Name: __________________________Date: __________

Grade: _____________________ Section: ___________

Activity 4a
Sum and Product of the Roots
Learning Competency: Describes the relationship
between the coefficients and the roots of a
quadratic equation. (M9AL – Ic – 2)

What I Need to Do
This activity is about quadratic equations. In this activity, you will be able to
understand the relationship between the roots of a quadratic equation and its
numerical coefficients and be able to apply the mathematical skills you have learned
in the previous lessons.
This activity is designed for you to:

1. describe the relationship between the coefficients and the roots of

quadratic equation; and
2. determine the sum and product of the roots of quadratic equations.

A. Try this out.

Recall how to add and multiply rational numbers by doing the following exercise.
Perform the operation indicated.

8 + (-4) −2 −5 (-9) (-12)

+
3 6

-5 + (-5) (8) (-4) 1 2

( )( )
2 5

12 + 14 (13) (7) −3 3
( )( )
8 4

B. Find My Roots

Find the roots of the following quadratic equations using quadratic formula.

1. x2 - 8x + 16 = 0 2. 2x2 – x = -5

Page 1
 Gearing Up

Relationship Between the Coefficient and Roots of Quadratic Equations

The sum and product of the roots of the quadratic equation ax2 + bx + c = 0
can be determined using the coefficients a, b, and c without solving for the roots.

Remember that the roots of a quadratic equation can be determined using the

−𝑏 ± √𝑏2 − 4𝑎𝑐 −𝑏+ √𝑏2 − 4𝑎𝑐

formula, x = . From the quadratic formula let 𝑥1 =
2𝑎 2𝑎

−𝑏− √𝑏2 − 4𝑎𝑐

and 𝑥2 = be the roots.
2𝑎

−𝑏+√𝑏2 −4𝑎𝑐 −𝑏−√𝑏2 −4𝑎𝑐 −2𝑏 𝒃

Sum: 𝑥1 + 𝑥2 = + = = −𝒂
2𝑎 2𝑎 2𝑎

Applying the sum and difference

−𝑏+√𝑏2 −4𝑎𝑐 −𝑏−√𝑏2 −4𝑎𝑐
Product: 𝑥1 • 𝑥2 = ( ) ( ) of the terms.
2𝑎 2𝑎

𝑏2 −𝑏2 +4𝑎𝑐 4𝑎𝑐 𝒄

= =- =
4𝑎2 4𝑎2 𝒂

Remember

If x1 and x2 are the solutions or roots of the equation ax2 + bx + c = 0,

where a ≠ 0 then:

𝒃 𝒄
x1 + x2 = −𝒂 (sum of the roots) and x1 • x2 = (product of the roots).
𝒂

Example

Find the sum and product of the roots for each of the following
equations: (a) 2x2 + 8x – 10 = 0

(b) x2 + 7x – 18 = 0

Solution:
(a) 2x2 + 8x – 10 = 0
1. Identify a, b, and c.
a=2 b=8 c = -10
2. Substitute and simplify.

Sum of the Roots Product of the Roots

𝒃 𝒄
x1 + x2 = −𝒂 x1 • x2 =
𝒂

𝟖 −𝟏𝟎
= −𝟐 =
𝟐

=-4 =-5

Page 2
 ✓ To check, find the roots of 2x2 + 8x – 10 = 0 using any of the
methods of solving quadratic equations. Then determine the
sum and product of the roots.

The roots of the equation are 1 and -5.

Let x1 = 1 and x2 = -5

Sum of the roots: x1 + x2 = 1 + (-5) = -4

Product of the roots: x1 • x2 = (1) (-5) = -5

(b) x2 + 7x – 18 = 0
1. Identify a, b, and c.
a=1 b=7 c = -18
2. Substitute and simplify.

Sum of the Roots Product of the Roots

𝒃 𝒄
x1 + x2 = − x1 • x2 =
𝒂 𝒂

𝟕 −𝟏𝟖
=− =
𝟏 𝟏

=-7 = - 18

✓ Checking

The roots of the equation x2 + 7x – 18 = 0 are -9 and 2.

Let x1 = -9 and x2 = 2

Sum of the roots: x1 + x2 = -9 + 2 = -7

Product of the roots: x1 • x2 = (-9) (2) = -18

Practice

Find the sum and product of the roots of each quadratic equation.

1. 2x2 + 12x – 3 = 0 2. x2 – 8x = -5

Sum of the roots: _________ Sum of the roots: _________

Product of the roots: _________ Product of the roots: _________

Page 3
 Getting Better

What Is A Smart Bird’s Favorite Type of Math?

To answer the question, use the values of a, b, and c of each of the following
quadratic equations in determining the sum and product of their roots.
Write the corresponding letter in the box matching the given sum and
product.

L x2 + x = 6 B 12x2 + x – 6 = 0 G x2 – 2x – 2 = 0

R A O
2x2 + 5x – 25 = 0 6x2 – 5x – 4 = 0 3x2 – 11x – 4 = 0

E W
4x – 5 = -4x2 8x2 + 12x = -1

Answer Box

answer

sum of 11 3 1 5 5
− -1 2 -1 − −
roots 3 2 12 2 6

4 1 5 1 25 2
product − -6 -2 − − − −
of roots 3 8 4 2 2 3

Gaining Mastery

A. Find the sum and product of the roots of each quadratic equation to
complete table below.

Quadratic Equation Sum of Roots Product of Roots

1. x2 + 8x - 18 = 0

2. 3x2 + 2x - 3 = 0

3. x2 - 5x - 8 = 0

Page 4
 B. What the Sum and Product Mean to Me.

Study the situation below and answer the questions that follow.

A rectangular plot has an area of 192 m2 and a perimeter of 56 m.

Questions:

1. What equation would describe the area of the garden? Write the equation
in terms of the width of the garden.
2. What can you say about the equation formulated in item 1?
3. Find the roots of the equation formulated in item 1. What do the roots
represent?
4. What is the sum of the roots and how is this related to the perimeter?
5. What is the product of the roots and how it this related to area of the
rectangle?

Rubrics for Scoring

5 points 4 points 3 points 1 point

The student The student’s work The student The student

showed all the is partly correct. attempted to attempted to
required work to There may be a answer the answer the
arrive at a correct computational problem and did problem and did
solution. error or a problem not finish it. The not finish it. The
with the format of student used the student failed to
the answer. correct method, use the correct
but did not method.
complete the
solution.

The work is neat, The work is neat, The work is neat The work
clear and clear and and organized but presented is
organized which organized which is hard to unorganized and
is easy to follow. easy to follow. understand. hard to know what
information goes
together.

Page 5
 What I Need to Remember

If x1 and x2 are the solutions or roots of the equation ax2 + bx + c = 0,

where a ≠ 0 then:

𝒃 𝒄
x1 + x 2 = − (sum of the roots) and x1 • x2 = (product of the roots).
𝒂 𝒂

Writer: ANA MILAFLOR B. PIAPE

School: F. BUSTAMANTE NATIONAL HIGH SCHOOL
Division: DAVAO CITY
Evaluator: ROMAN JOHN C. LARA
School: DAVA CITY NATIONAL HIGH SCHOOL
Division: DAVAO CITY

Page 6
 Answer Key

What I Need to Know

A.

3
4 − 108
2
1
-10 -32 5
9
26 −
91 32

B.
1. x = 4 (only one root)

1 ±2𝑖 √10
2. 𝑥= (two not-real roots)
4

Gearing Up

Practice
1. Sum of the roots: -6

−𝟑
Product of the roots:
𝟐

2. Sum of the roots: 8

Product of the roots: 5

Getting Better

What Is A Smart Bird’s Favorite Type Of Math? Ans: OWLGEBRA

Gaining Mastery

A. Find the sum and product of the roots of each quadratic equation to
complete table below.

Quadratic Equation Sum of Roots Product of Roots

x2 + 8x - 18 = 0 -8 -18

Page 7
 2 -1
3x2 + 2x - 3 = 0 −
3

x2 - 5x - 8 = 0 5 -8

B. What the Sum and Product Mean to Me…

1. Let x = width of the garden
Let 28 – x = length of the garden
The area of the garden is given by the equation x(28 – x) = 192
2. The equation is a quadratic equation.
3. x(28 – x) =192 x2 – 28x + 192 = 0
The roots of the equation are 12 and 16. These represent the dimensions of
the garden.
4. The sum of the roots is 28. This is half the perimeter of the garden.
5. The product of the roots is 192. This is equal to the area of the garden.

Writer: ANA MILAFLOR B. PIAPE

School: F. BUSTAMANTE NATIONAL HIGH SCHOOL
Division: DAVAO CITY
Evaluator: ROMAN JOHN C. LARA
School: DAVAO CITY NATIONAL HIGH SCHOOL
Division: DAVAO CITY

Page 8