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INTRODUCTION TO QUADRATIC EQUATIONS-GRADE 9 MATH MODULE 1 LESSON 1

hello and welcome to your students

today we are going to nourish your minds

with useful information

to improve your mathematical skills

[Applause]

[Music]

this video is for grade 9 mathematics

now before we start prepare the

following

a pen and a paper to write your

solutions

as we progress with our discussion and

remember you can always pause

and play this video whenever necessary

you can even go back to the portion or

the part of this video

that you want to revisit to attain

mastery

that's it we are all set so let's begin

in this video presentation i will be

guiding you

in the first module for the first

quarter

of this subject of our subject

mathematics for the grade nine

level the topic is quadratic

equation i am your math teacher mrs

rowan olofertis

specifically topic one or lesson one

of this module is all about the

introduction

to quadratic equation

what you need to know at the end of this

lesson

you are expected to illustrate and

identify quadratic equations

second write quadratic equations in

standard form

and determine the components of

quadratic equation

what's in currently you're a grade nine

student right

now let's recall and let's go back in

time

as we revisit your grade eight

let's recall your past lesson in the

first quarter

you were taught how to get the products

of various polynomials

and to get the special products let's

recall each

for example letter a product of binomial

now let's recall how to use foil method

so for example you're given two

binomials and multiplying a

plus b to the quantity of a plus c

let's identify the first terms followed

by

the outer terms that's a and c

we also have inner terms that's b and

a pointed on the screen right now then

we also have here the last terms that's

b and c

so that is your foil first outer

inner and last so how do we get the

answer or the product of this one the

result will be

the product of the first terms that's

a multiplied to a giving you a squared

okay then add it to the product of your

outer terms

that is your a multiplied to c therefore

you have ac

now we have here the third term that's

plus

a b where did we get a b that is the

result of your inner terms

b times a and having it alphabetically

that is

a b plus the last term is coming from

the last terms of our two binomials

which is b and c so this is now our

answer

a squared plus ac plus a b

plus bc now there's another pattern for

this

now let's have an example using the foil

method

for example we have here the quantity of

x plus two

multiplied to the quantity of x plus

five so we are going to get

the first terms the product of the first

terms that's

x multiplied to x for our f

followed by our outer terms that's x and

five

then we also have our inner terms that's

2 and x

then our last terms are 2 and 5

so following the foil method to get the

answer

we are going to multiply or get the

product of each terms

so for the first terms the outer terms

the inner terms and the last terms so

look at my laser pointer

for the first one that is x multiplied

to x

will give you x squared i add it to the

next

product that is the outer terms that's x

and 5 so that will give you yes

correct that's 5x then for the inner

terms that's 2

and x so the answer is good that's 2x

and finally our last terms are

two and five so what's the product of

two and five

nice that's ten so the final answer here

is

actually the answer using the foil

method is let's see

and let's compare so you have your x

squared plus 5x

plus 2x plus 10. now observed

that at the center you still have two

terms having

similar terms you have there two terms

with a variable x

so what do you need to do here is okay

you need to add similar terms

so the result now is x squared

plus seven x plus ten x plus ten

since you added 5x and 2x right here

giving you 7x for the middle term so the

final answer

is a trinomial which is x squared plus

7x

plus 10. that is correct

so that's how you use foil method just a

review

now this time how do we use patterns for

to get the product of binomials

say for example the same set we have

here quantity of a plus b

multiplied to the quantity of a plus c

so we can follow this pattern square the

first term a so we have a squared

next we have here

add it to the to the product of a

and the sum of b and c so add first

b and c then multiply it to

letter a then the last term is

the product of b and c so that would be

for the pattern

so how do we use that for example x plus

2

multiplied to x plus 5 same example for

the foil method

so here i'll use my laser pointer

observe

first terms are x so that will give you

x squared following the pattern as shown

above

here we have b plus c so our b

here is two our c is five so get the sum

two plus five seven multiply it with

a our a in this example is x so that was

seven right

seven times x will give you seven x

and our last term is the product of b

and c

so we have bs2 and c as five

so the product of two and five is ten so

that will give you ten

so the answer is x squared plus seven x

plus ten

having the same result when we use the

foil method

so that's it for the first pattern

correct now how about for the square of

a binomial

your binomial in this example is a plus

b

and you're getting the square of that so

to get that we can use this pattern

square the first term

and add it to twice

the product of a and b then the last

term will now be

the square of your given binomial mean

square of the last term of your given

binomial

so that would be the pattern a squared

plus twice

the product of a and b plus the square

of b

example we have here x

plus 2 or quantity of x plus 2 squared

so we will follow the pattern it will

give us

a squared which is x squared since our a

here

is x and our b is 2. so x squared

plus how did we get 4x we have here

multiply a and b then double it so you

have

x times 2 that will give you 2x

then double it you it will give you 4x

plus the last term here is the square of

your b

so our b is 2 so the square of 2 is 4

thus giving us this trinomial x squared

plus 4x

plus 4. that's it very good

the next is cube of a binomial this

would be the next on our list for the

special products

how do we get this now let's try to

recall the pattern

so we can get the value of this one the

cube of our a plus b

by cubing the first term that's a cubed

plus three times

the square of the first term

multiplied to the second term so that's

three a squared

times b followed by

three and then copy a

then square the second term b then our

last term for this polynomial is the

result is

more than three so it's not a trinomial

unlike the previous

example so here is a polynomial with

four terms

so we are final last or our final term

or the last term is

the cube of the second term so we have b

cube

so this is now the pattern for cubing a

binomial

so the next part and the last pattern

that we are going to tackle and review

is sum and difference of two squares so

let's have here

a plus b the quantity of a plus b

multiplied to the quantity of a

minus b so how do we get the product

it's a special product

the pattern is square the first term

like this a squared

minus square the second term that's it

you will get the square of b so the

answer is minus b

squared like that so that will be the

pattern for this

special product let's have an example

let's say you're given the quantity of x

plus 3

and you're going to multiply that to the

quantity of x

minus 3. now it follows the same pattern

as what we have earlier

now how do we get the answer for this we

are going to

square the first term giving us x

squared like this

minus the square of the second term

which

is three so that will give us nine so

the answer here

is x squared minus nine that's correct

very good i hope we have now

reviewed everything for our special

products

so what's new for us now on the screen

right now you can see here

five or six or six questions your task

is to get the product of the following

polynomials

i've asked you earlier to prepare your

pen and paper

so this time i want you to get that and

try to answer and get the product of the

following

polynomial shown on the screen now to

help you

out i prepared here the table

that summarizes our special products

and the result using or the patterns to

get the result

so do not forget to pause this video and

take your time

you have their six items okay are we

clear

again pause the video and you can play

it once you are ready to confirm your

answers

now let's do this you can pause the

video now

okay are you done all right

let's have guide questions first i know

you have already the answers on your

notebook or on your paper

so based on the answers before revealing

the complete answer the solutions

now what have you observed with each

resulting product

did it have the same number of terms and

which of the following products have the

same number of terms

now let's answer that by showing to

everyone

the solution in this table form

so this is the result showing this table

with three columns

wherein our first column shows the given

the method for the second the method and

how to get the answer and finally on our

last column it shows the result

now what do you observe

do all the results show the same number

of terms

observe that there are answers inside

the red box

what are the similarities good

so these terms or i mean these answers

inside the red box shows

three terms right the result is a

trinomial

while on number four we have here a

binomial and we have a multinomial for

number six

now which of the products have the same

number of terms that will be numbers one

two three and number five

number four is a binomial and number six

is a multinomial

i hope that's clear now this time what

we need to do is to focus

on the results inside the red box now

let's focus our attention for that four

results

now observe that this one shows the

following components look at the first

part

the first term of the trinomials so what

do you observe

it follows ax squared

wherein your first term is considered as

a quadratic term

the highest degree is true of your

variable so

here the highest degree for the first

term of the first answer is y

squared that's y with the variable y

with

the degree of 2. same with the second

third and fourth they differ with a

variable

second is r third variable is x

and here for our last example the

variable is z

now what do you mean by a a here

represents the numerical coefficient

of your x squared so here our y squared

has the numerical coefficient of one

so that will be a is equal to one same

here for the second one

pointed by my laser arrow my laser

pointer

so i have here 1 4a then here

the numerical coefficient of x squared

it's two therefore your a

is equal to two and for the last example

our a here is one okay

that's clear now i hope that's clear

so this time let's focus on the second

set

that means we are going to look at the

second term or

each of the second term of our examples

now it follows this format

it's bx where your b is the coefficient

of your x that's your second term and

it's a linear term

since the degree of this will be one

okay

so here our bx is 7y the value of b

is 7 with a variable y in the second

example

it's minus r so our b is take note it's

negative 1 since it has here negative so

again the value of our b

is negative one with a variable

r the third example here is negative

five x

so our b is negative five with a

variable x

and our final example here is 8z

our b is correct that's eight

and with the variable z so again this is

our second term

and it's a linear term meaning the

highest degree

is one okay now let's move on with our

third

set that is looking at the last terms

so here we are going to have c what does

it mean when we have c

now it's our third term and c represents

the constant number or the constant of

our expression

so here in this expression

it's in this expression look at this

this will be our value for c

that's negative 12 here it's still the

same

negative 12 and in the third example

our constant is positive 2 and in the

final example our letter c

or constant is positive 16.

so these are the three properties or i

mean the properties that we have to make

it a quadratic equation

take note that it has the first term

which is a quadratic term

the middle term or the second term as a

linear term

and we have here the last term of our

trinomial which is our constant

represented by letter c okay

speaking of quadratic equation now what

do we mean by that

hmm quadratic equation

is

it's a polynomial equation in one

variable

expressed in the form

ax squared plus bx plus c is equal to

zero

take note that we already have earlier

the three com the three properties being

mentioned that it consists of the

quadratic term

here it is your ax squared your middle

term which is your bx which

is a linear term and our last term here

is letter c which is our constant

take note a b and c are

real numbers a b and c

are real numbers these are numerical

coefficients of the different parts of

our quadratic equations

letter c by the way is the constant and

another thing that you need to remember

that in this quadratic equation

the value of our a must not be

equal to zero

okay now this quadratic equation

with the first term as ax squared

notice that our exponent there is two

that's why quadratic equation is also

termed as

a second degree equation again because

the highest degree

is 2.

now follow before going back to example

number four remember

that we have here m squared minus 36

using the special product for special

and so for sum and difference of two

binomials

now earlier we mentioned that there are

three properties right the three

parts of your quadratic equation now how

about for m squared minus 36

is that considered as a quadratic

equation what do you think

the answer for that is yes it's still a

quadratic equation

why because it has the first term which

is a quadratic term

wherein your first term here is m

squared

you do have your middle term but the

value of your b

is equal to zero so that will now be

zero m

and your constant is negative 36. so

again for clarification

we have here these values for our a

which is one

our b equal to zero and our letter c

equals negative thirty six

do you understand well that's great

and now i want you to try this

identify which of the following

equations are quadratic

keep in mind that this is the form of

your quadratic equation

with its first term as a quadratic term

its second term as is a linear term

and your last term here is a constant

now let's write these items

number one x squared plus 5x

is equal to 24. second

eight x squared minus eight is equal to

zero

third item is x minus three y

squared is equal to zero number four

square root of x plus four is equal to

seven

and our last item number five is

x over x plus two is equal to one over

x so we have five items you may pause

this video

and identify carefully if these items

are considered as quadratic equations

take your time again pause and take your

time

go ahead

you're done now this time let's reveal

the answers

so here let's confirm if what you have

identified earlier

is that is true or not so let's confirm

if these are quadratic equations or not

so we have here the five items right so

let's have the first one

x squared plus 5x is equal to 24. is

that quadratic

again is that a quadratic equation let's

see

that is a quadratic equation

very good why is this a quadratic

equation even though at the left side

it's only or it only consists of two

terms

and the other one or the right side is

not equal to zero

now let's see it's a quadratic equation

since the given equation is in the

highest degree of two

right here and the terms are complete

now if you're going to use addition

property of equality in both sides

adding negative

24 to the left and right side of the

equation

it will give you x squared plus 5x minus

24

is equal to 0 following the form of your

quadratic equation

and again the degree is to confirming

that it's really a quadratic equation

so if you've got it right tap your

shoulders and said

and say good job so that's it good job

for the first item now how about for the

second item

eight x squared minus eight is equal to

zero

is that a quadratic equation or not that

is

still the quadratic equation

why since it contains the quadratic term

similar to example number four that we

have explained

we've explained earlier even though it

only consists

of the first and the last term but still

it's a quadratic equation

since it contains the quadratic term and

the middle term here is

zero x where your b is zero and your

variable is still x

okay do you understand

good now here how about for the third

item

that is not a quadratic equation why

because it contains two variables

you all have here take note the first

term is x and your second term is y

even though the highest degree is two

but take note that for your first

this would be your first term right

because our arrangement here is from

highest degree

to the degree and and then following the

decreasing order

so highest degree first term second

but take note the variables are not the

same so it must have the same variable

to consider it as a quadratic equation

with the highest degree of two

that's why this number three or this

item is not a quadratic equation

how about for number four let's confirm

your answer

this is still not a quadratic equation

because there is no quadratic term

rather it contains

the rational exponent which is one half

for this square root of x now how about

for item number five

do you think this is a quadratic what do

you think is this a quadratic equation

or not a quadratic equation

let's reveal the answer is

this is a quadratic equation because

using proportion it could give us the

equation

x squared minus x minus 2 is equal to

zero

following the form of your quadratic

equation

with the first term as the quadratic

term the middle term

and your last term which is a constant

so that's it one two and three are

quadratic equations

while number three while numbers three

and four

are not quadratic equations so i hope

this one is clear now and you've done a

very good job if you're able to have

all the answers correctly fantastic work

grade nine student well done

now this time let's have the second part

of your task

and that is this one looking at our five

equations

your tasks here are as follows number

one

you're going to write the following in

standard form

using this this is the standard form of

your quadratic equation i'll be using my

laser pointer for clarification

follow this right now here this one

pointing

point i'm pointing it right now so this

one second task is

you're going to identify the quadratic

the linear and the constant term of each

equation

remember the first the second or the

middle term and the last term

of your trinomial and your last task is

you will identify the values of a

b and c so again you have three tasks

the first one is to write it in standard

form

second one is to identify the quadratic

linear and constant term

and the last one is you're going to

identify the values of

a b and c okay

so i hope that's clear now so this is

now

our solution in this table form so let's

try to have a look on our first item

so right here i'll be using my laser

pointer so that you can follow

so this was the given and using

symmetric property of equality then we

can have this

equation now from this we can get our

quadratic term which is 9x squared

our linear term which is negative 9x and

our constant is negative 28 with these

values of a b

and c okay so how about for the second

item

so inside this red box right here so

this was the given

and it's already in standard form of our

quadratic equation

so it's easier for us to determine the

different parts

so 7x squared is our quadratic term

followed by negative 6x as our linear

term

and our constant will be negative 20

with these values of our a b

and c okay that's good

now moving on our third item is this

now observe that there are only two

terms unlike our previous examples where

in the middle term is missing or there's

no value for our middle term

but right now we do have our middle term

so from here we can have x squared as

our quadratic term

our linear term is a negative 3x but

this time our constant is zero

so with these values of a b and c

okay now for our fourth one this fourth

item

here we still have negative four need

that needs to be multiplied to the

quantity of 2 plus x

so based on the given we need to apply

our different properties of equality

like distributive property of equality

and addition property of equality

to have this line of our solution which

is now

x squared minus 8 that is coming from

here

minus 4 x that is still coming from

negative 4 times

x minus 13 that's using the addition

property of equality

so we have this line after that we are

going to add like terms

thus our answer here is x squared

minus 4x minus 21 is equal to zero

this will now be our basis to get our

components or different parts of our

our terms so this would be our quadratic

term x squared

negative 4x is our middle term and our

constant as negative 21. so with these

values of a

b and c now finally our last item here

is this one

now same procedure for going back sorry

going back right here

so we have your last item for number

five so same procedure for number four

wherein we are going to apply different

properties of equality to end up with

our quadratic equation

in this standard form so we need to use

distributive property of equality

at the same time we will be using

addition property of equality and

symmetric property of equality

so here we have 6 minus

6x coming from here plus 2

then copy the right side of the equation

now

what we did in this part is we have

addition property of equality

thus we have everything on the right

side of our equation

so we have here 3x squared plus 6x minus

17

minus 2 minus 6. now combining like

terms we'll have 3x squared plus 6x

minus 25.

now to make it in standard form you'll

be using symmetric property of equality

thus we have 3x squared plus 6x minus 25

is equal to 0.

so it's easier for us now to identify

the quadratic term which is 3x squared

our

linear term 6x and our constant term

negative 25

with these values of a b and c

so that's it that those are the answers

for our five items for

the different quadratic equations so i

hope you're able to get it

so again you can go back to that part of

the video that you find

that you just need clarifications or you

can revisit for to attain mastery

so what's more i prepared here a seat

work for you

for you to work on to assess if you've

learned our topic

or to assess your understanding on our

topic

for today's for today's lesson so we

have here two parts

the first part here is you are going to

tell whether the equation is quadratic

or not

and you are going to justify your answer

similar to what we did on our first part

for the example

first set of examples so for the second

set

we have you write the formula or write

the following in standard form

and identify values for a b and c like

what we did for the second

set of examples so we have here five

items for letter b

and same we have five items per letter a

so you can pause this video copy the

given and work on it on your paper

and using your paper and your pen and

just take your time

and submit your answers on the agreed

date with your teacher

are we good with that okay

pause this video and copy the given for

both a

and b and take your time in answering do

not forget that you need to submit your

answers

to your teacher on your agreed date

so what you need to remember quadratic

equation

is a polynomial equation in one variable

that is expressed in the form of

a x squared plus b x plus c is equal to

zero

where your a for the values of a b and c

are real numbers such that your a is not

equal to zero

another important thing that you need to

remember is or another important point

is that equations can still be

considered quadratic

even if it has only two terms

if it contains the quadratic term a x

squared like our examples like the

examples that we had earlier

so that's it you have completed our

lesson

for today congratulations to you and see

you

in the next lesson