Mathematics

Quarter 1-Module 6
Nature of Roots of a Quadratic Equation
Week 2
Learning Code - M9AL-Ib-3
 GRADE 9
Learning Module for Junior High School Mathematics

Quarter 1 – Module 6 – New Normal Math for G9

First Edition 2020
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Published by the Department of Education

Secretary: Leonor Magtolis Briones
Undersecretary: Diosdado M. San Antonio

Development Team of the Module

Writers: Rowena F. Reyes – T1 Analynn M. Argel - MTII

Editors: Sally C. Caleja – Head Teacher VI

Maita G. Camilon – Haed Teacher VI
Melody P. Rosales – 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 NATURE OF ROOTS OF A QUADRATIC EQUATION

6 SQUARE ROOTS

From your previous modules, you learned how to get the roots of a quadratic
equation. At this point, you will explore on describing the characteristics of the roots of
a quadratic equation without solving for the roots. This knowledge would come in handy
on some real-life scenarios especially in decision making.

WHAT I NEED TO KNOW

PPREPREVIER!
LEARNING COMPETENCY

The learners will be able to:

• characterize the roots of a quadratic equation using the discriminant. M9AL-Ib-3

WHAT I KNOW
PPREPREVIER
! Find out how much you already know about the nature of roots of quadratic
equation. Write the letter that you think is the best answer to each question on your
answer sheet. Answer all items. After taking and checking this short test, take note of
the items that you were not able to answer correctly and look for the right answer as
you go through this module.

1. Which of the following is the discriminant of the quadratic equation: ax 2 − bx +

c = 0, where a, b, c are real number and a ≠ 0?
A. b2 C. √b 2 − 4ac
B. b − 4ac
2
D. −b ± √b 2 − 4ac

2. How many real roots does the quadratic equation 𝑥 2 + 8𝑥 + 12 = 0 have?

A. 0 B. 1 C. 2 D. 3

3. Which of the following quadratic equations has no real roots?

A. 2x 2 + 4x = 3 C. 3m2 − 2m + 5 = 0
B. z − 8z − 4 = 0
2 D. −2f 2 + f = −4

4. Which of the following is the nature of the roots of the quadratic equation if the
value of its discriminant is zero?
A. The roots are not real.
B. The roots are irrational and not equal.
C. The roots are rational and not equal.
D. The roots are rational and equal.

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Learning Module for Junior High School Mathematics

5. Which of the following is the nature of the roots of the quadratic equation if the
value of its discriminant is negative?
A. The roots are not real.
B. The roots are rational and equal.
C. The roots are rational and not equal.
D. The roots are irrational and not equal.

6. Which of the following is the nature of the roots of the quadratic equation if the
value of its discriminant is positive and a perfect square?
A. The roots are not real.
B. The roots are rational and equal.
C. The roots are irrational and not equal.
D. The roots are rational and not equal.

7. Which of the following is the nature of the roots of the quadratic equation if the
value of its discriminant is positive and not a perfect square?
A. The roots are not real.
B. The roots are rational and equal.
C. The roots are rational and not equal.
D. The roots are irrational and not equal.

8. What is the value of 𝑟 in the quadratic equation 𝑥 2 − 10𝑥 − 𝑟 = 0 whose roots are
equal rational and real?
A. −25
B. −10
C. 10
D. 25
9. Which of the following quadratic equations has equal roots?
A. 𝑥 2 + 6𝑥 + 9 = 0
B. 𝑥 2 + 5𝑥 + 10 = 0
C. 2𝑥 2 − 10𝑥 + 8 = 0
D. 3𝑥 2 − 2𝑥 − 5 = 0

10. What are the values of 𝑘 in the quadratic equation 6𝑥 2 − 5𝑥 − 𝑘 = 0 whose roots
are imaginary?
−25 −25
A. 𝑘 < C. 𝑘 >
24 24
25 25
B. 𝑘 < . D. 𝑘 > .
24 24

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Learning Module for Junior High School Mathematics

WHAT’S IN
PPREPREV Communication, Collaboration
IER!
Before we begin, let us unlock the following vocabularies. Get your dictionary
or any math book and find the meaning of the following words. Give at least two
examples of each.
1. Real numbers 3. Rational Numbers
2. Imaginary numbers 4. Irrational Numbers
Did you find the meaning of the words above? Now apply these words in the next
activity.
Let us FIND and INVESTIGATE!

Real or Imaginary? Rational or Irrational?

This is a classroom hidden number game. In this game you must find out
the hidden numbers and place it in the appropriate trash bin.
Reminder: a number can be written in two trash bins.

2
− 7
3 2 + √6
√7
−
√16
√−2
3
4 ± √−12
2

1 ± √−4

5 5
2
−√2

1Source: http://clipart-library.com/clean-classroom-cliparts.html
Added

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Learning Module for Junior High School Mathematics
WHAT’S NEW

The quadratic equations listed can be solved by using quadratic formula as

discussed from the previous topic. Using this method, find and examine the roots
obtained to complete the table below. In this case, let 𝑟1 and 𝑟2 be the roots.
Put a check in the cell that corresponds to the characteristics of the roots attained in
each equation.
ROOTS Real or Rational Equal or The value
Imaginary? or unequal? of b2-4ac
irrational?
EQUATIONS

Imaginary

Irrational
Rational
𝒓𝟏 𝒓𝟐
Real

Unequal
Equal
(a) 𝑥 2 − 7𝑥 + 10 = 0
9
(b) 𝑥 2 − 4𝑥 + 7 = 0
-12
(c) 𝑥 2 − 10𝑥 + 25 = 0
0
(d) 𝑥 2 − 4𝑥 − 2 = 0
24

If the quadratic formula is used to get the roots, the values of b − 4ac (quantity
2

under the radical sign) for those equations are 9,−12,0 and 24 respectively.

Communication, Critical
WHAT IS IT Thinking, and Collaboration

When describing the natures or characters of the roots of a quadratic equation,

it can be one of each of the following:
(a) Real or Imaginary
(b) Rational or irrational
(c) Equal or unequal
Given the form of the equation, 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0, where 𝑎, 𝑏, 𝑐 are real numbers and
𝑎 ≠ 0, then using the quadratic formula, the roots are
−𝑏+√𝑏2 −4𝑎𝑐 −𝑏−√𝑏2 −4𝑎𝑐
and
2𝑎 2𝑎

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Looking at the two roots, we can say that they are real or imaginary by using the
quantity under the radical sign. It depends whether it is positive, zero or negative.

Similarly, if the value inside the radical sign is a perfect square, then the roots
are rational; if not, they are irrational.

In order to have equal roots, the quantity under the radical sign must be zero.

Therefore, the nature of the roots can be decided by using the quantity
𝑏 2 − 4𝑎𝑐, which is called the discriminant of the equation 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0.
Thus:
(a) If 𝑏 2 − 4𝑎𝑐 = 0; the roots are real, equal and rational.
(b) If 𝑏 2 − 4𝑎𝑐 < 0; the roots are unequal and imaginary.
(c) If 𝑏 2 − 4𝑎𝑐 >0 and a perfect square; the roots are real, unequal and rational.
(d) If 𝑏 2 − 4𝑎𝑐 > 0 but not a perfect square; the roots are real, unequal, and irrational.
Applying this conclusion, you may check your answer on the last activity in the “WHAT’S
NEW” part.

Example 1: By inspection, determine the nature of the roots of the following equations:
(a) 𝑥 2 − 7𝑥 − 8 = 0
(b) 2𝑥 2 − 4𝑥 + 1 = 0
(c) 3𝑥 2 + 9𝑥 + 11 = 0
(d) 𝑥 2 − 12𝑥 + 36 = 0
Solutions:
(a) The equation 𝑥 2 − 7𝑥 − 8 = 0 gives 𝑎 = 1, 𝑏 = −7, and 𝑐 = −8
Substituting the given values of 𝑎,𝑏, and 𝑐 in the expression we have

𝑏 2 − 4𝑎𝑐 = (−7)2 − 4(1)(−8)

= 49 + 32
= 81
Since 81 is greater than 0 and a perfect square, then the equation 𝑥 2 − 7𝑥 − 8 =
0 has 2 real, rational, and unequal roots.

(b) The equation 2𝑥 2 − 4𝑥 + 1 = 0 gives 𝑎 = 2, 𝑏 = −4, 𝑎𝑛𝑑 𝑐 = 1

Substituting the given values of 𝑎,𝑏, and 𝑐 in the expression we have

𝑏 2 − 4𝑎𝑐 = (−4)2 − 4(2)(1)

= 16 − 8
=8

Since 8 is greater than 0 and not a perfect square, then the equation
2𝑥 2 − 4𝑥 + 1 = 0 has real, irrational, and unequal roots.

(c) The equation 3𝑥 2 + 9𝑥 + 11 = 0 gives 𝑎 = 3, 𝑏 = 9, and 𝑐 = 11

Substituting the given values of 𝑎,𝑏, and 𝑐 in the expression we have

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Learning Module for Junior High School Mathematics

𝑏 2 − 4𝑎𝑐 = (9)2 − 4(3)(11)

= 81 − 132
= −44

Since -44 is less than 0, then the equation 3𝑥 2 + 9𝑥 + 11 = 0 has imaginary and
unequal roots.

(d) The equation 𝑥 2 − 12𝑥 + 36 = 0 gives 𝑎 = 1, 𝑏 = −12, and 𝑐 = 36

Substituting the given values of 𝑎,𝑏, and 𝑐 in the expression we have

𝑏 2 − 4𝑎𝑐 = (−12)2 − 4(1)(36)

= 144 − 144
=0

Since the discriminant is equal to zero, then the equation 𝑥 2 − 12𝑥 + 36 = 0

has real, rational and equal roots.

Example 2: Answer the following

(a) Given the equation (3𝑘 + 1)𝑥 2 + 2(𝑘 + 1)𝑥 + 𝑘 = 0, find the values of 𝑘 for which
the roots are equal.
(b) For what values of 𝑘 are the roots of 6𝑥 2 − 5𝑥 − 𝑘 = 0 is: i.) imaginary? ii.) real and
unequal?
(c) Find 𝑟 so that the roots of the equation (𝑟 − 3)𝑥 2 − (𝑟 − 6)𝑥 − 4 = 0 are equal?
Solutions:
(a) Based on the given equation, (3𝑘 + 1)𝑥 2 + 2(𝑘 + 1)𝑥 + 𝑘 = 0, the expressions for
𝑎, 𝑏, 𝑐 are 𝑎 = 3𝑘 + 1, 𝑏 = 2(𝑘 + 1), and 𝑐 = 𝑘.
For equal roots, the discriminant must be equal to zero. Therefore,
𝑏 2 − 4𝑎𝑐 = 0
2
[2(𝑘 + 1)] − 4(3𝑘 + 1)(𝑘) = 0
(2𝑘 + 2)2 − (12𝑘 + 4)(𝑘) = 0
4𝑘 2 + 8𝑘 + 4 − 12𝑘 2 − 4𝑘 = 0
−8𝑘 2 + 4𝑘 + 4 = 0
8𝑘 2 − 4𝑘 − 4 = 0
Dividing both sides by 4, we have:
2𝑘 2 − 𝑘 − 1 = 0
To solve for k, we have:
(𝑘 − 1)(2𝑘 + 1) = 0
Equating both factors to zero and solving for k,
k–1=0 or 2k + 1 = 0
k=1 2k = -1
𝟏
k=−
𝟐
𝟏
Thus, the values of k for which the roots are equal are 1 and − .
𝟐

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Learning Module for Junior High School Mathematics
(b) Given the equation of 6𝑥 2 − 5𝑥 − 𝑘 = 0, we have 𝑎 = 6, 𝑏 = −5, and 𝑐 = −𝑘
(i) In order to have imaginary roots, 𝑏 2 − 4𝑎𝑐 must be less than zero which is
denoted by:
𝑏 2 − 4𝑎𝑐 < 0
Hence,
(−5)2 − 4(6)(−𝑘) < 0
25 + 24𝑘 < 0
24𝑘 < −25
−𝟐𝟓
Therefore, 𝒌 < .
𝟐𝟒

(ii) In order to have real and unequal roots, 𝑏 2 − 4𝑎𝑐 must greater than 0 which
is denoted by:
𝑏 2 − 4𝑎𝑐 > 0

−𝟐𝟓
Therefore, based on (i), 𝒌 > . [complement of the answer in (i)]
𝟐𝟒

(c) Given the equation (𝑟 − 3)𝑥 2 − (𝑟 − 6)𝑥 − 4 = 0, we have a = r – 3, b = -(r – 6), and
c = -4.
For equal roots, the discriminant must be equal to zero. Therefore,

𝑏 2 − 4𝑎𝑐 = 0
[-(r – 6)]2 – 4(r – 3)(-4) = 0
r – 12r + 36 + 16r -48 = 0
2

r2 + 4r -12 = 0
Solving for r, we have
(r – 2) (r + 6) = 0
r–2=0 or r+6=0
r=2 r = -6
Therefore, the values of r for which the roots are equal are 2 and -6.

Critical Thinking,
Collaboration
WHAT’S MORE

I. Using the discriminant, characterize the roots of the following quadratic

equations:
1. 𝑥 2 − 3𝑥 − 5 = 0
2. 𝑥 2 = 5𝑥 − 11
3. 5𝑥 2 − 6𝑥 + 13 = 0

II. Given (3𝑘 + 1)𝑥 2 + (11 + 𝑘)𝑥 + 9 = 0. Find the values of 𝑘 for which the roots are:
(a) equal;
(b) imaginary;
(c) real and unequal

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 GRADE 9
Learning Module for Junior High School Mathematics
III. Write a quadratic equation of each of the following types:
(a) two rational roots
(b) two imaginary roots
(c) two irrational roots
(d) one rational root

WHAT I HAVE LEARNED

The nature of the roots of a quadratic equation using the discriminant without
solving the equation. To characterize the roots of the quadratic equation 𝑎𝑥 2 + 𝑏𝑥 +
𝑐 = 0, where 𝑎, 𝑏, 𝑐 are real numbers and 𝑎 ≠ 0, we have the following:
1. If 𝑏 2 − 4𝑎𝑐 = 0, the roots are real, rational and equal.
2. If 𝑏 2 − 4𝑎𝑐 > 0 and a perfect square, the roots are real, rational, and unequal.
3. If 𝑏 2 − 4𝑎𝑐 > 0 and not a perfect square, the roots are real, irrational and
unequal.
4. If 𝑏 2 − 4𝑎𝑐 < 0, the roots are imaginary and unequal.

WHAT I CAN DO Critical Thinking

I. Characterize the roots of the following quadratic equations using the

discriminant.
1. 6𝑥 2 − 𝑥 + 2 = 0 6. 7𝑥 2 − 10𝑥 + 1 = 0
2. 𝑥 − 3𝑥 + 7 = 0
2
7. 𝑡 2 − 3𝑡 + 2 = 0
3. 𝑚2 + 8𝑚 + 16 = 0 8. 8𝑥 2 − 16𝑥 + 9 = 0
4. 3𝑥 + 8𝑥 − 8 = 0
2 9. 𝑟 2 − 𝑟 = 0
5. 3𝑥 − 4𝑥 + 2 = 0
2 10. 3𝑡 2 + 4𝑡 = 8

II. Solve the following:

1. Find the values of 𝑘 that will make the roots of 4𝑥 2 − 𝑘𝑥 + 4 = 0 equal.
2. For what values of 𝑘 are the roots of 𝑘𝑥 2 − 4𝑥 + 5 = 0 are imaginary?
3. Determine the value of 𝑐 if 4𝑥 2 − 5𝑥 + 𝑐 = 0 has real and unequal roots.
4. The equation 7𝑥 2 − 3𝑥 + 𝑚 = 0 has imaginary roots. Determine the value of
𝑚.
5. Find 𝑘 so that the roots of the equation 𝑤 2 + (𝑘 + 3)𝑤 + 3𝑘 = 0 are equal.

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 GRADE 9
Learning Module for Junior High School Mathematics

ASSESSMENT
Write the letter of the correct answer on your answer sheet. If your answer is not
among the choices, write E together with your final answer.

1. Which of the following is the discriminant of the quadratic equation: 𝑥 2 + 2𝑥 + 5 =

0?
A. -16 C. 4
B. -4 D. 16
2. How many real roots does the quadratic equation 5𝑥 2 − 8𝑥 + 6 = 0 have?
A. 0 C. 2
B. 1 D. 3

3. Which of the following quadratic equations has real roots?

A. 𝑥 2 + 𝑥 + 1 = 0 C. 2𝑥 2 − 6𝑥 + 6 = 0
2
B. 2𝑥 + 5𝑥 + 3 = 0 D. 4𝑥 2 + 3𝑥 + 3 = 0

4. The roots of a quadratic equation are imaginary. Which of the following statement
is true about the discriminant of equation?
A. The discriminant is negative.
B. The discriminant is equal to zero.
C. The discriminant is positive and a perfect square.
D. The discriminant is positive and not perfect square.

5. The roots of a quadratic equation are 2 and 5.Which of the following statement is
true about the discriminant of equation?
A. The discriminant is negative.
B. The discriminant is equal to zero.
C. The discriminant is positive and not perfect square.
D. The discriminant is positive and a perfect square.

6. Given the equation, 4𝑥 2 + 9𝑥 − 13 = 0, which of the following is the nature of its

roots?
A. The roots are real, rational and equal.
B. The roots are imaginary and unequal.
C. The roots are real, irrational and unequal.
D. The roots are real, rational, and unequal.

7. The discriminant of a quadratic equation is 34. Which of the following is true

about its roots?
A. The roots are real, rational and equal.
B. The roots are imaginary and unequal.
C. The roots are real, irrational and unequal.
D. The roots are real, rational, and unequal.

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Learning Module for Junior High School Mathematics
8. What is the value of 𝑡 that will make the roots of the equation 8𝑥 2 − 4𝑡𝑥 + 3 = 0
equal?
A. 𝑡 = ±6 C. 𝑡 = ±√6
B. 𝑡 = ±96 D. 𝑡 = ±4√6
9. Which of the following quadratic equations has rational, equal and real roots?
A. 9 − 6𝑥 + 𝑥 2 = 0 C. 𝑥 2 + 4𝑥 − 21=0
B. 𝑥 + 2𝑥 + 2 = 0
2 D. 4𝑥 2 + 12𝑥 + 10

10. What is/are the value/s of 𝑘 in the quadratic equation 2𝑥 2 − 5𝑥 − 𝑘 = 0 whose

roots are imaginary?
25 25
A. 𝑘 < C. 𝑘 > −
8 8
25 25
B. 𝑘 < − D. 𝑘 >
8 8

Critical Thinking,
ADDITIONAL ACTIVITIES Collaboration, Creativity

The shortest way of describing the nature of the roots of a quadratic equation is
not by solving for the roots but by just using the discriminant of the given quadratic
equation.

Below is a specific scenario on how this lesson may help you in some decision
making.

A. Apply the idea of the nature of roots of a quadratic equation to visualize, answer if it
is possible or not, and then justify your answer.

SCENARIO 1
Is it possible to make a rectangular park of perimeter 80 m and area 400 m2 ?

SCENARIO 2
The sum of the ages of two girls is 20 years. The product of their ages four
years ago was 48.Is this possible?

SCENARIO 3
Is it possible to design a rectangular rice field whose length is twice its width
10
and the area is 800 m²?

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Learning Module for Junior High School Mathematics

PROBLEM – BASED WORKSHEET

SPOT THE ERROR

Maria and Jose wanted to determine the number of real solutions of the
quadratic equation 5x2 – 3x = 2.

Maria Jose
5x2 - 3x = 2 5x2 - 3x = 2
b2- 4ac = (-3)2 - 4(5)(2) 5x2 - 3x -2 = 0
= -31 b2- 4ac = (-3)2 - 4(5)(-2)
Since the discriminant is -31, = 49
there are no real roots. Since the discriminant is 49,
positive and perfect square,
there are two real roots.

Who is correct? Justify your answer.

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Learning Module for Junior High School Mathematics

E-Search
You may also check the following link for your reference and further learnings on
nature of the roots of quadratic equations:

• https://www.khanacademy.org/math/in-in-grade-10-
ncert/x573d8ce20721c073:in-in-chapter-4-quadratic-
equation/x573d8ce20721c073:in-in-10-quadratic-discriminant-and-number-of-
solutions/v/discriminant-for-types-of-solutions-for-a-quadratic
• https://www.toppr.com/guides/maths/quadratic-equations/nature-of-roots/

REFERENCES

Bautista, E. P., & O'dell, I. C. (n.d.).Numlock III. pp 368- 372 Quezon City:
Trinitas Publishing, INC.
Dilao, S. J., & Bernabe, J. G. (2009). Intermediate Algebra.pp 52-54. Quezon CIty: SD
Publicaton.
Morgan, F. M., & Paige, B. L. (n.d.). Algebra 2. pp 98-102 .America: Henry Holt and
Company.
Oronce, O. A., Santos, G. C., & Ona, M. I. (n.d.). Interactive Mathematics III
(Concepts,Structures, adn Metods for High School. pp 250-256Manila: Rex Book
Store.
https://www.onlinemath4all.com/solving-word-problems-with-nature-of-roots-of-
quadratic-equation.html
https://ya-webdesign.com/explore/trashcan-drawing-basura/
http://clipart-library.com/clean-classroom-cliparts.html
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

https://www.freepik.com/free-vector/illustration-with-kids-taking-lessons-online-
design_7574030.htm

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WHAT I KNOW
1. B
2. C
3. C
4. D
5. A
6. D
7. D
8. A
9. A
10. A
ASSESSMENT
1. A
2. A
3. B
4. A
5. D
6. D
7. C
8. C
9. A
10. B
PISA-BASED WORKSHEET
Solutions:
Maria made a mistake by getting the values of a, b and c without re-writing the
equation in the form ax2 + bx + c = 0.
Jose did the correct solution. He first re-write the equation in the form ax2 + bx + c = 0
and came up with a = 5, b = -3 and c = 2.
Thus, the discriminant is 49. It is a positive number and a perfect square, therefore
there are two real roots.
Learning Module for Junior High School Mathematics
GRADE 9
 17
WHAT’S MORE
I.
1. The roots are real, irrational and unequal.
2. The roots are imaginary and unequal.
3. The roots are imaginary and unequal.
II.
a. 𝑘 = 1 𝑜𝑟 𝑘 = 85
b. 1 < 𝑘 < 85
c. 𝑘 > 85 𝑜𝑟 𝑘 < 1
ADDITIONAL ACTIVITIES
1. Yes, the dimension that will lead to an area of 400m 2 from a perimeter of 80m
is a 20m by 20m square. Since a square is a rectangle, thus the scenario is
possible.
2. No, if the product of the girl’s ages 4 years ago is 48, then the sum of their ages
is 20 less 8.
Let x and y be the girls’ ages
The equation will be,
x + y = 12
xy = 48
Solving for x and y,
If x + y = 12 , then y = 12 – x.
Substitute the value to the second equation, we have
x (12 – x) = 48
-x2 + 12x – 48 = 0
Solving for x using quadratic formula,
−𝑏± 𝑏2 −4𝑎𝑐
x=
2𝑎
−12± 12 2 −4(−1)(−48)
x=
2(−1)
−12± 12 2 −4(−1)(−48)
x=
2(−1)
−12±√−48
x=
−2
Since the radicand is negative, the roots are imaginary. Thus, the
product of the ages four years ago is impossible.
3. Yes, if we let x = width, then 2x = length. The area is:
x (2x) = 800
2x2 = 800
x2 = 400
Hence x = 20. This gives a width of 20 m, and a length of 40 m.
Learning Module for Junior High School Mathematics
GRADE 9
 18
MODULE
ANSWER KEY6
WHAT’S IN
Real Numbers Imaginary Numbers Rational Numbers Irrational Numbers
2 −√7 4±√−12 √−2 2 9 5 −√7
0, − , 7, , , , √−3, 0, − , 7, , , √49 2+√6, , √5,
3 √16 2 3 3 49 2 √16
9 1 ± √−4 √35, √2
2+√6, √5, √35, ,
49
5
, √49, -√2
2
WHAT’S NEW
ROOTS Real or Rational or Equal or
Imaginary? irrational? unequal?
EQUATIONS
l
al

𝒓𝟏 𝒓𝟐
ary

Real
Equal

Imagin
Rationa
Irration
Unequal

(a) 𝑥 2 − 7𝑥 + 10 = 0 5 2 √ √ √
(b) 𝑥 2 − 4𝑥 + 7 = 0 4 + √−12 4 − √−12 √ √
2 2
(c) 𝑥 2 − 10𝑥 + 25 = 0 5 5 √ √ √
(d) 𝑥 2 − 4𝑥 − 2 = 0 2 + √6 2 − √6 √ √ √
WHAT I CAN DO
I.
1. imaginary and unequal 6. real, irrational and unequal
2. imaginary and unequal 7. real, rational and unequal
3. real, rational and equal 8. imaginary and unequal
4. real, irrational and unequal 9. real, rational and unequal
5. imaginary and unequal 10. real, irrational and unequal
II.
1. 𝑘 = ±8
4
2. 𝑘 >
5
25
3. 𝑐 <
16
9
4. 𝑚 >
28
5. 𝑘 = 3
Learning Module for Junior High School Mathematics
GRADE 9