FACTORING DIFFERENCE OF TWO SQUARES || GRADE 8 MATHEMATICS Q1

Summary

This educational video teaches viewers how to identify perfect squares and the concept of a difference of two squares in algebra. It demonstrates the process of factoring such expressions by providing step-by-step examples, including rewriting perfect squares in exponential form and applying the formula for factoring differences of squares. The video aims to clarify these mathematical concepts with clear explanations and practical demonstrations, ensuring that viewers can apply these techniques in their own problem-solving.

Takeaways

• 📚 The video aims to teach viewers how to identify if an expression is a difference of two squares and how to factor it completely.
• 🔍 It starts by explaining the concept of perfect squares and how to determine if a number is a perfect square by multiplying a number by itself.
• 📉 The script provides examples of numbers that are and are not perfect squares, such as 25, 81, 144, and 100 being perfect squares, while others like 4 and 88 are not.
• 📝 Perfect squares are then rewritten in exponential form, for instance, 25 is rewritten as 5 squared (5^2).
• 📈 The script introduces the formula for a difference of two squares as x^2 - y^2 and explains it can be factored into (x + y)(x - y).
• 📚 It clarifies that the difference of two squares is the product of the sum and difference of the terms x and y.
• 📝 The video provides a step-by-step method to factor expressions like x^2 - y^2 by identifying and using the square roots of x and y.
• 🔢 Examples are given to demonstrate the factoring process, such as 9x^2 - 100, which is factored into (3x + 10)(3x - 10).
• 📉 The video also covers how to handle expressions with variables and exponents, like m^2 - 4, which is factored into (3m + 2)(3m - 2).
• ❗ It emphasizes the importance of correctly identifying the square roots and the sum and difference of the terms for accurate factoring.
• 👍 The video concludes by encouraging viewers to like, subscribe, and hit the bell button for more educational content.

Q & A

What is the main topic of the video?

The main topic of the video is to identify whether an expression is a difference of two squares and to factor the difference of two squares completely.

What is a perfect square and how can you identify it?

A perfect square is a number that can be expressed as the product of an integer with itself. You can identify a perfect square by checking if the square root of the number is an integer.

How is the number 25 expressed in exponential form?

The number 25 is expressed in exponential form as 5 squared, or 5^2, because 5 multiplied by 5 equals 25.

What is the difference of two squares in algebraic terms?

In algebraic terms, the difference of two squares is expressed as x^2 - y^2, where x and y are real numbers or algebraic expressions.

What is the factored form of the difference of two squares?

The factored form of the difference of two squares is the product of the sum and difference of the terms, which is (x + y)(x - y).

Can you provide an example of factoring a difference of two squares from the video?

An example from the video is factoring 9x^2 - 100, which can be rewritten as (3x)^2 - 10^2 and then factored into 3x + 10 and 3x - 10.

What is the process for rewriting a perfect square in exponential form?

The process involves taking the square root of the perfect square to find the base number and then expressing it as that base number squared, or using the exponent 2.

How does the video demonstrate the factoring of non-perfect squares?

The video demonstrates that non-perfect squares cannot be factored as a difference of two squares because they do not have an integer square root.

What is the factored form of the expression 4x^2 - 81 according to the video?

The factored form of the expression 4x^2 - 81 is (2x + 9)(2x - 9), after rewriting it as a difference of two squares.

Can the expression -49x^8 + 25 be factored as a difference of two squares?

Yes, the expression -49x^8 + 25 can be factored as a difference of two squares, resulting in (5 - 7x^4)(5 + 7x^4).

What is the importance of recognizing a difference of two squares in algebra?

Recognizing a difference of two squares is important in algebra because it allows for the simplification and factoring of expressions, which can be useful in solving equations and other mathematical problems.

Outlines

00:00

📚 Introduction to Difference of Squares

In this segment, Hyman Akiyama introduces the concept of identifying perfect squares and the difference of two squares. He explains that a perfect square is a number that can be expressed as the product of an integer with itself. Akiyama demonstrates by listing numbers and determining whether they are perfect squares, showing the calculation for each. He then transitions into rewriting perfect squares in exponential form, setting the stage for discussing the difference of two squares, which is expressed as \( x^2 - y^2 \). Akiyama emphasizes the importance of recognizing perfect squares to factor expressions completely and introduces the formula for factoring the difference of two squares as \( (x + y)(x - y) \).

05:02

🔍 Factoring Examples of Difference of Squares

This paragraph delves into practical examples of factoring expressions that represent the difference of two squares. Akiyama begins by rewriting non-perfect squares as part of the difference of two squares formula. He then systematically factors several expressions, such as \( 9x^2 - 100 \), \( 4x^2 - 81 \), and \( 81m^2 - 4 \), by identifying the square roots and applying the sum and difference method. Akiyama also addresses an expression that is not a difference of two squares, \( -49x^8 + 25 \), and shows how to rewrite it appropriately before factoring. The summary concludes with the factored forms of the given examples, illustrating the process of identifying and utilizing perfect squares to simplify algebraic expressions.

Keywords

💡Perfect Square

A perfect square is a number that can be expressed as the product of an integer with itself. In the context of the video, identifying perfect squares is crucial for recognizing and factoring expressions that represent the difference of two squares. For instance, the script mentions that 25 is a perfect square because it is 5 times 5.

💡Difference of Two Squares

This term refers to an algebraic expression of the form x^2 - y^2, where x and y are real numbers or algebraic expressions. The video's main theme revolves around identifying such expressions and factoring them into the product of two binomials, (x + y)(x - y). An example from the script is 9x^2 - 100, which is factored into (3x + 10)(3x - 10).

💡Factoring

Factoring is the process of breaking down a complex expression into a product of simpler, more fundamental expressions. The video demonstrates how to factor the difference of two squares completely, which is a specific method of factoring. The script shows the factoring of expressions like 4x^2 - 81 into (2x + 9)(2x - 9).

💡Exponential Form

Exponential form, or notation, is a way of expressing a number as a power of another number. In the script, perfect squares are rewritten in exponential form, such as 25 being rewritten as 5^2, indicating that 5 is squared.

💡Square Root

The square root of a number is a value that, when multiplied by itself, gives the original number. The script uses square roots to identify perfect squares and to rewrite them in exponential form. For example, the square root of 81 is 9, so 81 is expressed as 9^2.

💡Algebraic Expressions

Algebraic expressions are mathematical phrases that contain variables, numbers, and operation signs. In the video, x and y are used as variables in algebraic expressions to demonstrate the difference of two squares and the process of factoring.

💡Binomial

A binomial is an algebraic expression with two terms, typically in the form of (ax + b). The video explains that the factored form of a difference of two squares results in the product of two binomials, such as (x + y) and (x - y).

💡Sum and Difference

The terms 'sum' and 'difference' refer to the addition and subtraction of two numbers, respectively. In the context of factoring the difference of two squares, the script mentions the sum and difference of x and y, which are used to form the binomials in the factored expression.

💡Variable

A variable is a symbol, often a letter, that represents an unknown or variable quantity in an equation or expression. In the video, x and y are used as variables in the algebraic expressions that are factored as the difference of two squares.

💡Non-Perfect Square

A non-perfect square is a number that cannot be expressed as the product of an integer with itself. The script uses non-perfect squares to illustrate numbers that do not fit the criteria for the difference of two squares, such as the number 49 in the expression -49x^8 + 25.

💡Exponent

An exponent indicates the number of times a base number is multiplied by itself. In the script, exponents are used to express perfect squares in exponential form and to demonstrate the process of rewriting expressions like m^4 - 4 as a difference of two squares.

Highlights

Introduction to identifying expressions as a difference of two squares and factoring them completely.

Explanation of perfect squares and how to identify them.

Demonstration of determining if numbers like 25, 81, 144, and 121 are perfect squares.

Rewriting perfect squares in exponential form for clarity.

Definition of the difference of two squares in algebraic terms.

Illustration of the formula for the difference of two squares: x^2 - y^2 = (x + y)(x - y).

Process of factoring the difference of two squares with an example.

Example of factoring 9x^2 - 100 into (3x + 10)(3x - 10).

Another example of factoring 4x^2 - 81 into (2x + 9)(2x - 9).

Factoring a more complex expression involving exponents: 81m^2 - 4^4.

Handling negative expressions and factoring -49x^8 + 25.

Clarification on when an expression is not a difference of two squares.

The importance of recognizing the sum and difference in the factoring process.

Final example showcasing the factored form of a complex expression.

Encouragement to like, subscribe, and hit the bell for more educational content.

The video concludes with a summary of the key points covered.

Transcripts

[Music]

Hyman Akiyama in this video we will

identify whether or not an expression is

a difference of two squares and we will

factor the difference of two squares

completely so first let us tell whether

the following number is a perfect square

or not so identify munna net income it

don't wanna numbers Bennetto a perfect

square I know about boxing having

perfect square and two umaña product

pneumonia number nami multiply more by

itself

okay let's start 25 this is a perfect

square because 5 times 5 75 this is not

a perfect square each one this is a

perfect square because of nine times

nine forty this is not a perfect square

144 this is a perfect square because of

12 times 12 and then 60 this is not a

perfect square seven 100 because of ten

times ten it is a perfect square

64 this is also a perfect square because

of eight times eight 88 this is not a

perfect square and 121 this is a perfect

square because of eleven times eleven

okay next let us try to rewrite each

perfect squares in exponential form or

notation so Marana Homam a perfect

squares neoneun rewrite not inside into

exponential form an open attention is a

solid numero exponent okay so since 25

is 5 times 5 so we can rewrite this as 5

squared okay Kayla Manhattan to Paris a

difference of two squares so DARPA and

I'm not income

no I'm square root yeah so the square

root of 25 is 5 that's why it's 5

squared next since this is 9 times 9 the

square root of 81 is 9 so 9 squared 144

the square root of 144 is 12 so that is

12 squared the square root of 100 is 10

so that is 10 squared 64 is 8 times 8 so

8 squared 121 is 11 times 11

so that is 11 squared so we need to know

how to rewrite or a nappy not income I

know you square root no mana perfect

squares NATO basic a laminate inches a

difference of two squares okay

what is difference of two squares if x

and y are real numbers variables or

algebraic expressions then we will have

x squared minus y squared so the

difference of two squares is the product

of sum and difference of those terms

Inaba ito button the end and difference

of two squares now x squared minus y

squared a to us our goal or Prada kapag

maroon tie on sum and difference a new

hobby on so adam x squared minus y

squared a - ASA goat or Prada topic

Marin tae-young X plus y times X minus y

comma Poppins in uu is assembling Shyam

is a difference okay so on your bottom x

squared back it a toe and prata

u x squared minus y squared because x

times X is x squared ya know Maggie you

first term Nathan y times y that is y

squared Janna Maggie Maggie Ginn's

second term Nathan now bucket difference

because it is sum and difference so

positive times negative that is negative

kaya dunes

squared minus y squared minus ax okay

now it don't X plus y at X minus y a toy

anti-natal magnet a factoid form

so since on gagawin that in a on a mug

for factoring given an difference of two

squares

epilim and piling and Addington the end

so again parramatta Yoda but NASA is

shannon x squared minus y squared let's

have an example if we have 9x squared

minus 100 since 9 is a perfect square

and 100 so we can factoid is 2 so we

will have let us first rewrite as the

difference of two squares so a guy in

Agena why not in canina a non-perfect

square new 9 we have 3 and then x

squared that is X and um square root new

100 that is then ok and then we will

just copy the first and second term so

our first term you know Gilligan

attendance at 11 binomial so since

Nahanni 10 C 3 X you know in a lagina 10

doing sir some indifference and then

then for our second term and then do not

forget the plus and minus that bad

cement differentia so we will know how

that factored form which is 3x Plus 10

times 3x minus 10 another example 4x

squared minus 81 rewrite as difference

of two squares so we have an analogy not

enzyme and knowing square with me for

that is 2x

since Marin time x squared so that I get

nothing on X and then a non square root

81 that is nine okay so we will now copy

the first and second term

so your first term and your second term

so we now have 2x plus 9 into X minus 19

don't forget the sum and difference so

we now have the factored form of 2x plus

9 times 2x minus 9 it's so easy next 81m

squared minus 4 and raise to 4 P raise

to 6 so we will rewrite this this first

as the difference of two squares so what

is the square root of 81 that is 9 and

then since my M Squared is so we will

write M what is the square root of 4

that is 2 now Buffett's ax n raised to 2

because 2 times 2 is raised to 4 yay and

then 3 times 2 that is P raise to 6

that's why it's n raise to 2 P raise to

3 take note of your exponent

hey so let us now copy the first and

second term our first term will be 9m

our second term will be 2 and raise to 2

P raise to 3 so you will just copy don't

forget the sum and difference so our

factored form is 9m plus 2 and raise to

2 P raise to 3 x 9m minus 2 and raise to

P raise to 3 it's always the sum and

difference next I have here negative 49

X raise to 8 plus 25 so

subbing a nap in the but now a

difference of two squares munna

but as you can see and Mahalo gave ito i

nos so it's not the difference of two

squares so rewrite Moonen attention as

difference of two squares therefore we

will have 25 minus 49 x raised to a near

a arranged lung attention so lily pad

lugnut

25 and negative 49 so we now have the

difference of two squares

take note had a pad - ax Bogota yarmulke

Thor

okay so what is the square root of 25

that is 5 the square root of 49 is 7 now

back it's a X raise to 4 because 2 times

4 is 8

so let's now copy the first and second

term we have what is our first term

fight

what is our second term we have 7 X

raise to 4 so we now have the factored

form of 5 + 7 X raise to 4 times 5 minus

7 X raise to 4 take note of the sum and

difference thank you for watching this

video I hope you learned something don't

forget to Like subscribe and hit the

bell button so our Walmart channel just

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