, section 176 states that: No copyright shall subsist in any work of the Government of the Philippines. However, prior approval of the government agency or office wherein the work is created shall be necessary for exploitation of such work for profit. Such agency or office may, among other things, impose as a condition the payment of royalties. Borrowed materials (i.e., songs, stories, poems, pictures, photos, brand names, trademarks, etc.) included in this book are owned by their respective copyright holders. Every effort has been exerted to locate and seek permission to use these materials from their respective copyright owners. The publisher and authors do not represent nor claim ownership over them.
Published by the Department of Education Secretary: Br. Armin A. Luistro FSC Undersecretary: Dina S. Ocampo, Ph.D.
Department of Education-Instructional Materials Council Secretariat (DepEd-IMCS) Office Address: 5 th^ Floor Mabini Building, DepEd Complex Meralco Avenue, Pasig City Philippines 1600 Telefax: (02) 634-1054 or 634- E-mail Address: [email protected]
MODULE NO. 1 : QUADRATIC EQUATIONS AND INEQUALITIES
Was there any point in your life when you asked yourself about the different real life quantities such as costs of goods or services, incomes, profits, yields and losses, amount of particular things, speed, area, and many others? Have you ever realized that these quantities can be mathematically represented to come up with practical decisions?
Find out the answers to these questions and determine the vast applications of quadratic equations and quadratic inequalities through this module.
In this module, you will examine the above questions when you take the following lessons: Lesson 1 – ILLUSTRATIONS OF QUADRATIC EQUATIONS
Lesson 2 – SOLVING QUADRATIC EQUATIONS EXTRACTING SQUARE ROOTS FACTORING COMPLETING THE SQUARE QUADRATIC FORMULA
Lesson 3 – NATURE OF ROOTS OF QUADRATIC EQUATIONS
Lesson 4 – SUM AND PRODUCT OF ROOTS OF QUADRATIC EQUATIONS
Lesson 5 – EQUATIONS TRANSFORMABLE TO QUADRATIC EQUATIONS (INCLUDING RATIONAL ALGEBRAIC EQUATIONS)
Lesson 6 – APPLICATIONS OF QUADRATIC EQUATIONS AND RATIONAL ALGEBRAIC EQUATIONS
Lesson 7 – QUADRATIC INEQUALITIES
In these lessons, you will learn to: Lesson (^1) illustrate quadratic equations; Lesson 2 (^) solve quadratic equations by: (a) extracting square roots; (b) factoring; (c) completing the square; (d) using the quadratic formula; Lesson 3 (^) characterize the roots of a quadratic equation using the discriminant; Lesson 4 (^) describe the relationship between the coefficients and the roots of a quadratic equation; Lesson 5 (^) solve equations transformable to quadratic equations (including rational algebraic equations); Lesson 6 solve problems involving quadratic equations and rational algebraic equations; Lesson 7 illustrate quadratic inequalities; solve quadratic inequalities; and solve problems involving quadratic inequalities.
PRE-ASSESSMENT
Part I
Directions: Find out how much you already know about this module. Choose the letter that you think best answers the question. Please answer all items. Take note of the items that you were not able to answer correctly and find the right answer as you go through this module.
A. 7 4
w w C. 4
2 q 1 q
p p
m m m
A. I and II B. II and III C. I and III D. I, II, and III
y
x
y
x
y
x
y
x
A. I and II B. I and III C. II and III D. I, II, and III
The figure on the right shows the graph of y 2 x^2 4 x 1. Which of the following is true about the solution set of the inequality? I. The coordinates of all points on the shaded region belong to the solution set of the inequality. II. The coordinates of all points along the parabola as shown by the broken line belong to the solution set of the inequality. III. The coordinates of all points along the parabola as shown by the broken line do not belong to the solution set of the inequality.
y
x
A. 8 cm B. 12 cm C. 14 cm D. 18 cm
6 cm
4 cm
s
s
Rubric for Equations Formulated and Solved 4 3 2 1 Equations and inequalities are properly formulated and solved correctly.
Equations and inequalities are properly formulated but not all are solved correctly.
Equations and inequalities are properly formulated but are not solved correctly.
Equations and inequalities are properly formulated but are not solved.
Rubric on Problems Formulated and Solved Score Descriptors
Poses a more complex problem with 2 or more correct possible solutions and communicates ideas unmistakably, shows in-depth comprehension of the pertinent concepts and/or processes and provides explanations wherever appropriate.
Poses a more complex problem and finishes all significant parts of the solution and communicates ideas unmistakably, shows in- depth comprehension of the pertinent concepts and/or processes.
Poses a complex problem and finishes all significant parts of the solution and communicates ideas unmistakably, shows in-depth comprehension of the pertinent concepts and/or processes.
Poses a complex problem and finishes most significant parts of the solution and communicates ideas unmistakably, shows comprehension of major concepts although neglects or misinterprets less significant ideas or details.
Poses a problem and finishes some significant parts of the solution and communicates ideas unmistakably but shows gaps on theoretical comprehension.
Poses a problem but demonstrates minor comprehension, not being able to develop an approach. Source: D.O. #73 s. 2012
After going through this module, you should be able to demonstrate understanding of key concepts of quadratic equations, quadratic inequalities, and rational algebraic equations, formulate real-life problems involving these concepts, and solve these using a variety of strategies. Furthermore, you should be able to investigate mathematical relationships in various situations involving quadratic equations and quadratic inequalities.
What to KNOW:
Activity 1: Do You Remember These Products?
Directions : Find each indicated product then answer the questions that follow.
Questions: a. How did you find each product?
b. In finding each product, what mathematics concepts or principles did you apply? Explain how you applied these mathematics concepts or principles.
c. How would you describe the products obtained?
Start Lesson 1 of this module by assessing your knowledge of the different mathematics concepts previously studied and your skills in performing mathematical operations. These knowledge and skills will help you in understanding quadratic equations. As you go through this lesson, think of this important question: “How are quadratic equations used in solving real-life problems and in making decisions?” To find the answer, perform each activity. If you find any difficulty in answering the exercises, seek the assistance of your teacher or peers or refer to the modules you have gone over earlier. You may check your work with your teacher.
In the equation, a x^2 is the quadratic term, b x is the linear term, and c is the constant term.
Example 1: 2 x^2 + 5 x – 3 = 0 is a quadratic equation in standard form with a = 2, b = 5, and c = -3.
Example 2: 3 x ( x – 2) = 10 is a quadratic equation. However, it is not written in standard form.
To write the equation in standard form, expand the product and make one side of the equation zero as shown below.
3 x x 2 10 3 x^2 6 x 10
3 x^2 6 x 10 10 10
3 x^2 6 x 10 0
The equation becomes 3 x^2 – 6 x – 10 = 0, which is in standard form.
In the equation 3 x^2 – 6 x – 10 = 0, a = 3, b = -6, and c = -10.
How did you find the preceding activities? Are you ready to learn about quadratic equations? I’m sure you are!!! From the activities done, you were able to describe equations other than linear equations, and these are quadratic equations. You were able to find out how a particular quadratic equation is illustrated in real life. But how are quadratic equations used in solving real-life problems and in making decisions? You will find these out in the activities in the next section. Before doing these activities, read and understand first some important notes on quadratic equations and the examples presented.
A quadratic equation in one variable is a mathematical sentence of degree 2 that can be written in the following standard form.
a x^2 + b x + c = 0 , where a , b , and c are real numbers and a ≠ 0
Why do you think a must not be equal to zero in the equation a x^2 + b x + c = 0?
Example 3: The equation (2 x + 5)( x – 1) = -6 is also a quadratic equation but is not written in standard form.
Just like in Example 2 , the equation (2 x + 5)( x – 1) = -6 can be written in standard form by expanding the product and making one side of the equation zero as shown below.
2 x 5 x 1 6 2 x^2 2 x 5 x 5 6
2 x^2 3 x 5 6
2 x^2 3 x 5 6 6 6
2 x^2 3 x 1 0
The equation becomes 2 x^2 + 3 x + 1 = 0, which is in standard form.
In the equation 2 x^2 + 3 x + 1 = 0, a = 2, b = 3, and c = 1.
When b = 0 in the equation a x^2 + b x + c = 0 , it results to a quadratic equation of the form a x^2 + c = 0.
Examples: Equations such as x^2 + 5 = 0, -2 x^2 + 7 = 0, and 16 x^2 – 9 = 0 are quadratic equations of the form a x^2 + c = 0. In each equation, the value of b = 0.
What to PROCESS:
Your goal in this section is to apply the key concepts of quadratic equations. Use the mathematical ideas and the examples presented in the preceding section to answer the activities provided.
Learn more about Quadratic Equations through the WEB. You may open the following links.
http://library.thinkquest.org/20991/alg2/quad .html
http://math.tutorvista.com/algebra/quadratic- equation.html
http://www.algebra.com/algebra/homework/ quadratic/lessons/quadform/
Activity 6: Set Me To Your Standard!
Directions : Write each quadratic equation in standard form, a x^2 + b x + c = 0 then identify the values of a , b , and c. Answer the questions that follow.
Questions: a. How did you write each quadratic equation in standard form?
b. What mathematics concepts or principles did you apply to write each quadratic equation in standard form? Discuss how you applied these mathematics concepts or principles.
c. Which quadratic equations did you find difficult to write in standard form? Why?
d. Compare your work with those of your classmates. Did you arrive at the same answers? If NOT, explain.
How was the activity you have just done? Was it easy for you to write quadratic equations in standard form? It was easy for sure!
In this section, the discussion was about quadratic equations, their forms and how they are illustrated in real life.
Go back to the previous section and compare your initial ideas with the discussion. How much of your initial ideas are found in the discussion? Which ideas are different and need revision?
Now that you know the important ideas about this topic, let’s go deeper by moving on to the next section.
What to REFLECT ON and FURTHER UNDERSTAND:
Activity 7: Dig Deeper!
Directions : Answer the following questions.
Edna: a = 2; b = 3; c = -
Luisa: a = -2; b = -3; c = 5
Who do you think got the correct values of a, b, and c? Justify your answer.
Do you agree that the equation 4 3 x 2 x^2 can be written in standard form in two different ways? Justify your answer.
a. How are you going to represent the number of Mathematics Club members?
b. What expression represents the amount each member will share?
c. If there were 25 members more in the club, what expression would represent the amount each would share?
d. What mathematical sentence would represent the given situation? Write this in standard form then describe.
Your goal in this section is to take a closer look at some aspects of the topic. You are going to think deeper and test further your understanding of quadratic equations. After doing the following activities, you should be able to answer this important question: How are quadratic equations used in solving real-life problems and in making decisions?
The members of the school’s Mathematics Club shared equal amounts for a new Digital Light Processing (DLP) projector amounting to Php25,000. If there had been 25 members more in the club, each would have contributed Php50 less.
