Solving Quadratic Equations by Factoring: Module 2B, Exercises of Mathematics

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Mathematics

First Quarter

Module 2B: Solving Quadratic

Equations by Factoring

Module 2 B

Find My X

What This Module Is All About

This learning material deals with solving quadratic equations by factoring. As you

go through this lesson your skill in finding the solutions of a quadratic equation by

factoring will be developed.

What You Are Expected To Learn

After going through this module, the learners should be able to solve quadratic

equations by: (b) factoring (M9AL-Ia-b- 1 )

How Much Do You Know (Pre-test)

Directions: Choose the letter that corresponds to the correct answer.

A. 1. Which of the following equation can be solved by factoring?

A. 2 4 1

2

x + r − =0 C. 2 7 3

2

x − x 

B. 𝑥

2

− 11 𝑥 + 10 = 0 D. 3 𝑥

2

2. Which of the following are the solutions of 3 𝑥

2

A. 5, 1 C. 5 , - 1

B. - 5, 1 D. – 5 , - 1

3. Which of the following quadratic equations have 5 and - 2 as the solutions?

A. 3x

2

• 8x + 15 = 0 C. 5x

2

• 7x - 51 = 0

B. x

2

• 3x - 10 = 0 D. 4x

2

• 64 = 0

4. What are the solutions of the equation 𝑥

2

A. 7, 3 C. - 7, 3

B. 7, - 3 D. - 7, - 3

B. Factor the following quadratic equations:

2

2

2

2

Can you factor and find the solutions to these quadratic equations? How did

you factor and find the solutions of each equation? Well, let’s see if your answers are

correct.

Discussion :

What is factoring?

Factoring is the process of finding factors that when multiplied will give us the

quadratic equation as a product.

Some quadratic equations can be solved by factoring (only quadratic equations

that are factorable). To solve such quadratic equations, 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 fo the quadratic

expression equal to zero

Note:

Part 1: Where a = 1.

Example 1: Find the solutions of the quadratic equation 𝑥

2

− 7 𝑥 + 10 = 0 by factoring_._

Solution:

Our equation here is already in standard form and a = 1. Next is identify the

values of b and c. Our b = - 7 and c = 10.

2

Write the factors of 𝑥

2

(𝑥 ___)(𝑥 ___) = 0

List and write the factors of c which when added up will get the value of b. That will be

the conditions that we have to meet.

Our c is 10. Factors of 10: Our b is - 7. Sum of factors of 10:

Based on that conditions, the factors of c that must be used are (-5 ) and (-2)

since our b = - 7. Note that these factors can be writen interchangeably.

If the product of the two real numbers is zero, then either of the two is equal to zero or

both numbers are equal to zero.

Write the factors of c (𝑥 − 5 )(𝑥 − 2 ) = 0 or (𝑥 − 2 )(𝑥 − 5 ) = 0

Next step is apply the zero product property. If ab = 0 , then either b = 0 or a = 0

Check: Use the original equation and substitute the values obtained of x.

Both answers we had are correct and valid.

Example 2: Solve the equation 𝑟

2

− 3 𝑟 − 3 = 5 𝑟 + 6 by factoring.

Solution:

Transform the equation in standard form by adding - 5r and – 6 to both sides of

the equation. Then combine like terms.

2

2

Since a = 1 , the next step is to identify the values of b and c. Our b = - 8 and c = - 9.

2

Write the factors of 𝑟

2

(𝑟 ___)(𝑟 ___) = 0

List and write the factors of c which when added up will get the value of b. That will be

the conditions that we have to meet.

Our c is - 9. Factors of - 9 : Our b is - 8. Sum of factors:

Based on the conditions, the factors of c that must be used are (- 9 ) and ( 1 )

since our b = - 8. Note that these factors can be writen interchangeably

Write the factors of c

= 0 or

For x = 5

2

2

For x = 2

2

2

Next step is to apply the zero product property. If ab = 0 , then either b = 0 or a = 0

To check use the original equation and substitute the values of x we obtained.

Both our answers are correct and valid.

Example 4: Solve 9 𝑥

2

− 9 = 0 by factoring.

To solve the equation, simply factor the equation

2

Set each factor to 0.

Check the values of the variable obtained by substituting each value in the

original equation.

1

2

For 𝑥 = − 1

2

2

For 𝑥 = 2

2

2

1

3

1

3

Both our answers are correct and valid.

Activity 1

Factor the following:

2

2

2

2

2

1

4

2

2

2

Activity 2

Solve each quadratic equations by factoring:

2

2

2

2

2

2

2

2

For 𝑥 = 1

2

2

For 𝑥 = − 1

2

2