This document discusses algebraic expressions and identities. It defines an algebraic expression as an expression made up of variables and constants along with algebraic operations. Expressions are made up of terms. It discusses terms, factors, coefficients, like and unlike terms, and addition and subtraction of algebraic expressions. It also discusses multiplication of algebraic expressions and what an identity is in algebra. Several examples are provided of using identities to simplify or rearrange algebraic expressions, including expanding expressions using formulas like (a + b)2 = a2 + 2ab + b2.

1. CHAPTER-9
ALGEBRAIC EXPRESSIONS
AND
IDENTITIES

2. What are Expressions?
An algebraic expression is an expression that is made up of
variables and constants, along with algebraic operations (like
subtraction, addition, multiplication, etc.). Expressions are
made up of terms.
Example: 5x+7, 6-8x.

3. Terms, factors and coefficients

5. LIKE AND UNLIKE TERMS

6. ADDITION AND SUBTRACTION OF ALGEBRAIC EXPRESSIONS

11. MULTIPLICATION OF ALGEBRAIC EXPRESSIONS

20. WHAT IS AN IDENTITY IN ALGEBRAIC EXPRESSION?
An identity is an equality that holds true regardless
of the values chosen for its variables. They are used
in simplifying or rearranging algebra expressions. By
definition, the two sides of an identity are
interchangeable, so we can replace one with the other
at any time.

22. 1)Find the product of (x + 1)(x + 1) using standard algebraic identities.
Solution: (x + 1)(x + 1) can be written as (x + 1)2. Thus, it is of the form
Identity I where a = x and b = 1. So we have,
(x + 1)2 = (x)2 + 2(x)(1) + (1)2 = x2 + 2x + 1
2)Solve (3x + 5)2 using algebraic identities.
Solution: We know, by algebraic identity number 1,we can write the given
expression as;
(3x + 5)2 = (3x)2 + 2*3x*5 + 52
(3x + 5)2 = 9x2 + 30x + 25

23. Example 2: Simplify ( 7x +4y )2 + ( 7x - 4y )2
Solution: To solve this, we need to use the following algebraic
identities:
(a + b)2 = a2 + 2ab + b2
(a - b)2 = a2 - 2ab + b2
Adding the above two formulas we have:
(a + b)2 + (a - b)2 = a2 + 2ab + b2 + a2 - 2ab + b2
(a + b)2 + (a - b)2 = 2a2 +2 b2
Here we have a = 7x and b = 4y. Substituting this in the above
expression we have:
( 7x +4y )2 + ( 7x - 4y )2 = 2(7x)2 + 2(4y)2
= 98x2 + 32y2
Answer: (7x + 4y)2 + (7x - 4y)2 = 98x2 + 32y2

24. Question 1) Find the product of (x-1) (x-1)
Solution) We need to find the product (x-1) (x-1),
(x-1) (x-1) can also be written as (x-1)2.
We know the formula for (x-1)2, expand it
(a-b)2 = a2- 2ab+b2 where a= x, b=1
(x-1)2 = x2- 2x+1
Therefore, the product of (x-1) (x-1) is x2- 2x+1
Question 2) Find the product of (x+1) (x+1)
Solution) We need to find the product (x+1) (x+1),
(x+1) (x+1) can also be written as (x+1)2.
We know the formula for (x+1)2, expand it
(a+b)2 = a2+ 2ab+b2 where a= x, b=1
(x+1)2 = x2+ 2x+1
Question 3) Solve the following (x+2)2 using the concept of
identities.
Solution) According to the identities and algebraic expression
class 8,
We know the formula,
(a+b)2 = a2+2ab+b2
Where, a= x, b= 2
Let’s expand the given (x+2)2,
Therefore, (x+2)2 = x2+4x+4 is the solution.

25. Question 1: Find the product of (x + 2)(x + 2) using standard algebraic identities.
Solution: We can write (x + 2)(x + 2) as (x + 2)2. We know that (a + b)2 = a2 + b2 +
2ab.
So putting the value of a = x and b = 2, we get
(x + 2)2 = x2 + 22 + 2.2.x
= x2 + 4 + 4x
Question 2: Find the value of (x + 6)(x + 6) using algebraic
identities
Solution:(x+6)(x+6) can be re-written as (x + 6)2.
It can be rewritten in this form, (a + b)2 = a2 + b2 + 2ab.
(x + 6)2 = x2 + 62 + 2(6x)
= x2 + 36 + 12x

26. 1) Expand (5x – 3y)2.
Solution:
This is similar to expanding (a – b)2 = a2 + b2 – 2ab.
where a = 5x and b = 3y,
So (5x – 3y)2 = (5x)2 + (3y)2 – 2(5x)(3y)
= 25x2 + 9y2 – 30xy
2) Factorize (x6 – 1) using the identities
Solution:
(x6 – 1) can be written as (x3)2 – 12.
This resembles the identity a2 – b2 = (a + b)(a – b).
where a = x3, and b = 1.
So, x6 – 1 = (x3)2 – 1 = (x3 + 1) (x3 – 1).

31. 2) Solve the following using the standard identity: a 2 - b
2 = (a+b) (a-b)
A) 88 2 - 12 2 B) 89 2 - 11 2 C) 986 2 - 14 2 D) 997
2 - 3 2
1) Solve (2x + 3) (2x – 3) using algebraic identities.
Solution: By the algebraic identity number 3, we can write the
given expression as;
(2x + 3) (2x – 3) = (2x)2 – (3)2 = 4x2 – 9
Example 1: Using identities, solve 297 × 303.
Solution: 297 × 303 can be written as ( 300 - 3 ) × ( 300 + 3 )
And this is based on the algebraic identity (a + b)(a - b) = a2 - b2
Here we have a = 300, and b = 3
Substituting the values in the above identity, we get:
(300 - 3)(300 + 3) = 3002 - 32
= 90000 - 9
= 89991
Answer: Therefore 297 × 303 = 89991

32. EX: 9.5 Q.4,5 & 6 Simplify ( 7x +4y )2 + ( 7x - 4y )2
6th December, 2021

33. H.W- 6th Dec, 2021
Find the value of the following using suitable algebraic
identities:
1) (a-6)2
2) (3m+9)2
3) (c-d)(c+d)
4) (82-32)
5) (f+5g)(f+5g)
6) (q-r)(q-r)
7) (x3)2-1
8) (4y2-6x)2
9) (5/7t+3/4m)2
10)(0.2x-0.9z)2
11)(60-50)2
12)452
13)98x102
14)(36)2-(25)2
15)(x/2+y/3)2