INTRODUCTION TO QUADRATIC EQUATIONS | GRADE 9

Summary

This educational video script targets grade 9 students delving into quadratic equations. It introduces quadratic equations as those with the highest variable exponent of 2. The script defines the equation's degree, explains the standard form ax^2 + bx + c = 0, and distinguishes between complete and incomplete quadratic equations. It guides students through identifying quadratic terms, linear terms, and constants in equations. The video also includes exercises to test understanding and concludes with a summary of key concepts, promising further exploration in upcoming videos.

Takeaways

• 📘 The first topic in grade 9 math is quadratic equations.
• 🔢 A quadratic equation is defined as an equation with the highest exponent of the variable being 2.
• 📚 The objectives of the video include understanding what a quadratic equation is, recognizing examples, and solving exercises with accuracy.
• 📐 The degree of an equation is determined by the highest exponent of the variable, which for a quadratic equation is 2.
• 🔍 Examples are provided to differentiate between quadratic and non-quadratic equations based on the highest exponent.
• 📝 The standard form of a quadratic equation is ax^2 + bx + c = 0, where a, b, and c are real numbers and 'a' cannot be zero.
• 🔑 The terms in a quadratic equation are identified as the quadratic term (ax^2), the linear term (bx), and the constant term (c).
• 🧩 Incomplete quadratic equations can lack one or more of the terms (bx or c), but 'a' must not be zero.
• 📉 The video provides exercises to test the understanding of identifying quadratic equations.
• ✅ The video concludes with a summary emphasizing the definition, standard form, and components of a quadratic equation.

Q & A

What is the definition of a quadratic equation?

A quadratic equation is an equation where the highest exponent of the variable is 2.

What is the significance of the term 'quadratic' in quadratic equations?

The term 'quadratic' comes from the word 'square' because the variable gets squared, like x squared.

What is the standard form of a quadratic equation?

The standard form of a quadratic equation is written as ax squared plus BX plus C equals 0, where a, B, and C are real numbers and a must not be equal to 0.

What are the three terms in a quadratic equation?

The three terms in a quadratic equation are the quadratic term (ax squared), the linear term (BX), and the constant term (C).

Why must 'a' not equal zero in a quadratic equation?

In a quadratic equation, 'a' must not equal zero because if 'a' is zero, the equation is no longer quadratic.

Can you provide an example of a quadratic equation?

An example of a quadratic equation is 3x squared minus X minus 5 equals 0.

How do you identify if an equation is quadratic by looking at its highest exponent?

An equation is identified as quadratic if the highest exponent of the variable is 2.

What is an incomplete quadratic equation?

An incomplete quadratic equation is a quadratic equation that may lack one of the terms (BX or C), such as ax squared plus BX equals 0 or ax squared equals 0.

What happens to the degree of an equation if the highest exponent is not 2?

If the highest exponent is not 2, the degree of the equation is determined by that highest exponent, making it not a quadratic equation.

How do you find the values of 'a', 'B', and 'C' in a quadratic equation?

In a quadratic equation, 'a' is the coefficient of the x squared term, 'B' is the coefficient of the x term (or 0 if there is no x term), and 'C' is the constant term.

What is the significance of the equal sign in a quadratic equation?

The equal sign in a quadratic equation signifies that the expression on the left is equal to the expression on the right, making it an equation.

Outlines

00:00

📚 Introduction to Quadratic Equations

This paragraph introduces the topic of quadratic equations for grade 9 mathematics students. It emphasizes the importance of understanding the concept of quadratic equations, which are equations with the highest exponent of the variable being 2. The video aims to teach students to recognize and provide examples of quadratic equations. The learning objectives include knowledge, skills, and attitude towards solving quadratic equations with honesty and accuracy. The explanation begins with the definition of a quadratic equation, its degree, and the significance of the exponent in determining the degree of an equation.

05:02

🔍 Identifying Quadratic Equations

The paragraph continues with examples to help students identify which equations are quadratic. It explains that the highest exponent of the variable must be 2 for an equation to be considered quadratic. The script walks through several equations, discussing why some are quadratic and others are not, based on the highest exponent present. It also clarifies that even if an equation appears not to have an exponent of 2, it may still be quadratic if simplified correctly.

10:04

📘 Expanding and Simplifying Quadratic Equations

This section delves into more examples of quadratic equations, showing how to simplify expressions to reveal the quadratic form. It demonstrates the use of the distributive property to expand expressions and how multiplying binomials results in a quadratic equation. The paragraph explains the standard form of a quadratic equation, ax^2 + bx + c = 0, and defines the terms: quadratic term (ax^2), linear term (bx), and constant term (c). It also stresses that 'a' must not be zero for the equation to remain quadratic.

15:06

📖 Incomplete Quadratic Equations

The paragraph discusses incomplete quadratic equations, which may lack one of the terms (bx or c) in the standard form. It provides examples and explains how to identify the values of a, b, and c in various scenarios. The importance of 'a' not being zero is reiterated, even if b or c can be zero. The section ensures students understand that a quadratic equation must have the highest variable exponent of 2.

20:08

📝 Practice and Summary

The final paragraph involves practice questions for students to apply their knowledge of identifying quadratic equations. It includes multiple-choice questions to test comprehension of the concept. The paragraph concludes with a summary of the key points learned: the definition of a quadratic equation, the standard form ax^2 + bx + c = 0, and the importance of the coefficient 'a' not being zero. The instructor also hints at further discussions on quadratic equations in upcoming videos.

Keywords

💡Quadratic Equation

A quadratic equation is a second-degree polynomial equation with one variable, where the variable is raised to the power of 2 at most. In the video, this concept is central as it defines the type of equations being discussed. The script provides examples such as '3x^2 - x - 5 = 0' to illustrate what constitutes a quadratic equation.

💡Exponent

The exponent in a mathematical expression indicates the number of times that a base quantity is multiplied by itself. In the context of the video, the highest exponent of the variable in a quadratic equation is 2, as seen in the explanation of 'x^2' where '2' is the exponent.

💡Degree of an Equation

The degree of an equation refers to the highest power of the variable present in the equation. The video emphasizes that the degree of a quadratic equation is 2, which is why it is called 'quadratic', as opposed to equations of other degrees.

💡Standard Form

The standard form of a quadratic equation is expressed as ax^2 + bx + c = 0, where a, b, and c are constants, and 'a' is non-zero. The video script uses this form to explain the structure of quadratic equations and to identify the coefficients a, b, and c in various examples.

💡Quadratic Term

The quadratic term in a quadratic equation is the term with the variable raised to the second power, such as 'ax^2'. The video explains that this term is crucial in defining the equation as quadratic and provides examples where 'a' is the coefficient of the quadratic term.

💡Linear Term

The linear term in a quadratic equation refers to the term with the variable raised to the first power, represented as 'bx'. The script clarifies that while the linear term is part of the quadratic equation, it is not the defining characteristic.

💡Constant Term

The constant term in a quadratic equation is the term without any variables, represented as 'c'. The video uses this term to complete the general form of a quadratic equation and provides examples to show where 'c' is located in the equation.

💡Incomplete Quadratic Equations

Incomplete quadratic equations are those that do not contain all three terms (quadratic, linear, and constant) of a standard quadratic equation. The video gives examples such as 'ax^2 + bx = 0' and 'ax^2 = 0' to illustrate incomplete forms.

💡Distributive Property

The distributive property is a mathematical rule that allows for the multiplication of a term by a parenthesis containing multiple terms. The video uses this property to expand expressions like 'x(x + 2)' into a standard quadratic form 'x^2 + 2x'.

💡FOIL Method

The FOIL method stands for First, Outer, Inner, Last and is used to multiply two binomials. The video script mentions this method when explaining how to expand expressions like '(x + 2)(x - 5)' into a quadratic equation 'x^2 - 3x - 10'.

💡Exercises

The exercises in the video are practical applications of the concepts taught, allowing students to test their understanding of identifying and working with quadratic equations. The script provides a series of questions for the viewers to determine whether certain equations are quadratic or not.

Highlights

Introduction to quadratic equations in grade 9 math.

Objectives of the video include understanding, illustrating, and accurately solving quadratic equations.

Definition of a quadratic equation: an equation with the highest variable exponent of 2.

Explanation of the term 'quadratic' and its relation to the variable being squared.

Identification of the degree of an equation based on the highest variable exponent.

Examples of quadratic equations and distinguishing them from non-quadratic equations.

Simplification of expressions to reveal the highest exponent and identify quadratic equations.

Use of the distributive property to simplify expressions and identify quadratic equations.

Explanation of the standard form of a quadratic equation: ax^2 + bx + c = 0.

Emphasis on the condition that 'a' must not be equal to zero in a quadratic equation.

Identification of the quadratic term, linear term, and constant term in a quadratic equation.

Examples of incomplete quadratic equations and how to identify their components.

Exercises to test the understanding of identifying quadratic equations.

Multiple-choice questions to reinforce the concept of quadratic equations.

Summary of the key points about quadratic equations learned in the video.

Anticipation for more detailed discussion on quadratic equations in upcoming videos.

Transcripts

[Music]

hello everyone especially to our

students who really love to learn

mathematics this video will discuss the

first topic in grade 9 math and it's all

about quadratic equation so please

listen carefully or if you want you can

also get your notebook and pen and take

down some important notes about our

topic

our objectives in this video will be the

first M DLC or the most essential

learning competency in mathematics 9 and

that is to illustrate quadratic equation

we will be able to know what is a

quadratic equation and for the KS a

objectives we have this for the

knowledge you will be able to recognize

and give examples of a quadratic

equation for the skills illustrate

quadratic equation and for the attitude

or values you will be answering a given

exercises with honesty and accuracy now

let's start what is a quadratic equation

what does it mean and what is it all

about the named quadratic comes from

what meaning square because the variable

gets squared like x squared therefore a

quadratic equation is an equation where

the highest exponent of the variable

usually X is 2 so as you can see from

the definition it is an equation meaning

to say it must have an equal sign and

remember that the highest exponent of

the variable is to just our recap our

recall about the exponent the exponent

is written on the upper right of a

certain number or a variable just like

this 1 X raise to 4 4 is our exponent so

here in a quadratic equation the highest

exponent is 2 that is why it is called

as equation of degree 2

the degree of an equation is determined

through the highest exponent of the

variable in an equation so called an

highest exponent now if some equation is

to the degree of equation is to if the

highest exponent is 3 the degree of

equation is 3 so it has to turn mean by

the highest exponent of the variable let

us now have the examples of a quadratic

equation

we have now five equations here and

we're going to identify which among

these equations are quadratic and which

among are not quadratic equation let's

begin with the first equation and that

is 3x squared minus X minus 5 is equal

to 0 what do you think is the highest

exponent here the highest exponent of

this equation is 2 therefore this is a

quadratic equation next how about this

one

5x minus of 3 is equal to 0 it says a

quadratic or not quadratic equation what

is the highest exponent of this equation

yes the highest exponent is not two but

one so therefore this is not a quadratic

equation next how about the third

equation here we have x cubed plus 4x

minus 3 x squared is equal to zero do

you think it is also a quadratic

equation no it is not a quadratic

equation because what is the highest

exponent here the highest exponent here

is 3 right not to although we can see

two here as our exponent 2 is not the

highest exponent but the highest

exponent is 3 therefore this is also not

a quadratic equation next is the year

four equation here 2x squared plus 12x

is equal to 0 what do you think is the

highest exponent of this equation yes

the highest exponent is 2 therefore this

is a quadratic equation and the last one

x squared minus 4x is equal to 0

you can see here that the highest

exponent is 2 and it is a quadratic

equation ok so you can notice here that

all the quadratic equation has the

highest exponent of 2 unlike on the nut

quadratic here the axe exponent here is

1 and the highest exponent here is 3 so

we can now easily say that a quadratic

equation must be the highest exponent of

a quadratic equation must be 2 but let

us have more examples

let us now have the examples of a

quadratic equation another example of a

quadratic equation is this x times the

quantity of X plus 2 is equal to

negative 1 but you will notice that

there is no 2 exponent here so how come

that this is an example of a quadratic

equation the reason is that we need to

simplify it first or find its product

and the product of this is a quadratic

equation let's see we will apply the

distributive property of multiplication

to get its product the first step here

is to multiply X to the first term

inside the parentheses and that is X at

some times X the answer will be x

squared we will just simply add the

exponent in multiplying this next step

we will multiply X and the second in to

the second term inside the quantity and

that is 2x times 2 the answer is 2x plus

2x and just copy negative 1 equals

negative 1 just like this x squared plus

2x is equal to negative 1 but a

quadratic equation must be equal to 0 so

we will transpose or move the negative 1

here to this side and as we move it we

will change the sign it will become

positive 1/4 plus 1 therefore the

product of this is x squared plus 2x

plus 1 is equal to 0 you can see now

that we have the highest exponent of 2

in its product therefore this is also an

example of a quadratic equation another

example we have X

quantity x squared X plus 2 times

quantity of X minus 5 is equal to 0 this

is a multiplication of two binomials and

this is also considered as a quadratic

equation because we all know that if we

multiply this the product is a quadratic

equation or it will become x squared and

in getting the product we will use the

foil method first term outer inner and

last turn okay let's see X plus 2 our

quantity X plus 2 times the quantity X

minus 5 is equal to 0 we will be

multiplying the first term ok x times X

the answer is x squared followed by the

all the outer term the outer term is

this X and negative 5 so x times

negative 5 we have negative 5 X followed

by the inner term and that is 2 and x2

times X the answer is positive 2x and

for the L the last term that is 2 and

negative 5 2 times negative 5 the answer

is and negative 10 and then copy equal

to 0 but we can simplify negative 5 and

negative 1 positive 2 right we can add

the product of oh and I so negative 5 X

plus 2 X since they are or they have sin

term we can simplify this so the answer

is 3 negative 3x therefore the final

product is x squared minus

3x minus 10 is equal to 0 you see here

that the highest exponent of the product

is 2 therefore that this example is also

a quadratic equation the standard form

of a quadratic is written as ax squared

plus BX plus C is equal to 0 where a B

and C are real numbers and always

remember that a must not be equal to 0

because if a is equal to 0 this is not

if I'd Radek equation anymore

okay let us find out more about it

the first term that we have in our

periodic equation ax squared is pulled

as our quadratic term okay so you marrow

x squared Union quadratic term Athena

power at the end while the BX or you

excellent or one exponent that is Lin

your turn and the last one young Salam

or your number lamb that is what we call

the constant term we have now three x

squared minus X plus five equals to 0

what will be our quadratic leaner and

constant term the quadratic term has the

x squared or the two exponent here we

have the x squared that is the quadratic

term the linear term okay that is the

negative x or it has a one exponent

right variable having a one exponent

this

our leader term and the last term the

number only that is what we call the

constant term this time we are going to

find the values of a B and C of a

quadratic equation we have the equation

3x squared minus X plus 5 is equal to 0

what will be the a B and C here a is

equal to 3 it can be seen beside the x

squared or the variable having this 2

exponent the number beside it is the

value of a how about the value of B the

value of B is found here the number

beside the X but you cannot see a number

here right but we can still remember

that it has an invisible one here and no

need for us to write it down in this

equation but if we are asked to find B B

is equal to negative 1 again it has an

invisible 1 here and if we want to find

the value of C the value of C here is

negative 5 I'm sorry positive 5 C is

positive 5 or the letter C the value of

C can be seen on the constant term how

about this one x squared plus 4x minus

21 is equal to 0

what is a and what is B and what is here

a is there is no number written here but

it has an invisible one right so X is a

is equal to 1

how about linear term or the B the BS

for B is equal to 4 and C the C here the

constant term is negative 21

maybe I'll include the negative here

again do not include the variable in

finding the values of a B and C we

understand now that the standard form of

a quadratic equation is ax squared plus

BX plus C is equal to 0 but there are

cases that we also have what we called

incomplete quadratic equations

what are those incomplete quadratic

equations example is this one ax squared

plus BX is equal to 0 as you notice

there is no C right example of this

equation 2x squared plus 8x is equal to

0 if we are asked to find the a B and C

this will be the answer is equal to 2 B

is equal to 8 and C is equal to 0 C is

equal to 0 because there is no C here

right

there's no see another incomplete

quadratic equation is in the form of AX

squared plus C is equal to 0 example x

squared minus 9 is equal to 0 if we are

asked to find the values of a B and C a

is equal to 1 B is equal to 0 because

there is no B or there's no linear term

here but we only have C C is equal to

negative 9 another case is that a x

squared

still this is and

the quadratic equation but is it is

incomplete and there is no B and C

example 2x squared is equal to zero if

we are asked to find the a B and C a is

equal to 2 B is equal to 0 and C is

equal to 0 both B and C are equal to 0

remember that in a quadratic equation B

and C can be equal to 0

yes it can be but it must not be a is is

not equal to 0

hey in a selector a dhaba indicia equal

to 0 because if we let a is equal to 0

then our equation is not a quadratic

anymore right

okay so let's have some exercise which

of the following equations are quadratic

equations number one 3x minus 2 is equal

to 0 what do you think is it put that

peak or not

hey the answer is not quadratic okay we

don't have to exponent right number 2 X

plus the 3x squared is equal to 0

I said quadratic or not the answer is

yes it is a quadratic we have here our 2

as the highest exponent next number 3 2

times the quantity of X minus 4 is equal

to 0 is it quadratic or not the answer

is not good body even though we multiply

this 2 times X the answer is 2x

ok the product of this is not equal

ratification

and the last one how about the last one

x times the quantity of X plus 3 minus 5

is equal to 0 is it quadratic or not the

answer if we get the product of this we

have a quadratic equation okay very good

so did you get it all correctly I hope

so

next question is a multiple-choice write

the letter in words of the correct

answer

number 5 question which of these

equations in a straight quadratic

equation is it a quantity X plus 3 plus

8 is equal to 0 B opps pair of x plus 3

is equal to 0 C X plus y is equal to 0

or a T X cube is equal to 2x what is the

answer the correct answer is B Square of

X plus 3 is equal to 0

next question number 6 it is a

polynomial equation of degree 2 I said a

quadratic equation B linear equation C

quadratic inequality the linear

inequality what will be the answer the

answer here is yes quadratic equation

number 7 question which of the following

is the standard form of a quadratic

equation is it a ax plus B greater than

or equal to 0 B ax plus B is equal to 0

C ax squared plus BX plus C is greater

than or equal to 0 or D ax squared plus

BX plus C is equal to 0 what will be the

answer the correct answer is yes you're

correct D ax squared plus BX plus C is

equal to 0

eight which of the following real number

in the quadratic equation from ax

squared plus BX plus C equals zero

cannot be zero

is it a b c or none of this ileenium top

at the Indies zero socratic okay the

answer is a number nine which of the

following is the quadratic term of our

equation x squared minus 10x plus 25 is

equal to zero what is the quadratic term

is if a is 0 B negative 10x see x

squared be 25 so the answer is okay the

answer is C and we also have the last

one what is the value of B in this

equation 3x squared minus 12x is equal

to zero

it said 83 be negative 12 see three x

squared or be negative 12 X again if we

are asked to find B the answer is only a

number so the answer here is okay the

answer is negative 12 B negative 12 and

that's it so how was your score did you

pass at the test did you get at least

eight and about out of ten I hope so

now let's summarize what we have learned

today we understand now that the

quadratic equation is an equation where

the highest exponent of the variable is

2 and we have the standard form of

quadratic equation we have ax squared

plus BX plus C is equal to 0 and

remember a must not be equal to 0 and a

quadratic equation has a quadratic term

that is ax squared a linear term the X

and the constant term is our C okay we

will talk about more about quadratic

equation in my next video so please

stand by and watch me again on my next

video that's it bye bye