Factoring a Quadratic Expression (eNotes Book 1)

The easy way to factorise nasty quadratics
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The design is made but not appropriate. 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.

Source: D. Furthermore, you should be able to investigate mathematical relationships in various situations involving quadratic equations and quadratic inequalities. Illustrations of Quadratic Equations 11 1 What to Know Start Lesson 1 of this module by assessing your knowledge of the different mathematics concepts previously studied and your skills in performing mathematical operations.

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These knowledge and skills will help you understand quadratic equations. 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. Find each indicated product then answer the questions that follow. How did you find each product? In finding each product, what mathematics concepts or principles did you apply? Explain how you applied these mathematics concepts or principles. How would you describe the products obtained?

Factoring quadratics: leading coefficient ≠ 1

Are the products polynomials? If YES, what common characteristics do these polynomials have? Were you able to find and describe the products of some polynomials? Were you able to recall and apply the different mathematics concepts or principles in finding each product? Why do you think there is a need to perform such mathematical tasks?

You will find this out as you go through this lesson. Below are different equations. Use these equations to answer the questions that follow. Which of the given equations are linear? How do you describe linear equations? Which of the given equations are not linear? How are these equations different from those which are linear? What common characteristics do these equations have? In the activity you have just done, were you able to identify equations which are linear and which are not?

Were you able to describe those equations which are not linear? These equations have common characteristics and you will learn more of these in the succeeding activities. Jacinto asked a carpenter to construct a rectangular bulletin board for her classroom. Draw a diagram to illustrate the bulletin board. What are the possible dimensions of the bulletin board? Give at least 2 pairs of possible dimensions. How did you determine the possible dimensions of the bulletin board? Suppose the length of the board is 7 ft. What equation would repre-sent the given situation?

How would you describe the equation formulated? Do you think you can use the equation formulated to find the length and the width of the bulletin board? Justify your answer. How did you find the preceding activities? Are you ready to learn about quadratic equations? 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. In the equation, ax2 is the quadratic term, bx is the linear term, and c is the constant term.

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. You may open the following links. Use the mathematical ideas and the examples presented in the preceding section to answer the activities provided. Identify which of the following equations are quadratic and which are not. If the equation is not quadratic, explain.

Some of the equations given are not quadratic equations. Were you able to explain why? In the next activ-ity, you will identify the situations that illustrate quadratic equations and represent these by mathematical statements. Tell whether or not each of the following situations illustrates quadratic equations. Justify your answer by representing each situation by a mathematical sentence. The length of a swimming pool is 8 m longer than its width and the area is m2.

Edna paid at least Php1, for a pair of pants and a blouse. The cost of the pair of pants is 15 Php more than the cost of the blouse. A motorcycle driver travels 15 kph faster than a bicycle rider. The motorcycle driver covers 60 km in two hours less than the time it takes the bicycle rider to travel the same distance. A realty developer sells residential lots for Php4, per square meter plus a processing fee of Php25, One of the lots the realty developer is selling costs Php, A garden 7 m by 12 m will be expanded by planting a border of flowers.

The border will be of the same width around the entire garden and has an area of 92 m2. Did you find the activity challenging? Were you able to represent each situation by a mathematical statement? For sure you were able to identify the situations that can be represented by quadratic equations. In the next activity, you will write quadratic equations in standard form. Answer the questions that follow. How did you write each quadratic equation in standard form?

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. Which quadratic equations did you find difficult to write in standard form? 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? What to Reflect and understand Your goal in this section is to take a closer look at some aspects of the topic.

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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? Answer the following questions. How are quadratic equations different from linear equations? How do you write quadratic equations in standard form? Give at least 3 examples. If there had been 25 members more in the club, each would have contributed Php50 less. How are you going to represent the number of Mathematics Club members?

What expression represents the amount each member will share? If there were 25 members more in the club, what expression would represent the amount each would share? What mathematical sentence would represent the given situation? Write this in standard form then describe. In this section, the discussion was about your understanding of quadratic equations and how they are illustrated in real life. What new realizations do you have about quadratic equations? How would you connect this to real life? How would you use this in making decisions? Now that you have a deeper understanding of the topic, you are ready to do the tasks in the next section.

You will be given a practical task which will demonstrate your understanding of quadratic equations. Give 5 examples of quadratic equations written in standard form. Identify the values of a, b, and c in each equation. Name some objects or cite situations in real life where quadratic equations are illustrated.

Formulate quadratic equations out of these objects or situations then describe each. In this section, your task was to give examples of quadratic equations written in standard form and name some objects or cite real-life situations where quadratic equations are illustrated. How did you find the performance task?

How did the task help you realize the importance of the topic in real life? The les-son provided you with opportunities to describe quadratic equations using practical situations and their mathematical representations. Moreover, you were given the chance to formulate qua-dratic equations as illustrated in some real-life situations. Your understanding of this lesson and other previously learned mathematics concepts and principles will facilitate your learning of the next lesson, Solving Quadratic Equations.

Solving Quadratic Equations by Extracting Square Roots 18 2A What to Know Start Lesson 2A 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 solving quadratic equations by extracting square roots. You may check your answers with your teacher. Find the following square roots. How did you find each square root?

How many square roots does a number have? Explain your answer. Does a negative number have a square root? Describe the following numbers: 8 , — 40 , 60 , and — Are the numbers rational or irrational? How do you describe rational numbers? How about numbers that are irrational? Were you able to find the square roots of some numbers? Did the activity provide you with an opportunity to strengthen your understanding of rational and irrational numbers?

In the next activity, you will be solving linear equations. Just like finding square roots of numbers, solving linear equations is also a skill which you need to develop further in order for you to understand the new lesson. Solve each of the following equations in as many ways as you can. How did you solve each equation? What mathematics concepts or principles did you apply to come up with the solution of each equation? Explain how you applied these. Compare the solutions you got with those of your classmates.

Did you arrive at the same 19 answers? If not, why? Which equations did you find difficult to solve? How did you find the activity? Were you able to recall and apply the different mathematics concepts or principles in solving linear equations? In the next activity, you will be representing a situation using a mathematical sentence.

Such mathematical sentence will be used to satisfy the conditions of the given situation. Use the situation below to answer the questions that follow. He asked a carpenter to make a square opening on the wall where the exhaust fan will be installed. The square opening must have an area of 0. Draw a diagram to illustrate the given situation. How are you going to represent the length of a side of the square-shaped wall? How about its area? Suppose the area of the remaining part of the wall after the carpenter has made the square opening is 6 m2. What equation would describe the area of the remaining part of the wall?

How will you find the length of a side of the wall? The activity you have just done shows how a real-life situation can be represented by a mathematical sentence. Were you able to represent the given situation by an equation? Do you now have an idea on how to use the equation in finding the length of a side of the wall?

To further give you ideas in solving the equation or other similar equations, perform the next activity. Use the quadratic equations below to answer the questions that follow. Describe and compare the given equations. What statements can you make? Solve each equation in as many ways as you can. Determine the values of each variable to 20 make each equation true.

How did you know that the values of the variable really satisfy the equation? Aside from the procedures that you followed in solving each equation, do you think there are other ways of solving it? Describe these ways if there are any. Were you able to determine the values of the variable that make each equation true? Were you able to find other ways of solving each equation? Let us extend your understanding of quadratic equations and learn more about their solutions by performing the next activity.

Find the solutions of each of the following quadratic equations, then answer the questions that follow. How did you determine the solutions of each equation? How many solutions does each equation have? What can you say about each quadratic equation based on the solutions obtained? Are you ready to learn about solving quadratic equations by extracting square roots? From the activities done, you were able to find the square roots of numbers, solve linear equations, represent a real-life situation by a mathematical sentence, and use different ways of solving a quadratic equation.

But how does finding solutions of quadratic equations facilitate solving real-life problems and in making decisions? Before doing these activities, read and understand first some important notes on solving quadratic equations by extracting square roots and the examples presented. Since t2 equals 0, then the equation has only one solution. There is no real number when squared gives —9.

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Solve the resulting equation. Check the obtained values of x against the original equation. Solve the following quadratic equations by extracting square roots. How did you find the solutions of each equation? What mathematics concepts or principles did you apply in finding the solutions? Explain 24 how you applied these. Compare your answers with those of your classmates.

Did you arrive at the same solutions? Was it easy for you to find the solutions of quadratic equations by extracting square roots? Did you apply the different mathematics concepts and principles in finding the solutions of each equation? I know you did! Write a quadratic equation that represents the area of each square. Then find the length of its side using the equation formulated. How did you come up with the equation that represents the area of each shaded region?

How did you find the length of side of each square? Do all solutions to each equation represent the length of side of the square? In this section, the discussion was about solving quadratic equations by extracting square roots. What to Reflect and Understand 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 solving quadratic equations by extracting square roots. After doing the following activities, you should be able to answer this important question: How does finding solutions of quadratic equations facilitate in solving real-life problems and in making decisions?

Solve each of the following quadratic equations by extracting square roots. Answer the ques-tions 25 that follow. How did you find the roots of each equation? Which equation did you find difficult to solve by extracting square roots? Which roots are rational? Which are not? How will you approximate those roots that are irrational? Were you able to find and describe the roots of each equation? Were you able to approximate the roots that are irrational? Deepen further your understanding of solving quadratic equations by extracting square roots by doing the next activity.

Answer the following. Do you agree that a quadratic equation has at most two solutions? Justify your answer and give examples. Give examples of quadratic equations with a two real solutions, b one real solution, and c no real solution. Do you agree with Sheryl? Cruz asked Emilio to construct a square table such that its area is 3 m2. Is it possible for Emilio to construct such table using an ordinary tape measure?

A 9 ft2 square painting is mounted with border on a square frame. If the total area of the border is 3. In this section, the discussion was about your understanding of solving quadratic equations by extracting square roots. What new realizations do you have about solving quadratic equations by extracting square roots?

You will be given a practical task in which you will demonstrate your understanding of solving quadratic equations by extracting square roots. Describe quadratic equations with 2 solutions, 1 solution, and no solution. Give at least two examples for each. Give at least five quadratic equations which can be solved by extracting square roots, then solve.

Collect square tiles of different sizes. Using these tiles, formulate quadratic equations that can be solved by extracting square roots. Find the solutions or roots of these equations. How did the task help you see the real-world use of the topic? The lesson provided you with opportunities to describe quadratic equations and solve these by extracting square roots. You were also able to find out how such equations are illustrated in real life.

Moreover, you were given the chance to demonstrate your understanding of the lesson by doing a practical task. Your understanding of this lesson and other previously learned mathematics concepts and principles will enable you to learn about the wide applications of quadratic equations in real life. Solving Quadratic Equations by Factoring 27 2B What to Know Start Lesson 2B 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 solving quadratic equations by factoring. Factor each of the following polynomials. How did you factor each polynomial? What factoring technique did you use to come up with the factors of each polynomial? Explain how you used this technique. How would you know if the factors you got are the correct ones? Which of the polynomials did you find difficult to factor? Were you able to recall and apply the different mathematics concepts or principles in factoring polynomials?

This mathematical sentence will be used to satisfy the conditions of the given situation. A rectangular metal manhole with an area of 0. The length of the pathway is 8 m longer than its width. How are you going to represent the length and the width of the 28 pathway? What expression would represent the area of the cemented portion of the pathway?

Suppose the area of the cemented portion of the pathway is What equation would describe its area? How will you find the length and the width of the pathway? Do you now have an idea on how to use the equation in finding the length and the width of the pathway? Use the equations below to answer the following questions.

How would you compare the three equations? What value s of x would make each equation true? How would you know if the value of x that you got satisfies each equation? Compare the solutions of the given equations. What statement can you make? Are you ready to learn about solving quadratic equations by factoring? From the activities done, you were able to find the factors of polynomials, represent a real-life situation by a mathematical statement, and interpret zero product.

But how does finding solutions of quadratic equations facilitate solving real-life problems and making decisions? Before doing these activities, read and understand first some important notes on solving quadratic equations by factoring and the examples presented. Some quadratic equations can be solved easily by factoring. To solve such quadratic equations, 29 the following procedure can be followed. Transform the quadratic equation into standard form if necessary. Factor the quadratic expression.

Apply the zero product property by setting each factor of the quadratic expression equal to 0. Zero Product 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. Solve each resulting equation. Check the values of the variable obtained by substituting each in the original equation. To solve the equation, factor the quadratic expression 9x2 — 4. Solve the following quadratic equations by factoring. Was it easy for you to find the solutions of quadratic equations by factoring?

The quadratic equation given describes the area of the shaded region of each figure. Use the equation to find the length and width of the figure. How did you find the length and width of each figure?

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Can all solutions to each equation be used to determine the length and width of each figure? In this section, the discussion was about solving quadratic equations by factoring.

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Find each indicated product then answer the questions that follow. If we completely factor a number into positive prime factors there will only be one way of doing it. However, this is usually easy enough to do in our heads and so from now on we will be doing this solving in our head. The way to factorise is to find two numbers that multiply together to make 18 but add to make I do, but you can check your work by expanding your factorised expressions which is probably better practice. To form the box, a square of side 5 cm will be removed from each corner of the cardboard.

You are going to think deeper and test further your understanding of solving quadratic equations by fac-toring. After doing the following activities, you should be able to answer this important question: How does finding solutions of quadratic equations facilitate solving real-life problems and making decisions? Answer each of the following. Which of the following quadratic equations may be solved more appropriately by factoring? Do you agree with Patricia? Do you agree that not all quadratic equations can be solved by factoring?

Justify your answer 33 by giving examples.

Find the solutions of each of the following quadratic equations by factoring. Explain how you arrived at your answer. A computer manufacturing company would like to come up with a new laptop computer such that its monitor is 80 square inches smaller than the present ones. Suppose the length of the monitor of the larger computer is 5 inches longer than its width and the area of the smaller computer is 70 square inches. What are the dimensions of the monitor of the larger computer?

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In this section, the discussion was about your understanding of solving quadratic equations by factoring. What new insights do you have about solving quadratic equations by factoring? What to transfer Your goal in this section is to apply your learning to real-life situations. You will be given a practical task which will demonstrate your understanding of solving quadratic equations by factoring.

Lakandula would like to increase his production of milkfish bangus due to its high demand in the market. He is thinking of making a larger fishpond in his sq m lot near a river. Help Mr. Lakandula by making a sketch plan of the fishpond to be made. Out of the given situation and the sketch plan made, formulate as many quadratic equations then solve by factoring. You may use the rubric below to rate your work. Rubric for the Sketch Plan and Equations Formulated and Solved 4 3 2 1 The sketch plan is accurately made, presentable, and appropriate.

The sketch plan is accurately made and appropriate. The sketch plan is not accurately made but appropriate. The sketch plan is made but not appropriate. Quadratic equations are accurately formulated and solved correctly. Quadratic equations are accurately formulated but not all are solved correctly. Quadratic equations are accurately formulated but are not solved correctly. Quadratic equations are accurately formulated but are not solved. The lesson provided you with opportunities to describe quadratic equations and solve these by factoring. You were able to find out also how such equations are illustrated in real life.

Your understanding of this lesson and other previously learned mathematics concepts and principles will facilitate your learning of the wide applications of quadratic equations in real life. Solving Quadratic Equations by Completing the Square 35 2C What to Know Start Lesson 2C 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 understand Solving Quadratic Equations by Complet-ing the Square.

How did you find the solution s of each equation? Which of the equations has only one solution? Which of the equations has two solutions? Which of the equations has solutions that are irrational? Were you able to simplify those solutions that are irrational? How did you write those irrational solutions? In the next activity, you will be expressing a perfect square trinomial as a square of a binomial. I know that you already have an idea on how to do this. This activity will help you solve qua-dratic equations by completing the square. How do you describe a perfect square trinomial? How did you express each perfect square trinomial as the square of a binomial?

What mathematics concepts or principles did you apply to come up with your answer? Compare your answer with those of your classmates. Did you get the same answer? Observe the terms of each trinomial. How is the third term related to the coefficient of the middle term? Is there an easy way of expressing a perfect square trinomial as a square of a binomial? If there is any, explain how. Were you able to express each perfect square trinomial as a square of a binomial? Let us further strengthen your knowledge and skills in mathematics particularly in writing perfect square trinomials by doing the next activity.

Determine a number that must be added to make each of the following a perfect square trinomial. Were you able to figure out how it can be easily done? Such a mathematical sentence will be used to satisfy the conditions of the given situation. The shaded region of the diagram at the right shows the portion of a square-shaped car park that is already cemented.

The area of the cemented part is m2. Use the diagram to answer the following questions. How would you represent the length of the side of the car park? How about the width of the cemented portion? What equation would represent the area of the cemented part of the car park? Using the equation formulated, how are you going to find the length of a side of the car park? Are you ready to learn about solving quadratic equations by completing the square? From the activities done, you were able to solve equations, express a perfect square trinomial as a square of a binomial, write perfect square trinomials, and represent a real-life situation by a mathematical sentence.

Before doing these activities, read and understand first some important notes on Solving Quadratic Equa-tions by Completing the Square and the examples presented. Another method of solving quadratic equations is by completing the square. Can you tell why the value of k should be positive? Divide both sides of the equation by a then simplify.

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. 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. Express the perfect square trinomial on the left side of the equation as a square of a binomial. Solve the resulting quadratic equation by extracting the square root. Solve the resulting linear equations.

Then to check,. Setting all terms equal to zero,. Setting each factor to 0,. A quadratic with a term missing is called an incomplete quadratic as long as the ax 2 term isn't missing. Many quadratic equations cannot be solved by factoring. This is generally true when the roots, or answers, are not rational numbers. A second method of solving quadratic equations involves the use of the following formula:. When using the quadratic formula, you should be aware of three possibilities.

These three possibilities are distinguished by a part of the formula called the discriminant. The discriminant is the value under the radical sign, b 2 — 4 ac. A quadratic equation with real numbers as coefficients can have the following:. Two different real roots if the discriminant b 2 — 4 ac is a positive number. Setting all terms equal to 0,. Then substitute 1 which is understood to be in front of the x 2 , —5, and 6 for a , b , and c, respectively, in the quadratic formula and simplify. Because the discriminant b 2 — 4 ac is positive, you get two different real roots. Example produces rational roots.