FACTORING DIFFERENCE OF TWO SQUARES || GRADE 8 MATHEMATICS Q1
Summary
TLDRThis 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
📚 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 \( (x + y)(x - y) \).
🔍 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.
Mindmap
Keywords
💡Perfect Square
💡Difference of Two Squares
💡Factoring
💡Exponential Form
💡Square Root
💡Algebraic Expressions
💡Binomial
💡Sum and Difference
💡Variable
💡Non-Perfect Square
💡Exponent
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
keep on watching
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