Introduction to Algebraic Formulas for Class 9
Algebra is just like assigning values to variables and also understanding relationships between values in a given statement. Algebra employs four mathematical operations, such as addition, subtraction, multiplication, and division.
How Are Maths Identities Helpful for Class 9 Students?
Maths identities for Class 9 usually are used to make complex calculations simpler without using automated electronic calculators. There are many maths identities that Class 9 students can use to lessen their efforts in complex mathematical problems. The 9th class algebra has a huge list of algebraic identities for Class 9 students to make use of. Most of the algebraic identities are used in evaluating polynomial equations and expressions. Algebraic identities for Class 9 maths can also be used to find the product of larger numbers and also their squares.
Algebraic Identities for Class 9
Algebraic identities for Class 9 is 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 generalised identities that can be used in many computations.
Identities of Maths Class 9 with Proofs:
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
Identities of Maths Class 9 Problems
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 - 2 x 100 x 1 + 12
= 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 + 2ab + b2
(100 + 1)2 = 1002 + 2 x 100 x 1 + 12
= 10000 + 200 + 1
= 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.
FAQs on Algebraic Formulas for Class 9 – Basic Concepts and Identities
1. What are the algebraic identities of Class 9? Why are they used?
Algebraic identities are the equalities that are eternally true irrespective of the true values chosen by the user. These identities use variables that can take any values. They are the generalised equality statements. Algebraic identities are used in the factorisation 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 methods.
2. Distinguish between algebraic expressions and algebraic identities.
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.
