mathematics first quarter Module 2, Exercises of Mathematics

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Mathematics

First Quarter

Module 1: Illustrations of

Quadratic Equations

Module 1

My X To The Second Degree

What This Module Is All About

This learning material is about illustrating quadratic equations. As you go

through this module you will be able to define and identify equations which are

quadratic and which are not. You will also learn to rewrite quadratic equations into

standard form and determine the values of a , b , and c.

What You Are Expected To Learn

After going through this module, the learners should be able to illustrate

quadratic equations (M9AL-Ia- 1 ). Specifically, this aims to let the learners (a) identify

equations that are quadratic, (b) rewrite quadratic equations in standard form, and (c)

determine the values of a, b, and c in a quadratic equation.

How Much Do You Know (Pre-test)

A. Directions: Choose the letter that corresponds to the correct answer.

1. What is a polynomial equation of degree two that can be written in a form

2

ax + bx + c = , where a, b, c are real numbers and a  0?

A. Linear Equation C. Quadratic Equation

B. Quadratic Inequality D. Linear Inequality

2. Which of the following is a quadratic equation?

A.

2

x + r −

C.

2

s + s − =

B.

3 t − 7 = 2

D.

2

x − x 

3. In the quadratic equation

2

x + x − =

, which is the quadratic term?

A. 3

2

x

C.

2

x

B.

7 x

D. − 4

4. What is the standard form of 3 𝑥 − 2 𝑥

2

A. 3 𝑥 − 2 𝑥

2

− 7 = 0 C. − 2 𝑥

2

B. 2 𝑥

2

+ 3 𝑥 − 7 = 0 D. − 2 𝑥

2

5. In the equation 5 𝑟

2

− 𝑥 + 2 = 0 , what are the values of a, b, and c?

A. a = 5, b = - 1, c = 2 C. a = 5, b = - 1 , c = - 2

B. a = 5, b = 0, c = 2 D. a = 5, b = 1, c = 2

Discussion

What have you noticed with the answer of examples 1 and 3 above? And how

about example 2?

Examples 1 and 3 are quadratic equations because the highest exponent or

degree of the variable x is 2 while axample 2 is a linear equation since its x is in degree

So, how do you define quadratic equation?

Distinguishing Quadratic Equation from Linear Equation:

Try this out:

A. Identify whether the following equations are linear, quadratic or neither:

1. x

2

• 4x + 7 = 0 6. x² - 5x = 3

2. 2x – 5 = 0 7. x² - 4x – x² = 0
3. – 4 x

2

• 4x + 5 = 0 8. 2x – 5 = 4

4. 3 x

2

• 12 = 0 9. 3x – 3x = 5

x + 1 = 0 10.

2

x

x

By this time you are already familiar with quadratic equation. Quadratic equation

is to be in standard form when it’s written in the form 𝒂𝒙

𝟐

• 𝒃𝒙 + 𝒄 = 𝟎, where a , b , and

c are real numbers and a ≠ 0.

The equation x

2

• 7x + 4 = 0 is a quadratic equation in standard form where, a

=1, b = - 7 and c = 4.

Let’s try to look at the following equations and determine whether it is a quadratic

equation or not. If it does, then set it to standard form and give the values of a, b, and

c.

2

2

2

4. (2x + 3)(x – 1) = - 6

2

Solutions:

2

• 5 𝑥 − 3 = 0 is a quadratic in standard form with a = 2, b = 5 and c = - 3.

2. 3 𝑥 + 4 = 0 not a quadratic equation

A quadratic equation in one variable is a mathematical sentence of degree 2

that can be written in (standard form) the form 𝒂𝒙

𝟐

• 𝒃𝒙 + 𝒄 = 𝟎, where a , b ,

and c are real numbers and a ≠ 0. In the equation 𝒂𝒙

𝟐

𝟐

is the quadratic term

• 𝒃𝒙 is the linear term
• c is the constant term

3. 3 𝑥(𝑥 − 2 ) = 10 is quadratic equation. However, it is not written in standard form.

To write the equation in standard form, expand the product and make one side

of the equation equal to zero as shown below.

2

− 6 𝑥 = 10 Apply distributive property

2

− 6 𝑥 − 10 = 10 − 10 Apply subtraction property

2

− 6 𝑥 − 10 = 0 Final form

The equation becomes 3 𝑥

2

− 6 𝑥 − 10 = 0 , which is the standard form.

In the equation 3 𝑥

2

− 6 𝑥 − 10 = 0 , a = 3, b = - 6, and c = - 10

4. The equation (2x + 3 )(x – 1) = - 6, is also a quadratic equation but is not written in

standard form.

Just like in number 2 it can be written in standard form by expanding its product

and making one side of the equation equal to zero as shown:

(2x + 3 )(x – 1 ) = − 6 , 2 𝑥

2

− 2 𝑥 + 3 𝑥 − 3 = − 6 Expand the product

2

• 𝑥 − 3 = − 6 Apply addition property

2

• 𝑥 − 3 + 6 = − 6 + 6 Addition proeprty

2

• 𝑥 + 3 = 0 Final form

The equation becomes 2 𝑥

2

• 𝑥 + 3 = 0 , which is in standard form. In the equation

a = 2, b = 1 and c = 3.

Note: When b = 0 in the equation 𝒂𝒙

𝟐

• 𝒃𝒙 + 𝒄 = 𝟎 , it results to a quadratic

equation of the form 𝒂𝒙

𝟐

5, 6, and 7. Equations such as 𝒙

𝟐

𝟐

• 𝟕 = 𝟎, and 𝟏𝟔𝒙

𝟐

− 𝟗 = 𝟎 , are quadratic

equations of the form 𝒂𝒙

𝟐

• 𝒄 = 𝟎. In each equation, the value of b = 0.

Activity 1

Identify which of the following are quadratic and which are not. If the equation is not a

quadratic, explain.

2

2

1

2

2

2

Were you able to identify which equations are quadratic and which are not? I’m

sure you did.