Algebraic identities for class 9 is an equality that holds good irrespective of the true values chosen for the calculation. It is very useful in many mathematical and scientific computations involving either a very large number or a very small number. Algebraic formulas for class 9 are generally represented in the form of variables that can take any desired value. 9th class algebra has generalized identities that can be used in many computations.

Identity 1:

The square of sum of any two numbers ‘a’ and ‘b’ is given by (a + b)2 = a2 + b2 + 2ab

Proof:

(a + b)2 = (a + b) (a + b) = a2 + ab + ba + b2 = a2 + b2 + 2ab

Algebraic identities can be proved using an activity method.

Identity 2:

The square of the difference of any two numbers ‘a’ and ‘b’ is given by (a - b)2 = a2 + b2 - 2ab

Proof:

(a - b)2 = (a - b) (a - b) = a2 - ab - ba + b2 = a2 + b2 - 2ab

Identity 3:

The product of sum and difference of 2 numbers ‘a’ and ‘b’ is given as (a + b) (a - b) = a2 - b2

Proof:

(a + b) (a - b) = a2 - ab + ab - b2 = a2 - b2

Identity 4:

(x + a) (x + b) = x2 + x (a + b) + ab

Proof:

(x + a) (x + b) = x2 + xb + ax + ab = x2 + x (a + b) + ab

Identity 5:

(x - a) (x + b) = x2 + x (b - a) - ab

Proof:

(x - a) (x + b) = x2 + xb - ax - ab = x2 + x (b - a) - ab

Identity 6:

(x - a) (x - b) = x2 - x (a + b) + ab

Proof:

(x - a) (x - b) = x2 - xb - ax + ab = x2 - x (a + b) + ab

Identity 7:

The cube of sum of any two numbers is given by (a + b)3 = a3 + b3 + 3ab (a + b)

Proof:

(a + b)3

= (a + b) (a + b)2

= (a + b) (a2 + b2 + 2ab)

= a3 + ab2 + 2a2b + ba2 + b3 + 2ab2

= a3 + b3 + 3a2b + 3ab2

= a3 + b3 + 3ab (a + b)

Identity 8:

The cube of difference of any two numbers is given by (a - b)3 = a3 - b3 - 3ab (a - b)

Proof:

(a - b)3

= (a - b) (a - b)2

= (a - b) (a2 + b2 - 2ab)

= a3 + ab2 - 2a2b - ba2 - b3 + 2ab2

= a3 - b3 - 3a2b + 3ab2

= a3 - b3 - 3ab (a - b)

Identity 9:

The square of sum of three numbers ‘a’, ‘b’ and ‘c’ is given by

(a + b + c)2 = a2 + b2 + c2 + 2ab +2bc + 2ca

Proof:

(a + b + c)2

= (a + b + c) (a + b + c)

= a2 + ab + ac + ba + b2 + bc + ca + cb + c2

= a2 + b2 + c2 + 2ab +2bc + 2ca

1. Evaluate the square of 99 and 101 using appropriate algebraic identity.

Solution:

992 = (100 - 1)2 and 1012 = (100 + 1)2

992 can be solved using the algebraic identity (a - b)2 = a2 + b2 - 2ab.

a = 100 and b = 1

(a - b)2 = a2 + b2 - 2ab

(100 - 1)2 = 1002 + 12 - 2 x 100 x 1

= 10000 + 1 - 200

= 9801

1012 can be solved using the algebraic identity (a + b)2 = a2 + b2 + 2ab.

a = 100 and b = 1

(a + b)2 = a2 + b2 + 2ab

(100 + 1)2 = 1002 + 12 + 2 x 100 x 1

= 10000 + 1 + 200

= 10201

2. If the sum of two numbers is 12 and the sum of their cubes is 468. Find the product of these numbers using algebraic identities.

Solution:

Let us consider the two numbers as ‘a’ and ‘b’.

Sum of the numbers = a + b = 12

Sum of their cubes = a3 + b3 = 468

Product of these two numbers = ab =?

The product of two numbers can be found using the algebraic identity:

(a + b)3 = a3 + b3 + 3ab (a + b)

(12)3 = 468 + 3ab (12)

1728 = 468 + 3ab (12)

36 ab = 1728 - 468

36 ab = 1260

ab = 1260 / 36

ab = 35

The product of two numbers is 35.

Which of the following 9th class algebra identities cannot be used to find the product of 75 and 65?

(x - a) (x - b) = x2 - x (a + b) + ab

(a + b) (a - b) = a2 - b2

(a + b)3 = a3 + b3 + 3ab (a + b)

One of the merits of using algebraic identities is

Lowers the complexity of 9th class algebra problem

Facilitates the use of automated calculators

Not suitable for school level students

FAQ (Frequently Asked Questions)

1. What are the Algebraic Identities of Class 9? Why are they Used?

A. Algebraic identities are the equalities that are eternally true irrespective of the true values chosen by the user. Algebraic identities use variables that can take any values. They are the generalized equality statements. Algebraic identities are used in the factorization of polynomial equations. They are also used in finding the squares and cubes of larger numbers such as numbers in several hundreds and thousands and also smaller numbers of few decimal points. Algebraic identities can be proved by theoretical calculations and also by activity method.

2. Distinguish Between Algebraic Expressions and Algebraic Identities.

A. An algebraic expression is a mathematical representation of statements in the form of variables and constants. An expression may not take the same values in all instances. The value of an expression changes with the value of the variable. An algebraic identity is a mathematical representation of statements that can be equated to the other statements. They are general equality statements used to solve various types of algebraic expressions. The value of a particular algebraic identity gives the same equality relationship irrespective of the values assigned for the variable.