UNIT 1: Quadratic Equations
Lesson 1
Introduction to Quadratic Equations
Lesson 2
Solving Quadratic Equations by Extracting
Square Roots
Lesson 1
Introduction to Quadratic
Equations
Warm Up!
Before we start studying quadratic equations, let’s have a quick
look on its history and possible applications by watching the
following video.
(Click on the link to access the video.)
“Quadratic Equations by Shmoop”. Shmoop.
Retrieved 24 January 2019 from
https://www.youtube.com/watch?v=fHH1FF9RWY0
Guide Questions
● How are quadratic equations different from linear equations?
● What are some of the possible applications of quadratic
equations?
● What is the degree of a quadratic expression and how does it
affect the number of solutions of a quadratic equation?
Learn about It!
Quadratic Equation
1 a second-degree polynomial equation that can be written in the standard form
ax 2 bx c 0
where 𝑎, 𝑏, and 𝑐 are real numbers and 𝑎 ≠ 0
Example:
3𝑥 2 + 4𝑥 + 15 = 0 is a quadratic equation in standard form where
𝑎 = 3, 𝑏 = 4, and 𝑐 = 1.
Try It!
Example 1: Determine whether the equation
4𝑥 − 2 = −4𝑥 2 − 5
is a quadratic equation or not.
Try It!
Example 1: Determine whether the equation
4𝑥 − 2 = −4𝑥 2 − 5
is a quadratic equation or not.
Solution: Write the equation in standard form.
Recall that the standard form of a quadratic equation is
𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0.
Try It!
4 x 2 4 x 5
2
4x2 4x 2 5 0
4x 4x 3 0
2
The equation follows the standard form of a quadratic equation
where 𝑎 = 4, 𝑏 = 4, and 𝑐 = 3.
Try It!
Example 2: Write the quadratic equation 5𝑥 2 − 6𝑥 = 7𝑥 + 3𝑥 2 in
standard form.
Try It!
Example 2: Write the quadratic equation 5𝑥 2 − 6𝑥 = 7𝑥 + 3𝑥 2 in
standard form.
Solution: The standard form of a quadratic equation is
ax bx c 0
2
1. Transpose all the terms to one side of the equation.
5 x 6 x 7 x 3x
2 2
5 x 3x 6 x 7 x 0
2 2
Try It!
2. Combine similar terms.
5 x 2 3x 2 6 x 7 x 0
2 x 13x 0
2
3. Arrange the terms following the standard form of quadratic
equation.
The equation is already arranged in standard form where 𝑎
= 2, 𝑏 = −13, and 𝑐 = 0.
Let’s Practice!
Individual Practice:
1. Write your own example of a quadratic equation in standard
form. Determine the values of 𝑎, 𝑏, and 𝑐.
2. Write each equation in standard form.
a. 6 x 9 x 5
2
b.10 5 x 2 x 2
c. 3x 10 x 2
Let’s Practice!
Group Practice: To be done in groups of five.
The length of a rectangle is one less than twice its width. The
area of the rectangle is 45 square units.
Translate the above mathematical sentence into a mathematical
equation and write the equation in standard form.
Lesson 2
Solving Quadratic
Equations by Extracting
Square Roots
Warm Up!
Before we solve equations by extracting square roots, let us have
a short drill on square roots by working as a class on the following
interactive online exercise!
(Click on the link to access the exercise.)
“Square Root Concentration Game”. Math-Play.
Retrieved 24 January 2019 from http://www.math-
play.com/Square-Root-Concentration-Game/square-root-
concentration-game_html5.html
Guide Questions
● How many square roots does a positive real number have?
● Is there any real number with exactly one square root?
● Why can’t a negative real number have a square root?
Learn about It!
Perfect Square
1 a number that is obtained by multiplying an integer by itself
Example:
The number 25 is a perfect square since 5•5 = 25 and
(-5)•(-5) = 25.
Learn about It!
2 Solution
A number 𝑥 is a solution of the quadratic equation if it satisfies the equation 𝑎𝑥 2
+ 𝑏𝑥 + 𝑐 = 0. This is also called a root of the quadratic equation.
Example:
In the quadratic equation 𝑥 2 − 1 = 0, 𝑥 = 1 is a solution since
12 − 1 = 0 is true.
Try It!
Example 1: What are the solutions of 𝑥 2 − 121 = 0?
Try It!
Example 1: What are the solutions of 𝑥 2 − 121 = 0?
Solution: To determine the solution of the quadratic equation 𝑥 2
− 121 = 0, follow the steps below.
1. Isolate the constant on one side of the equation.
By APE, add 121 to both sides of the equation.
𝑥 2 − 121 + 121 = 0 + 121
𝑥 2 = 121
Try It!
2. Extract the square roots on both sides of the equation.
Do not forget the ± symbol in getting the square root of the
constant.
𝑥 2 = ± 121
3. Solve for the value(s) of the variable.
𝑥 = ±11
Therefore, the solutions are 11 and −11.
Try It!
Example 2: Determine the solutions of 𝑥 2 + 11 = 35.
.
Try It!
Example 2: Determine the solutions of 𝑥 2 + 11 = 35.
Solution: To determine the solution of the quadratic equation 𝑥 2
+ 11 = 35, follow the steps below.
1. Isolate the constant on one side of the equation.
By APE, add −11 to both sides of the equation.
𝑥 2 + 11 + −11 = 35 + −11
𝑥 2 = 24
Try It!
2. Extract the square root of both sides of the equation
𝑥 2 = ± 24
3. Solve for the values of the variable.
Since 24 is not a perfect square, we need to factor it such that
one of the factors is a perfect square.
𝑥2 = ± 4 ⋅ 6 4 is a perfect
square
Try It!
Since 4 is a perfect square, we may take its square root, which
is 2.
𝑥2 = ± 4 ⋅ 6
𝑥 2 = ±2 6
Thus, the solutions are 2 6 and −2 6. .
Let’s Practice!
Individual Practice:
1. The area of a square lot is 288 sq. meters. What is its
perimeter?
2. Solve for the values of xin the equation 3𝑥 2 + 147 = 0.
Let’s Practice!
Group Practice: To be done in groups of five.
2
Find the solutions of the equation 2𝑥 + 1 − 25 = 0.
Key Points
Quadratic Equation
1 a second-degree polynomial equation that can be written in the standard form
ax 2 bx c 0
where 𝑎, 𝑏, and 𝑐 are real numbers and 𝑎 ≠ 0
Synthesis
● How do we write a quadratic equation in its standard form?
● How can you apply the concept of quadratic equation in your
daily life as a student?
● How will you determine whether a certain value is a solution
of a quadratic equation?
Key Points
Perfect Square
1 a number that is obtained by multiplying an integer by itself
2 Solution
A number 𝑥 is a solution of the quadratic equation if it satisfies the equation 𝑎𝑥 2
+ 𝑏𝑥 + 𝑐 = 0 .This is also called a root of the quadratic equation.
Synthesis
● How do you solve quadratic equations by extracting square
roots?
● Why is it important to follow the necessary steps in extracting
square roots? What will happen if these steps are jumbled?
● How would you know if extracting square roots cannot be
used in a particular quadratic equation?
