mathematics first quarter Module 1, Exercises of Mathematics

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Mathematics

First Quarter

Module 2A: Solving Quadratic

Equations by Extracting

Square Roots

Module 2 A

Find My X

What This Module Is All About

This learning material deals with solving quadratic equations by extracting

square roots. As you go through this lesson your skill in finding the solutions of a

quadratic equation by extracting square roots will be developed.

What You Are Expected To Learn

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

equations by: (a) extracting square roots (M9AL-Ia-b- 1 )

How Much Do You Know (Pre-test)

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

1. Which of the following equation can be solved by extracting square roots?

A. 2 4 1

2

x + r − =0 C. 2 7 3

2

x − x 

B. 3 t − 7 = 2 D. 9 𝑥

2

2. Which of the following is/are the solution/s of 3 𝑥

2

A. ±5 C. 5 only

B. 5,

5

3

D. – 5 only

3. Which of the following quadratic equations can be solved appropriately by extracting

square roots?

A. 3x

2

• 8x + 15 = 0 C. 5x

2

• 7x - 51 = 0

B. x

2

• 3x - 10 = 0 D. 4x

2

4. What are the solutions of the equation 3x

2

A. - 4 C. ±

B. ±2 D. 2

B. Determine the value/s of x of the following quadratic equations:

2

2

2

2

1

4

2

1

9

1

4

2

1

9

3

9

2

2

Example 1: Find the solutions of the quadratic equation 𝑥

2

− 16 = 0 by extracting

square roots_._

Solution:

Write the equation in the form 𝑥

2

2

2

2

Since 16 is greater than 0, then the first property can be applied to find the

values of x that will make the equation 𝑥

2

− 16 = 0 true.

2

2

16 Apply extracting square roots

To check, substitute these values in the original equations.

Both values of x satisfy the given equation. So the equations 𝑥

2

− 16 = 0 is true

when 𝑥 = 4 or 𝑥 = − 4.

Answer: The equation 𝑥

2

− 16 = 0 has two solutions: 𝑥 = 4 or 𝑥 = − 4.

Example 2: Solve the equation 𝑟

2

Since 𝑟

2

= 0 , then the equation has one real solution or root.

That is, 𝑟 = 0

To check: 𝑟

2

2

Answer: The equation 𝑟

2

= 0 has one real solution or root that is 𝑟 = 0

Example 3: Find the solution of the quadratic equation 𝒙

𝟐

Solution:

Write the equation in the form 𝑥

2

2

2

2

Since 𝒌 < 𝟎 , so the equation has no real solution or root.

Answer: Has no real solution.

For 𝑥 = 4 :

2

2

For 𝑥 = − 4 :

2

2

But, if you want to have the answer(Not indicated in our book), you need to

simplify the equation and get the root of this eqaution by simply applying the square

root method.

2

2

− 4 Apply extracting square roots

𝑥 = ±(√ 4 )(√− 1 ) Factor of √− 4 is (√ 4 )(√− 1 )

𝑥 = ± 2 𝑖 Note: 𝑖 = √− 1

Example 4: Find the solution of (𝒙 + 𝟓)

𝟐

To solve the equation simply add – 9 to both sides of the equation

𝟐

𝟐

𝟐

= ±√𝟏𝟔 Apply extracting square roots

Solve for x in the equation 𝒙 + 𝟓 = ±𝟒

The equation will result to two values of x.

and

To check the obtained solutions or values of x substitute the values of x in the

original equation.

The values of x satisfy the equation. So, the equation (𝒙 + 𝟓)

𝟐

• 𝟗 = 𝟐𝟓 is true

when 𝒙 = −𝟏 and 𝒙 = −𝟗.

Answer: The equation (𝒙 + 𝟓)

𝟐

• 𝟗 = 𝟐𝟓 has two real solutions or roots: 𝒙 = −𝟏 or

For 𝒙 = −𝟏

𝟐

𝟐

𝟐

For 𝒙 = −𝟗

𝟐

𝟐

𝟐

What Have You Learned (Post-test)

A. Directions: Choose the letter that corresponds to the correct answer. (Please show

your solution on a separate sheet of paper.)

1. Which of the following are the solutions of the quadratic equation

2

A. 3 , 4 C. ± 2

B. 6, 2 D. ± 4

2. How many solutions has 𝑥

2

A. 1 C. 0

B. 2 D. can’ t be determined

3. Which of the following equations CAN’T be solved by extracting square roots?

A. 11 − 11 𝑥 + 2 𝑥

2

= 0 C. 𝑡

2

B. 2 𝑥

2

= 50 D. 𝑥

2

1

4

4. What is the next step in solving 𝑡

2

− 16 = − 15 by extracting square roots?

A. combine like terms C. add +15 to both sides

B. square both sides D. add +16 to both sides

B. Find the solutions of the following quadratic equations:

2

2

2

2

2

2

2

1

9

2