9
1
I
ILLUSTRATING QUADRATIC
EQUATIONS
2
Mathematics – Grade 9
Quarter 1 – Module I: ILLUSTRATING QUADRATIC EQUATIONS
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Regional Director: Gilbert T. Sadsad
Assistant Regional Director: Jessie L. Amin
Development Team of the Module
Writer: Elizabeth T. Tablizo – Catanduanes National High School
Editors: Zoren I. Añonuevo – Baras Rural Development High School
Validators: Jonel G. Aznar – Bato Rural Development High School
Lyra C. Tusi – San Andres Vocational School
Ludy M Avila – Catanduanes National High School
Riza B. Benavidez – Bato Ruran Development High School
Romel G. Petajen – EPS-Math, Catanduanes Division
Illustrator: Edwin T. Tomes – Dororian National High School
Layout Artist: Edwin T. Tomes – Dororian National High School
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ILLUSTRATING QUADRATIC
EQUATIONS
LESSON
Start Lesson 1 of this module by assessing your knowledge of the
different Mathematics concepts previously studied and your skills in
performing mathematical operations. These knowledge and skills will
help you understand quadratic equations. As you go through this lesson,
think of this important question: “How are quadratic equations used in
solving real-life problems and in making decisions?’’ To find the answer,
perform each activity. If you find any difficulty in answering the exercises,
seek the assistance of your teacher or peers or refer to the modules you
have gone over earlier.
• illustrates quadratic equations.
A quadratic equation in one variable is a mathematical
sentence of degree 2 that can be written in standard form ax2
+ bx + c = 0, where a, b and c are real numbers and a ≠ 0.
To easily
understand
the lesson,
let us recall
the skill on
finding the
product of
some
polynomials.
Find each indicated product. Match column A to column B.
A
B
1.
2.
3.
4.
5.
3x(x2 + 7)
(x + 3)(x + 7)
(x + 4)(x + 4)
2(x – 8)
(2x – 1)(x + 5)
a. x2 + 8x + 16
b. 2x2 + 9x -5
c. 2x – 16
d. 3x3 + 21x
e. x2 + 10x + 21
From the given above, what factors give a polynomial of degree 2?
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ANOTHER KIND OF EQUATION!!!
x2 – 5x + 3 = 0
r2 = 144
8k – 3 = 0
c = 12n - 5
3m = 2m2 - 6
0
x2 – 5xWrite
+ 3 = 0your answer on the table.
x2 – 5x + 3 = 0
– 5x + 3of=the
0 given
x2 equations
– 5x + 3 = 0are linear?
1. x2Which
2
– 5x + 3EQUATION
=0
NOT xLINEAR
LINEAR EQUATION
Example: c – 8 = c2
Example: 2x + 5 = 0
2. How do you describe linear equations?
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___________________________________________________________________
3. How do you differentiate linear equation from those equations which are not linear?
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Quadratic Equations in a Real Step
Mrs. Jacinto asked a carpenter to construct a
rectangular bulletin board for her classroom. She
told the carpenter that the board’s area must be 18
square feet. The length of the board is 7 ft. longer than its
width(x).
Based from the situation:
Width is represented by x ft.
Length is x + 7 ft
Area is 18 ft2
Can you formulate an equation?
Can you describe the
formulated equation?
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The equation in the given situation is x(x + 7) = 18, since Area of rectangle
= lw. The result of the equation is x2 + 7x = 18 and this is an example of a quadratic
equation, since the equation has degree 2.
A quadratic equation in one variable is a mathematical sentences of
degree 2 that can be written in standard form ax2 + bx + c = 0, where a, b and
c are real numbers and a ≠ 0.
In the equation, ax2 is the quadratic term, bx is the linear term and c is the
constant term.
Example 1: 2x2 + 5x = 3
Standard form: 2x2 + 5x – 3 = 0
a=2
b=5
c = -3
Standard form: -2x2 - 5x + 3 = 0
a=-2
b=-5
c=3
or
Example 2: 3x(x – 2) = 10 is a quadratic equation. However, it is not written
in standard form.
Solution: 3x(x – 2) = 10
3x2 – 6x = 10
3x2 – 6x – 10 = 0
Given
Apply Distributive Property
Standard form: 3x2 - 6x – 10 = 0
a=3
b = -6
or
Standard form: -3x2 + 6x + 10 = 0
c = -10
a = -3
b=6
c = 10
Example 3: The equation (2x + 5)(x – 1) = -6 is also a quadratic equation but it
is not written in standard form.
Solution: (2x + 5)(x – 1) = -6
2x2 – 2x + 5x – 5 = -6
2x2 + 3x – 5 + 6 = -6 + 6
2x2 + 3x + 1 = 0
Standard form: 2x2 + 3x + 1 = 0
a=2
b=3
c=1
or
Given
By FOIL Method
Addition 2roperty of Equality
Standard form: -2x2 - 3x - 1 = 0
a = -2
b = -3
c = -1
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When b = 0 in the equation ax2 + bx + c = 0, it results to a quadratic
equation of the form ax2 + c = 0.
When c = 0 in the equation ax2 + bx + c = 0, it results to a quadratic
equation of the form ax2 + bx = 0.
Examples:
Equations such as x2 + 5 = 0, -2x2 – 6 = 0 and 12x2 + 8 = 0 are
quadratic equations of the form ax2 + c = 0. In each equation, the value of b
= 0.
Equations such as x2 + 2x = 0, -3x2 – 16x = 0 and 4x2 + 8x = 0 are
quadratic equations of the form ax2 + bx = 0. In each equation, the value of
c = 0.
I.
Fill in the blank. Write the appropriate word which
is the result of the given.
1. side x side = Area of a square
2. length x width = _________
3. rate x time = _________
II. Complete the given table by filling the blanks.
Given
1. width: x
length: x + 4
area: 16
2. rate: m
time: m + 8
distance: 42
Equation
ax2 + bx + c = 0
x(x + 4) = _____
x2 + 4x -16 = 0
m( ___ )= 42
____________
Good job in finishing the activity! Now, take time to process the
information you acquired by accomplishing the next task. It’s time
to be creative!
Your goal in this section is to apply the key concepts of quadratic equations. Use the
mathematical ideas and the examples presented in the preceding section to answer the
activities provided.
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Direction: Tell whether the given is a quadratic equation or not. If it is quadratic,
write it in the form ax2 + bx + c = 0.
Given Equation
Quadratic Equation
or NOT
ax2 + bx + c = 0
1. 4x(x + 8) = 15
Quadratic Equation
4x2 + 32x – 15 = 0
2. x3 + 5x – 26 = 0
3. 3(m – 14) = 0
4. 2t(t – 7) = - 3t
5. (x – 7)( x + 7) = 0
Congratulations, you have finished the first part of
this module! Now, it’s is time to answer the next
tasks to enhance your understanding about
illustrating quadratic equation. Go, go, go!
I. Direction: Identify which of the following equations are quadratic
and which are not. Write your answer on the space provided.
1. 3m +8 = 17 __________________
2. x2 – 5x + 10 = 0 ______________
3. 3x(x - 4) = -7 ________________
4. 2(x + 8) = 15 ________________
5. (x + 4)2 = 11 _________________
II. Direction: Write the quadratic equation in standard form ax 2 + bx + c = 0 and identify
the values of a, b and c.
Quadratic Equation
ax2 + bx + c = 0
a
b
c
3 + 4s2 = 15s
4s2 -15s + 3 = 0
-4s2 +15s - 3 = 0
4
-4
-15
15
3
-3
1. 6s – 2s2 = 0
2. 3x(x + 5) – 12 = 0
3. (x + 7)(x + 2) = 0
4. (x - 4)(x – 4) = 12
5. (2m + 2)(m + 3) = (m - 3)2
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Directions: Tell whether or not each of the following situations
illustrates quadratic equations. Justify your answer by representing
each situation by a mathematical sentence.
1. The length of a swimming pool in Virac Resort is 8 m longer than its width and
the area is 105 m2.
________________________________________________________________
________________________________________________________________
2. Edna paid at least Php 1,200 for a pair of pants and a blouse. The cost of the
pair of pants is Php 600 more than the cost of the blouse.
________________________________________________________________
________________________________________________________________
This time, share your final insights by answering the
following questions below.
1. How do you define quadratic equations?
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__________________________________________________________
2. How are quadratic equations different from linear equations?
__________________________________________________________
__________________________________________________________
3. Give 3 examples of quadratic equations written in standard form. Identify
the values of a, b and c in each equation.
a) ______________________________
b) ______________________________
c) ______________________________
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Key to Correction
Find each indicated product.
1. D
2. E
3. A
4. C
5. B
Factors give a polynomial degree of 2
(x + 3)(x + 7)
(x + 4)(x + 4)
(2x – 1)(x+ 5)
Another Kind of Equation!
1.
LINEAR EQUATION
NOT LINEAR EQUATION
8k – 3 = 0
x2 – 5x + 3 = 0
c = 12n -5
r2 = 144
3m = 2m2 – 6
2. Linear equations are equations in which variable has a highest exponent of 1
Linear equation has a degree of one while other equations has a degree of 2
or 3.
Let’s Do This!
I.
Length x width = Area of a rectangle
Rate x time = distance
II.
Given
1. width: x
length: x + 4
area: 16
2. rate: m
time: m + 8
distance: 42
Equation
ax2 + bx + c = 0
x(x + 4) =x2 + 4x
x2 + 4x -16 = 0
m( m + 8 )= 42
m2 + 8m – 42 = 0
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Given Equation
Quadratic Equation
or NOT
ax2 + bx + c = 0
1. 4x(x + 8) = 15
Quadratic Equation
4x2 + 32x – 15 = 0
2. x3 + 5x – 26 = 0
Not Quadratic Equation
3. 3(m – 14) = 0
Not Quadratic Equation
4. 2t(t – 7) = - 3t
Quadratic Equation
2t2 – 11t = 0
5. (x – 7)( x + 7) = 0
Quadratic Equation
x2 – 49 = 0
Direction: Tell whether the given is a quadratic equation or not. If it is quadratic,
write it in the form ax2 + bx + c = 0.
I.
Direction: Identify which of the following equations are quadratic
and which are not. Write your answer on the space provided.
3m +8 = 17 Not Quadratic Equation
2. x2 – 5x + 10 = 0 Quadratic Equation
3. 3x(x - 4) = -7 Quadratic Equation
1.
II.
4. 2(x + 8) = 15 Not Quadratic Equation
5. (x + 4)2 = 11 Quadratic Equation
Direction: Write the quadratic equation in standard form
ax2 + bx + c = 0 and identify the values
of a, b and c.
Quadratic Equation
ax2 + bx + c = 0
a
b
c
3 + 4s2 = 15s
4s2 -15s + 3 = 0
-4s2 +15s - 3 = 0
-2s2 + 6s = 0
2s2 - 6s = 0
2
3x + 15x - 12 = 0
-3x2 - 15x + 12 = 0
x2 + 9x + 14 = 0
-x2 - 9x - 14 = 0
x2 - 8x + 4 = 0
-x2 + 8x - 4 = 0
2m2 + 2m - 3 = 0
-2m2 - 2m + 3 = 0
4
-4
-2
2
3
-3
1
-1
1
-1
2
-2
-15
15
6
-6
15
-15
9
-9
-8
8
2
-2
3
-3
0
0
-12
12
14
-14
4
-4
-3
3
1. 6s – 2s2 = 0
2. 3x(x + 5) – 12 = 0
3. (x + 7)(x + 2) = 0
4. (x - 4)(x – 4) = 12
5. (2m + 2)(m + 3) = (m - 3)2
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Check Your Understanding
1. Quadratic Equation
x(x + 8) = 105
x2 + 8x – 105 = 0
x is the width, x + 8 the length
2. Not Quadratic Equation x + x + 600 ≥ 1200 or 2x + 600 ≥ 1200
x is the cost in pesos, of the blouse
REFERENCE
DepEd Materials: Mathematics Learner’s Material in Grade 9