Math9_Quarter1_Module2_FINAL-V3-1.pdf

Mathematics Quarter 1-Module 2 Solving Quadratic Equations by Extracting Square Roots Week 1 Learning Code - M9AL-Ia-2.1 Mathematics – Grade 9 Alternative Delivery Mode ...

Mathematics Quarter 1-Module 2 Solving Quadratic Equations by Extracting Square Roots Week 1 Learning Code - M9AL-Ia-2.1 Mathematics – Grade 9 Alternative Delivery Mode  GRADE 9 Learning Module for Junior High School Mathematics 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 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  GRADE 9 Learning Module for Junior High School Mathematics MODULE SOLVING QUADRATIC EQUATION BY EXTRACTING 2 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 PPREPREVIER! LEARNING COMPETENCY The learners will be able to: solve quadratic equations by extracting square roots. (M9AL-Ia-2.1) WHAT I KNOW PPREPREVIER ! Write the letter of the correct answer on your answer sheet. 1. A _____ produces a specific quantity when multiplied by itself. A. square root C. constant B. sum D. real number 2. What are the positive and negative square roots of 36? A. ±2 C. ±6 B. ±4 D. ±1 3. Which of the following denotes a square root expression? 3 A. √𝑏 C. 〈𝑏〉 B. 𝑏 3 D. √𝑏 4. How many real solutions does the equation 𝑥 2 = 𝑐 where 𝑐 < 0 have? A. no real solution C. three B. two D. one 5. How do you describe (the signs) of the square root of a positive real number? A. positive & positive C. positive & negative B. negative & negative D. none of them 6. What are the roots of the quadratic equation 𝑥 2 – 49 = 0? A. ± 49 C. ±1 B. ±4 D. ±7 7. Applying the method of extracting the square roots, solve for b in 4𝑏 2 – 9 = 71. A. 𝑏 = ±2√5 C. 𝑏 = ±5√2 B. 𝑏 = ±3√5 D. 𝑏 = ±5√3 1  GRADE 9 Learning Module for Junior High School Mathematics 8. Find the solution set of the equation 2𝑥 2 – 32 = 0. A. {2, 4} C. {2, −4} B. {−4, 4} D. {−1, 3} 9 9. Simplify ±√( ). 16 3 4 A. ± C. ± 4 3 1 2 B. ± D. ± 4 4 10. Simplify √50. A. 25 in C. 5 in B. 25√2 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 PPREPREV IER! 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: Positive square (𝟒)(𝟒) = 𝟒 = 𝟏𝟔→√𝟏𝟔 𝟐 =𝟒 root of 16 and (−𝟒)(−𝟒) = (−𝟒)𝟐 = 𝟏𝟔→√𝟏𝟔 = −𝟒 Negative square root of 16 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, Positive and negative ±√𝟏𝟔 = ±𝟒 square root of 16 The expression ±4 is read as ‘positive and negative 4’ 2  GRADE 9 Learning Module for Junior High School Mathematics Examples: 4 𝟐 1. √25 = 𝟓 3. ±√ = ± 9 𝟑 2. −√64 = −𝟖 4. √−4 = 𝒏𝒐𝒕 𝒂 𝒓𝒆𝒂𝒍 𝒏𝒖𝒎𝒃𝒆𝒓 You Try! 36 1. −√49 = _______ 2. √225 = ________ 3. ±√144 = _______ 4. −√49 = ________ WHAT’S NEW Communication, Collaboration and Critical Thinking Let’s Investigate! 1. √225 = __________ 2. √400 = __________ 1Comic generated using storyboard.com 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. 3  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.. This is one method that can be used to solve quadratic equations. It states that if 𝑥 2 = 𝑐, then 𝑥 = √𝑐 or 𝑥 = −√𝑐 , where c is a real number. WORDS NUMBERS ALGEBRA To solve for 𝒙 in the c𝑥 2 = 17 If 𝒙𝟐 = 𝒄 and c is a quadratic equation of c√𝑥 2 = √17 positive real number, the form 𝒙𝟐 = 𝒄, take c𝑥 = ±√17 then 𝒙 = ±√𝒄 the square root of both sides of the equation. Example 1: USING SQUARE ROOTS TO SOLVE 𝑥 2 = 𝑐 Solve by extracting the square roots. a. 𝑥 2 = 64 Solution: 𝑥 2 = 64 √𝑥 2 = √64 𝑥 = ±8 The solutions are 8 and -8 Example 2: USING SQUARE ROOTS TO SOLVE QUADRATIC EQUATIONS Solution: Solve by extracting the square roots. Quadratic a. 𝑥 2 + 6 = 6 equation with only one Solution: solution. 𝑥2 + 6 = 6 𝑥 2 + 6 − 6 = 6 − 6 Subtract 6 from both sides 𝑥2 = 0 4 √𝑥 2 = ±√0 Take the square root of both sides 𝑥 = 0 The solution is 0.  GRADE 9 Learning Module for Junior High School Mathematics b. 9𝑥2 + 16 = 0 Quadratic equation with no Solution: real solution. 9𝑥 2 + 16 = 0 9𝑥 2 + 16 − 16 = −16 Subtract 16 from both sides 9𝑥 2 = -16 9 9 Divide both sides by 9. 16 𝑥2 = − 9 16 √𝑥 2 = ±√− Take the square root of both sides. 9 16 𝑥 = ±√− The answer will not be a real number 9 There is no real solution. (Since the square root of a negative radicand is an imaginary number) c. 2(𝑥 + 4)2 = 18 Solution: 2(𝑥 + 4)2 = 18 2 2 Divide both sides by 2. Quadratic (𝑥 + 4)2 = 9 equation with two solutions. √(𝑥 + 4)2 = ±√9 Take the square root of both sides. 𝑥 + 4 = ±3 𝑥+4=3 or 𝑥 + 4 = −3 Write two equations using both the positive 5 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. 1. 𝑥 2 − 49 = 0 4. 5(𝑥 + 7)2 = 1 125 2. 9𝑥 2 − 25 = 0 5. 𝑥 2 = 43 3. 4𝑥 2 + 1 = 5 WHAT I HAVE LEARNED Solving quadratic equations by extracting roots is applicable if the equation is in the form 𝑎𝑥 2 + 𝑐 = 0 where 𝑎 and 𝑐 are real numbers and 𝑎 ≠ 0. Below are the steps in solving this type of quadratic equation. 𝑐 1. Write the equation in the form: 𝑥 2 = 𝑎 2. 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. 3. 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 6  GRADE 9 Learning Module for Junior High School Mathematics 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. 1. The __________ states that if x and c are real number and if 𝑥 2 = 𝑐, then 𝑥 = √𝑐 or 𝑥 = −√𝑐. A. Square Root Property C. Addition Property B. Multiplication Property D. Zero Product Property 2. What are the positive and negative square root of 81? A. ±8 C. ±9 B. ±16 D. ±7 3. What is the practical way to solve 𝑥 2 – 25 = 0? A. factoring C. completing the square B. extracting the square root D. quadratic formula 4. How many real number solutions does the equation 𝑥 2 = 𝑐, where 𝑐 > 0 have? A. no real solution C. three B. two D. one 5. What are the roots of 𝑥 2 – 144 = 0? A. ± 24 C. ± 12 B. ± 11 D. ± 13 6. Solve: 4𝑥 2 – 80 = 0 by extracting the square root. A. ± 5 C. ±5√2 B. ±2 D. ±2√5 144 7. Simplify ±√. 169 11 13 A. ± C. ± 12 14 12 14 B. ± D. ± 13 15 8. What is the solution set of the equation 𝑥 2 + 16 = 0 ? A. {2, 4} C. {2, −4} B. {−4, 4} D. 𝑛𝑜 𝑟𝑒𝑎𝑙 𝑟𝑜𝑜𝑡𝑠 7  GRADE 9 Learning Module for Junior High School Mathematics 256 9. Simplify −√. 16 A. −16 C. 4 B. −4 D. 18 4 10. 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. C. 4 cm B. 10 cm. D. 5√2 cm Critical Thinking, Creativity and Collaboration ADDITIONAL ACTIVITIES 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. 9  GRADE 9 Learning Module for Junior High School Mathematics 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=2ahUK Ewi5ze-AlLDpAhXRA6YKHUG_BrgQ2- cCegQIABAA&oq=handkerchiefs+collections&gs_lcp=CgNpbWcQAzoCCAA6BggA EAUQHjoGCAAQCBAeOgYIABAKEBhQgMcDWPnjA2D15gNoAHAAeACAAXuIAd QIkgEDNS42mAEAoAEBqgELZ3dzLXdpei1pbWc&sclient=img&ei=NIy7XrnjHdG HmAXB_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 8  GRADE 9 Learning Module for Junior High School Mathematics https://www.freepik.com/free-vector/illustration-with-kids-taking-lessons-online- design_7574030.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. x x 1 x 1 1 9  GRADE 9 Learning Module for Junior High School Mathematics 1. Model x2 + 4x + 4 using the algebra tiles and arrange the tile to form a rectangle. Make a sketch of your arrangement. 2. What special kind of rectangle is it? __________________________________________ 3. Use the dimension of your rectangle to complete the statement: x2 + 4x + 4 = (_______)(________) = (_________) 2 4. If the area of the figure formed is 225cm 2, what is the value of x? ________________ 10  GRADE 9 Learning Module for Junior High School Mathematics Answer Key 11