This is the Mathematics Grade 9 self-learning module for Quarter 1, Module 2, “Solving Quadratic Equations by Extracting Square Roots” (Week 1). It is an Alternative Delivery Mode module for Junior High School learners. It builds on Module 1, where learners learned to tell whether an equation is quadratic. After this module, learners can solve quadratic equations by taking the square root of both sides.
On this page 7 sections
What learners will learn
The module has one learning competency. Learners will be able to:
- solve quadratic equations by extracting square roots (M9AL-Ia-2.1)
Key ideas: square roots
Before solving equations, the module reviews square roots. A square root of a number is a value that gives that number when multiplied by itself. Both (4)(4) and (−4)(−4) equal 16, so every positive real number has two square roots.
- Principal (positive) square root: √x, for example √25 = 5
- Negative square root: −√x, for example −√64 = −8
- Both square roots: ±√x, for example ±√16 = ±4 (read as “positive and negative 4”) and ±√(4/9) = ±2/3
- Square root of a negative number: √−4 is not a real number
A short comic, “Mathew: The Math Person,” opens the lesson. A classmate knows her square lot has an area of 1,764 square meters and wants the length of its side. Since Area = s², the question becomes s² = 1,764, which is solved by working backward with square roots.
Key ideas: the Square-Root Property
The Square-Root Property says that if x² = c, then x = √c or x = −√c. In short, if c is a positive real number, then x = ±√c. For example, if x² = 17, then x = ±√17.
The module’s examples show the three possible outcomes:
| Equation | Steps | Result |
|---|---|---|
| x² = 64 | Take the square root of both sides: x = ±8 | Two solutions: 8 and −8 |
| x² + 6 = 6 | Subtract 6: x² = 0, so x = 0 | One solution: 0 |
| 9x² + 16 = 0 | Subtract 16, divide by 9: x² = −16/9 | No real solution, because the radicand is negative |
| 2(x + 4)² = 18 | Divide by 2: (x + 4)² = 9, so x + 4 = ±3 | Two solutions: −1 and −7 |
In the last example, the square root gives x + 4 = 3 or x + 4 = −3. Learners write two separate equations and solve each one.
Steps in solving by extracting square roots
The method works for equations of the form ax² + c = 0, where a and c are real numbers and a ≠ 0. The module summarizes it in three steps:
- Write the equation in the form x² = c/a.
- Take the square root of both sides. Put a ± sign before the square root, and use both roots to solve the resulting equation.
- Check each answer by substituting it into the original equation.
Parts and activities of the module
- What I Know: a 10-item multiple-choice pretest
- What’s In: a review of positive and negative square roots
- What’s New: the comic and a short “You Try!” set, such as −√49 and ±√144
- What is It: the Square-Root Property and worked examples
- What’s More: five equations to solve and check, such as 9x² − 25 = 0 and 5(x + 7)² = 1,125
- What I Have Learned: the three-step summary
- What I Can Do: ten equations, such as 3h² − 147 = 0 and (2s − 1)² − 225 = 0
- Assessment: a 10-item multiple-choice posttest
- Additional Activities: a graphic organizer where learners write and solve their own equations with no real solution, one solution, and two solutions, plus a reflection question
- PISA-Based Worksheet, “The Algebra Tiles”: learners model x² + 4x + 4 as a square, write it as (x + 2)², and find x when the area is 225 cm²
How to use this module
For learners: Start with the What I Know pretest and write your answers on a separate answer sheet. The module says that if you honestly get a perfect 10, you may skip it. Otherwise, go through the parts in order. When you solve, always remember the ± sign, and check your answers by substitution.
For parents: Help your child work through each part in order. Ask them to explain how they got both the positive and negative answers before they check the answer key.
For teachers: The answer key at the end covers the pretest, the practice activities, the assessment, and the PISA-based worksheet. The Additional Activities ask learners to create their own equations, so they need to be checked separately.
Frequently asked questions
What grade and quarter is this module for?
It is for Grade 9 Mathematics, Quarter 1, Module 2, and it is set for Week 1. Its learning code is M9AL-Ia-2.1.
What does extracting square roots mean?
It means solving an equation like x² = c by taking the square root of both sides, so x = ±√c. For example, x² = 64 gives x = 8 or x = −8.
Can a quadratic equation have no real solution?
Yes. If isolating x² gives a negative number, as in 9x² + 16 = 0 where x² = −16/9, there is no real solution because the square root of a negative number is not a real number.
Does the module have an answer key?
Yes. The answer key covers the pretest, the You Try items, What’s More, What I Can Do, the assessment, and the PISA-based worksheet.
Can learners skip this module?
The module says learners who honestly get a perfect 10 on the What I Know pretest may skip it.
Full text of the module
The text below is a transcription of the PDF above, as printed.
Mathematics Quarter 1-Module 2 Solving Quadratic Equations by Extracting Square Roots
Week 1 Learning Code – M9AL-Ia-2.1
[illustration: teachers and learners using digital devices and learning materials]
Mathematics – Grade 9 Alternative Delivery Mode
GRADE 9 Learning Module for Junior High School Mathematics
Development Team of the Module
Writers: Jhon Edward M. Valera – T1, Rowena F. Reyes – T1, Analyn M. Argel – MTII, Roderick F. Borja – TIII, Queenie Pearl E. Domasig – TII
Editor: Sally C. Caleja – Head Teacher VI, Bernard O. Tabang – Head Teacher VI, Maita G. Camilon – Head Teacher VI
Validators: Remylinda T. Soriano, EPS-Math; Angelita Z. Modesto, PSDS; George B. Borromeo, PSDS
Illustrator: Writers
Layout Artist: Writers
Management Team: Malcolm S. Garma, Regional Director; Genia V. Santos, CLMD Chief; Dennis M. Mendoza, Regional EPS in Charge of LRMS and Regional ADM Coordinator; Maria Magdalena M. Lim, CESO V, Schools Division Superintendent; Aida H. Rondilla, Chief-CID; Lucky S. Carpio, Division EPS in Charge of LRMS and Division ADM Coordinator
Quarter 1 – Module 2 – New Normal Math for G9 First Edition 2020 Copyright © 2020
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 module 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: Leonor Magtolis Briones Undersecretary: Diosdado M. San Antonio
GRADE 9 Learning Module for Junior High School Mathematics
Module 2: Solving Quadratic Equation by Extracting Square Roots
In the previous module, you have learned how to determine whether a given equation is quadratic or not. Every equation contains variables, the values of which need to be solved. A quadratic equation is a second-degree equation that has at most two solutions. In this module, you will learn how to solve quadratic equations by extracting square roots.
What I Need to Know
Learning Competency
The learners will be able to:
- solve quadratic equations by extracting square roots. (M9AL-Ia-2.1)
What I Know
Write the letter of the correct answer on your answer sheet.
A _____ produces a specific quantity when multiplied by itself.
- A. square root
- B. sum
- C. constant
- D. real number
What are the positive and negative square roots of 36?
- A. ±2
- B. ±4
- C. ±6
- D. ±1
Which of the following denotes a square root expression?
- A. ³√𝑏
- B. 𝑏³
- C. ⟨𝑏⟩
- D. √𝑏
How many real solutions does the equation 𝑥² = 𝑐 where 𝑐 < 0 have?
- A. no real solution
- B. two
- C. three
- D. one
How do you describe (the signs) of the square root of a positive real number?
- A. positive & positive
- B. negative & negative
- C. positive & negative
- D. none of them
What are the roots of the quadratic equation 𝑥² – 49 = 0?
- A. ± 49
- B. ±4
- C. ±1
- D. ±7
Applying the method of extracting the square roots, solve for b in 4𝑏² – 9 = 71.
- A. 𝑏 = ±2√5
- B. 𝑏 = ±3√5
- C. 𝑏 = ±5√2
- D. 𝑏 = ±5√3
GRADE 9 Learning Module for Junior High School Mathematics
Find the solution set of the equation 2𝑥² – 32 = 0.
- A. {2, 4}
- B. {−4, 4}
- C. {2, −4}
- D. {−1, 3}
Simplify ±√(9/16).
- A. ±3/4
- B. ±1/4
- C. ±4/3
- D. ±2/4
Simplify √50.
- A. 25 in
- B. 25√2 in.
- C. 5 in
- D. 5√2 in
*** If you got an honest 10 points (perfect score), you may skip this module
What’s In
Communication and collaboration
Knowing how to get the square root of real number is a prerequisite skill in order to understand the lesson in this module. Below is an illustrative example on how to get it.
We often see that:
(4)(4) = 4² = 16 → √16 = 4
and
(−4)(−4) = (−4)² = 16 → √16 = −4
[chart: annotated arrows; labels as printed]
- Positive square root of 16 (points to √16 = 4)
- Negative square root of 16 (points to √16 = −4)
The square root of a positive real number can be positive or negative. Thus, in order to avoid confusion on what square root is being asked, the positive square root or principal square root of a positive real number x is denoted √𝒙, while the negative square root of a positive real number x is denoted by −√𝒙. If both square roots are required, the notation becomes ±√𝒙.
For instance,
±√16 = ±4
[chart: annotated arrows; labels as printed]
- Positive and negative square root of 16 (points to ±√16 = ±4)
- The expression ±4 is read as ‘positive and negative 4’
GRADE 9 Learning Module for Junior High School Mathematics
What’s New
Let’s Investigate!
Communication, Collaboration and Critical Thinking
[illustration: four-row comic strip titled “Mathew: The Math Person” with speech bubbles; caption printed below: “1Comic generated using storyboard.com”]
Speech bubbles, in reading order:
- Panel 1: “Mathew: The Math Person” / “funny math comics”
- Panel 2: “Hello Mathew! I have a homework in Math, Do you know how to solve for the Area of our square house lot?” / “Hi Crush! Ofcourse I know that! You just need to identify the length of the side of your house then follow this formula Area = side x side or s²”
- Panel 3: “That’s good! I measured it already and it is 30 meters” / “Just square that 30 meters and the area of your house lot is 900 square meters” / “Thank you” / “Easy!”
- Panel 4: “Mathew! I have problem. My dad told me that the area of our square lot is 1,764 sq. meters. But my teacher ask for the measurement of the length of the side of our house…” / “Huh!?”
- Panel 5: “How can I get the measurement of the side when the area is given?” / “It is reversed???” / “Meh??”
- Panel 6: “I hope you have a meter stick at home …” / “Just measure the side, hehe” / “What?”
How can we solve for the value of the side? What is the value of the side?
If the floor area of a house is 1,764 sq. meters, and the formula for finding the area of a square is 𝐴𝑟𝑒𝑎 = 𝑠², where s is a side, then, 𝑠² = 1,764 sq. units.
Examples:
- √25 = 5
- −√64 = −8
- ±√(4/9) = ±2/3
- √−4 = not a real number
You Try!
- −√49 = _______
- √225 = _______
- ±√144 = _______
- −√(36/49) = _______
[transcriber’s note: the text layer of this page also contains the lines “1. √225 = ____” and “2. √400 = ____”, but they are not visible on the page image, so they are omitted here]
GRADE 9 Learning Module for Junior High School Mathematics
What is It
One way to help us answer the problem above is by extracting the square roots. How? Let us take a look at the properties and examples.
Square-Root Property
This is one method that can be used to solve quadratic equations. It states that if 𝑥² = 𝑐, then 𝑥 = √𝑐 or 𝑥 = −√𝑐, where c is a real number.
| WORDS | NUMBERS | ALGEBRA |
|---|---|---|
| To solve for 𝒙 in the quadratic equation of the form 𝒙² = 𝒄, take the square root of both sides of the equation. | 𝑥² = 17 √𝑥² = √17 𝑥 = ±√17 | If 𝒙² = 𝒄 and 𝒄 is a positive real number, then 𝒙 = ±√𝒄 |
Example 1: Using Square Roots to Solve 𝑥² = 𝑐
Solve by extracting the square roots.
a. 𝑥² = 64
Solution:
𝑥² = 64
√𝑥² = √64
𝑥 = ±8
The solutions are 8 and -8
Example 2: Using Square Roots to Solve Quadratic Equations
Solve by extracting the square roots.
a. 𝑥² + 6 = 6
Solution:
𝑥² + 6 = 6
𝑥² + 6 − 6 = 6 − 6 (Subtract 6 from both sides)
𝑥² = 0
√𝑥² = ±√0 (Take the square root of both sides)
𝑥 = 0
The solution is 0.
[transcriber’s note: the callout beside this part reads “Quadratic equation with only one solution.”]
GRADE 9 Learning Module for Junior High School Mathematics
b. 9𝑥² + 16 = 0
Solution:
9𝑥² + 16 = 0
9𝑥² + 16 − 16 = −16 (Subtract 16 from both sides)
9𝑥² = −16
9/9 (Divide both sides by 9.)
𝑥² = −16/9
√𝑥² = ±√(−16/9) (Take the square root of both sides.)
𝑥 = ±√(−16/9)
The answer will not be a real number
There is no real solution.
(Since the square root of a negative radicand is an imaginary number)
[transcriber’s note: the callout beside this part reads “Quadratic equation with no real solution.”]
c. 2(𝑥 + 4)² = 18
Solution:
2(𝑥 + 4)² = 18
2/2 (Divide both sides by 2.)
(𝑥 + 4)² = 9
√(𝑥 + 4)² = ±√9 (Take the square root of both sides.)
𝑥 + 4 = ±3
𝑥 + 4 = 3 or 𝑥 + 4 = −3
[transcriber’s note: the callout beside this part reads “Quadratic equation with two solutions.”]
Margin note: “Write two equations using both the positive and negative square roots and solve each equation.”
GRADE 9 Learning Module for Junior High School Mathematics
𝑥 + 4 − 4 = 3 − 4 (𝑥 + 4 − 4 = −3 − 4)
𝒙 = −𝟏 or 𝒙 = −𝟕
The solutions are −𝟏 and −𝟕
What’s More
Solve the following quadratic equations. Check the solutions.
- 𝑥² − 49 = 0
- 9𝑥² − 25 = 0
- 4𝑥² + 1 = 5
- 5(𝑥 + 7)² = 1 125
- 𝑥² = 43
What I Have Learned
Solving quadratic equations by extracting roots is applicable if the equation is in the form 𝑎𝑥² + 𝑐 = 0 where 𝑎 and 𝑐 are real numbers and 𝑎 ≠ 0. Below are the steps in solving this type of quadratic equation.
- Write the equation in the form: 𝑥² = 𝑐/𝑎
- Extract the square roots of both side of the equation. Put a ± sign before the square root of the number. Use the ± roots to solve for the resulting equation.
- Check your answer by substitution to see whether the equation is true.
What I Can Do
Solve the following quadratic equations by extracting square roots. ENCIRCLE your final answer.
| 1) 𝒙² = 16 | 6) 4𝒙² − 225 = 0 |
|---|---|
| 2) 𝒕² = 81 | 7) 3𝒉² − 147 = 0 |
| 3) r² − 100 = 0 | 8) (x − 4)² = 169 |
| 4) x² − 144 = 0 | 9) (2s − 1)² − 225 = 0 |
| 5) 2s² = 50 | 10) k² + 12 = 3 |
ASSESSMENT
Write the letter of the correct answer on your answer sheet.
The __________ states that if x and c are real number and if x² = c, then x = √c or x = −√c.
- A. Square Root Property
- B. Multiplication Property
- C. Addition Property
- D. Zero Product Property
What are the positive and negative square root of 81?
- A. ±8
- B. ±16
- C. ±9
- D. ±7
What is the practical way to solve x² – 25 = 0?
- A. factoring
- B. extracting the square root
- C. completing the square
- D. quadratic formula
How many real number solutions does the equation x² = c, where c > 0 have?
- A. no real solution
- B. two
- C. three
- D. one
What are the roots of x² – 144 = 0?
- A. ± 24
- B. ± 11
- C. ± 12
- D. ± 13
Solve: 4x² – 80 = 0 by extracting the square root.
- A. ± 5
- B. ±2
- C. ±5√2
- D. ±2√5
Simplify ±√(144/169).
- A. ±11/12
- B. ±12/13
- C. ±13/14
- D. ±14/15
What is the solution set of the equation x² + 16 = 0?
- A. {2, 4}
- B. {−4, 4}
- C. {2, −4}
- D. no real roots
Simplify −√(256/16).
- A. −16
- B. −4
- C. 4
- D. 18/4
Baby Brown has a piece of wood whose area is 25 square centimeters. What is the length of the side of the largest square that can be formed using the wood?
- A. 5 cm.
- B. 10 cm.
- C. 4 cm
- D. 5√2 cm
ADDITIONAL ACTIVITIES
Critical Thinking, Creativity and Collaboration
A.
Copy and complete the graphic organizer. In each box, write at least 3 quadratic equation, not on this module, with the given number of solutions. Solve each equation.
[chart: flowchart; untitled]
Solving Quadratic Equations by Using Square Roots When the Equation Has…
- No real solutions
- One solution
- Two solutions
B.
You already know that in solving quadratic equations of the form x² = c, it is possible to have a positive and a negative root of the variable x. Can you think of a problem in your life that after solving it resulted to a positive and a negative consequence?
How did the task help you see the real-world use of the topic?
E-Search
You may also check the following links for your reference and further learnings on solving quadratic equations by extracting roots:
Solving Quadratic Equations:
- https://www.youtube.com/watch?v=NnjVQRwAaMg&t=272s
- https://www.youtube.com/watch?v=ZFFDSHoZBVo
REFERENCES
Refer to the following links to further understand the lesson.
https://www.mathsisfun.com/square-root.html https://www.coursehero.com/file/37800346/math-9-lm-draft-3242014pdf/ https://www.stcs.org/view/11836.pdf https://www.storyboardthat.com/storyboard-creator https://charat.me/en/face/create/ https://www.google.com/search?q=handkerchiefs+collections&tbm=isch&ved=2ahUKEwi5ze-AlLDpAhXRA6YKHUG_BrgQ2-cCegQIABAAoq=handkerchiefs+collections&gs_lcp=CgNpbWcQAzoCCAA6BggAEAUQHjoGCAAQCBAeOgYIABAKEBhQgMcDWPnjA2D15gNoAHAAeACAAXuIAdQIkgEDNS42mAEAoAEBqgELZ3dzLXdpei1pbWc&sclient=img&ei=NIy7XrnjHdGHmAXB_prACw&bih=663&biw=1366#imgrc=HSkKooIAFxGDmM https://www.freepik.com/free-vector/woman-with-long-hair-teaching-online_7707557.htm https://www.freepik.com/free-vector/kids-having-online-lessons_7560046.htm
PISA – BASED WORKSHEET
The Algebra Tiles
Suppose you can build a rectangle with algebra tiles. The expression for the length and the width of the rectangle are the factors of the trinomial.
[illustration: algebra tiles of several sizes, arranged loosely; printed labels: x (on the large square and on the bars), 1 (on the small tiles)]
- Model x² + 4x + 4 using the algebra tiles and arrange the tile to form a rectangle. Make a sketch of your arrangement.
[transcriber’s note: blank rectangle for the student’s sketch]
What special kind of rectangle is it? __________________________________________
Use the dimension of your rectangle to complete the statement:
x² + 4x + 4 = ()() = (______)²
If the area of the figure formed is 225cm², what is the value of x? _______________
Answer Key
WHAT’S IN
- −7
- 15
- ±12
- −5/7
WHAT I CAN DO
- 4 and −4
- 9 and −9
- 10 and −10
- 12 and −12
- 5 and −5
- 15/2 and −15/2
- 7 and −7
- 17 and −9
- 8 and −7
- No real solution
WHAT’S MORE
- ±7
- ±5/3
- ±1
- 8, −22
- ±√43
WHAT I KNOW
- A
- C
- D
- A
- C
- D
- A
- B
- A
- D
ASSESSMENT
- A
- C
- B
- B
- C
- D
- B
- D
- B
- A
PISA-BASED WORKSHEET
- Arrange the tiles as shown:
[illustration: algebra tiles arranged into a square; labels x², x, 1]
- The figure formed is a square.
- x² + 4x + 4 = (x + 2)(x + 2) = (x + 2)²
- If the area of the square is 225cm², then:
(x + 2)² = 225
By extracting the square root, x + 2 = ±15
Thus, we have x + 2 = 15 and x + 2 = −15
x = 13 and x = −17
But a negative value of x does not fit the conditions of the problem, thus the value of x is 13.
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