Mathematics 11

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Mathematics 11

Quarter 1 - Module 4: Solving Equations Transformable into Quadratic Equations

Department of Education ● Republic of the Philippines

Math- Grade 9 Alternative Delivery Mode Quarter 1 - Module 4: Solving Equations Transformable into Quadratic Equations First Edition, 2020

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Published by the Department of Education – Division of Iligan City Schools Division Superintendent: Roy Angelo L. Gazo, PhD.,CESO V

Development Team of the Module

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Mathematics

Quarter 1 - Module 4 Solving Equations Transformable into Quadratic Equation

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Department of Education ● Republic of the Philippines

Table of Contents

What This Module is About ...... i

What I Need to Know ...... i

How to Learn from this Module ...... ii

Icons of this Module ...... ii

Lesson 1: Equations Transformable to Quadratic Functions ...... 1

What I Need to Know...... 1

What I Know ...... 1

What’s In ...... 2

What’s New ...... 3

What Is It ...... 3

What’s More…...... 6

What I Have Learned...... 6

What I Can Do ...... 6

Summary ...... 7

Assessment: (Post-Test) ...... 7

Key to Answers ...... 8

References ...... 9

What This Module is About

In this module, the lesson starts in assessing your understanding of the various mathematics principles and concepts studied previously, and enhance skills in performing mathematical operations. All these skills and knowledge may help you in solving Equations transformable into Quadratic Equations.

As you embrace through this lesson, be aware of this significant question: “How are these Algebraic Equations be transformed into Quadratic Equations?”. To discover the answers to this vital question, the set of activities must be performed. If difficulty ascends, you may ask your teacher, peer, or friends to help you in revisiting the modules completed over earlier. Your teacher can help you measure your answers.

What I Need to Know

In this lesson you will learn to:  Solve Algebraic Equations which can be transformed to Quadratic Equations;  Check for extraneous solutions.

How to Learn from this Module

To achieve the objectives cited above, you are to do the following:

• Take your time reading the lessons carefully.

• Follow the directions and/or instructions in the activities and exercises diligently.

• Answer all the given tests and exercises.

Icons of this Module

What I Need to This part contains learning objectives that Know are set for you to learn as you go along the module.

What I know This is an assessment as to your level of knowledge to the subject matter at hand, meant specifically to gauge prior related knowledge What’s In This part connects previous lesson with that of the current one.

What’s New An introduction of the new lesson through various activities, before it will be presented to you

What is It These are discussions of the activities as a way to deepen your discovery and understanding of the concept.

What’s More These are follow-up activities that are in- tended for you to practice further in order to master the competencies.

What I Have Activities designed to process what you Learned have learned from the lesson

What I can do These are tasks that are designed to show- case your skills and knowledge gained, and applied into real-life concerns and situations.

Lesson Equations Transformable into

Quadratic Equations

What I Need to Know

In this Lesson, you will have a chance to build up your skills in solving equations which can be transformed to quadratic equations.

What I Know

Pre-Assessment Test

Directions: Find out how much you already know about this module. Solve the following problems.

1. Express in simplest form.

A B. C. D.

2. Which of the following is the standard form of quadratic equations? A. C. B. D. y = mx +b 3. Express (w+7)(w-2) =0 in standard form A. C. B. D. 4. Find the roots of the equation . A. -5 , -2 B. 3 , 4 c. 5 , 2 D. 2 , -3

5. Which of the following rational algebraic equations is transformable to a quadratic equation?

A. = C. =

B. = 6m D. = 7

6. Which of the following equations have extraneous roots or solutions?

A. - x x x C. - x x x B. - D. x (x +3) = 28 x

What’s In

Activity 1: Who Said It?

Direction: Answer the puzzle below by simplifying the following expressions. Then shade the box containing the corresponding answer. The unshaded boxes will show the answer to this puzzle.

Which great mathematician and scientist said: “Do not worry about difficulties in Mathematics. I can assure you that mine are still greater.”

1. 7.

2. 8.

3. 9.

4. 10.

5. 11.

6. 12.

C E I A

N H S H

A T R E

W L I E

T N P O

What’s New

In learning this module, you will need your skills in adding and subtracting rational algebraic expressions. Because of that, let’s recall these first.

Example: Find the sum of and . x

Step 1: Write the expression

x x

Step 2: Find the Least Common Denominator (LCD) of the rational algebraic

expressions, and . x

2x = 2 · x 6x = 2 · 3 · x 2 · 3 · x = 6x

Therefore, the LCD is 6x.

Step 2: Rewrite the expression using LCD, 6x.

or x x x x x x

Now, you’re ready! Let’s try having the activity below.

What Is It

Activity 2: Let’s Add and Subtract! Direction: Perform the indicated operation then express your answer in simplest form.

1. 4.

Were you able to add or subtract the rational expressions and simplify the results? Suppose you were given a rational algebraic equation, how would you find its solution/s? You will learn this in the succeeding activities.

Solving Quadratic Equations That Are Not Written in Standard Form

2 Standard Form of Quadratic Equation: ax + bx + c = 0, where x is the variable and a, b and c are constants (a ≠ 0)

Example 1: Solve

This quadratic equation is not written in standard form. To write the quadratic equation in standard form:  write the given equation = 36  simplify the left side of the equation x2 - 5x = 36  transform to standard form of quadratic equation

Use any of the four methods in finding the solutions of the quadratic equation

Try factoring in finding the roots of the equation.  factor the left side of the equation  equate each factor to zero or  solve each resulting equation or

Check whether the obtained values of x make the equation true. If the obtained values of x make the equation true, then the solutions of the equation are: or .

Example 2: Find the roots of the equation

The given equation is a quadratic equation but it is not written in standard form. Transform this equation to standard form, then solve it using any of the methods of solving quadratic equations.  write the given equation  simplify the square of a binomial  combine like terms  transform to standard form of quadratic equation

 factor the left side of the equation  equate each factor to zero or  solve each resulting equation or The solutions of the equation are or . These values of make the equation true.

Solving Rational Algebraic Equations which can be transformed to Quadratic Equations

Example 3: Solve the rational algebraic equation

The given rational algebraic equation can be transformed into a quadratic equation. To solve the equation, the following procedure can be followed.

 write the given equation

 multiply both sides of the equation by the Least Common Denominator (LCD) 4x.

( )

 distribute 4x on the left side of the equation

 reduce the left side of the equation to its simplest form

 write the resulting quadratic equation in standard form

Find the roots of the resulting equation using any of the methods of solving quadratic equations. Try factoring in finding the roots of the equation.

 equate each factor to zero or  solve each resulting equation or

Check whether the obtained values of x make the equation true.

If the obtained values of x make the equation true, then the solutions of the equation are: or .

Extraneous Solution of Rational Quadratic Equations

Example 4: Solve =

The given rational algebraic equation can be transformed into a quadratic equation. To solve the equation, the following procedure can be followed.

 write the given equation =

 multiply both sides of the equation by

 simplify both sides of the equation to its simplest form

 write the resulting quadratic equation in standard form

 factor the left side of the equation

 equate each factor to zero 3x = 0, x - 4 = 0  solve each resulting equation x =0 , x = 4

Check whether the obtained values of x make the equation = true. x = 4 is the solution but x =0 is not a correct solution, thus 0 is an extraneous solution. Value of x variable that does not make the original equation true.

What’s More

Activity 3: View Me in Another Way!

Directions: Transform each of the following equations into a quadratic equation in the form

1. 2.

Process Questions: 1. How did you transform each equation into a quadratic equation? What mathematics concepts or principles did you apply? 2. Did you find any difficulty in transforming each equation into a quadratic equation? Explain.

Were you able to transform each equation into a quadratic equation? Why do you think there is a need for you to do such activity? Find this out in the next activity.

What I Have Learned

Activity 4: Let’s Be True!

Direction: Find the solution set of the following.

1. 2. 3. 4.

What I Can Do

Activity 5: My Understanding of Equations Transformable into Quadratic Equations.

Direction: Answer the following.

1. How do you transform a rational algebraic equation into a quadratic equation? Explain and give example. 2. How do you determine the solutions of quadratic equations? How about rational algebraic equations transformable into quadratic equations? 3. Suppose a quadratic equation is derived from a rational algebraic equation. How do you check if the solutions of the quadratic equation are also the solutions of the rational algebraic equation? 4. Which of the following equations have extraneous roots or solutions? Justify your answer.

a. c.

b. d.

Summary

This lesson was about the solutions of equations that are transformable into quadratic equations including rational algebraic equations. This lesson provided you with opportunities to transform equations into the form ax 2 + bx + c = 0 and to solve them. Moreover, this lesson served as your foundation skills to solve real-life problems involving rational algebraic equations. Your knowledge of this lesson and earlier mathematics concepts and principles will help you in understanding the succeeding lessons.

Assessment Test: (Post Test)

Directions: Find out how much you already know about this module. Solve the following problems.

1. Express in simplest form.

A B. C. D.

2. Which of the following is the standard form of quadratic equations? A. C. B. D. y = 2x +5 3. Express (x +2)2 + 9 =0 in standard form A. C. D. 4. Find the roots of the equation . A. -1 , -3 B. -3 , 4 C. 5 , 2 D. 3 , -3

5. Which of the following rational algebraic equations is transformable to a quadratic equation?

A. = C. =

B. = D.

6. Which of the following equations have extraneous roots or solutions? x x A. - C. -

- y 2 B. D. 2(3x -1) = 11x y y

Key to Answers

References:

Mathematics – Grade 9 Learner’s Material First Edition, 2014, Department of Education DepEd link: http://www.depednegor.net/uploads/8/3/5/2/8352879/math_9_lm_draft_3.24.2014.pdf

LINKS: https://brainly.ph/question/248888 https://saylordotorg.github.io/text_elementary-algebra/s10-05-solving-rational-equations.html https://www.chilimath.com/lessons/advanced-algebra/solving-rational-equations/ http://www.montereyinstitute.org/courses/Algebra1/COURSE_TEXT_RESOURCE/U11_L2_T1_text _final.html

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