Algebra Formula, Types, Expressions, Identities, Operations

Algebra Formula

Algebra Formula: Algebra is a branch of mathematics that deals with number theory, geometry, and analysis. The definition of algebra sometimes states that the study of the mathematical symbols and the rules involves manipulating these mathematical symbols. Algebra Formula includes almost everything right from solving elementary equations to the study of abstractions. Algebra Formula is very important for all competitive exams to solve the maximum number of topics which includes the implications, multiplication, division, subtraction, and addition.

We are providing you the detailed information on the Algebra Formula, algebra expressions, algebraic identities, algebra calculator, and algebra in maths. Read the detailed article for more information related to the formula of Algebra.

Types of Algebra

The algebra is divided into the following parts which are based on the complexity of algebra which is simplified by the use of numerous algebraic expressions. Here are some several types of Algebra:

• Pre-algebra: It helps in transforming real-life problems into an algebraic expression in mathematics.
• Elementary Algebra: Elementary algebra deals with solving the algebraic expressions for a viable answer. In elementary algebra, simple variables like x, and y, are represented in the form of an equation.
• Abstract Algebra: Abstract algebra deals with the use of abstract concepts like groups, rings, and vectors rather than simple mathematical number systems.
• Universal Algebra: All the other mathematical forms involving trigonometry, calculus, and coordinate geometry involving algebraic expressions can be called universal algebra. 

Algebra Expressions

The algebra expression is an expression that is made up of variables and constants, along with algebraic operations (addition, subtraction, etc.). There are three types of algebraic expression i.e.

Monomial Expression

An algebraic expression which is having only one term is known as a monomial.

Examples of monomial expressions include 2x⁴, 4xy, 6x, 8y, etc.

Binomial Expression

A binomial expression is an algebraic expression which is having two terms, which are unlike.

Examples of binomial include 4xy + 9, Xyz + y³, etc.

Polynomial Expression

An expression with more than one term with non-negative integral exponents of a variable is known as a polynomial.

Examples of polynomial expression include ax + by + ca,  2x³ + 3x + 6, etc.

Formula of Algebra

Formula of Algebra: Algebra is a combination of letters and numbers which is used to find any unknown quantity. When the numbers, letters, matrices, and vectors are used in the combined manner then an algebraic equation or expression is formed. In the algebraic expression the number whose values are known and letters whose values are unknown. The letters are used instead of numbers and the formulas are used to find the values of unknown quantities.

Algebra is a very important chapter for all the exams. The students faced some problems with the algebra questions they can get help from this article. Here we are going to mention the list of some important formulas of algebra that helps students to prepare for their exams easily.

• a² – b² = (a-b)(a+b)
• (a+b)² = a² + 2ab + b²
• (a-b)² = a² – 2ab + b²
• a² + b² = (a-b)² +2ab
• (a+b+c)² = a²+b²+c²+2ab+2ac+2bc
• (a-b-c)² = a²+b²+c²-2ab-2ac+2bc
• a³-b³ = (a-b) (a² + ab + b²)
• a³+b³ = (a+b) (a² – ab + b²)
• (a+b)³ = a³+ 3a²b + 3ab² + b³
• (a-b)³ = a³- 3a²b + 3ab² – b³
• “n” is a natural number, a^nd – bⁿ = (a-b) (a^n-1 + a^n-2b +….b^n-2a + b^n-1)
• “n” is an even number, aⁿ + bⁿ = (a+b) (a^n-1 – a^n-2b +….+ b^n-2a – b^n-1)
• “n” is an odd number aⁿ + bⁿ = (a-b) (a^n-1 – a^n-2b +…. – b^n-2a + b^n-1)
• (a^m)(aⁿ) = a^m+n (ab)^m = a^mn

Algebraic Identities

The algebraic equations which satisfy all the values of variables are termed algebraic identities. Algebraic identities should be valid for all the values of variables. Algebraic identities are the math expressions that include numbers, variables (unknown values), and mathematical operations (addition, subtraction, multiplication, and division). The algebraic identities are used in geometry, algebra, trigonometry and other chapters to solve complex questions easily.

Algebraic Operations

There are four algebraic operations that are mostly used for solving and algebraic expressions. Algebra Calculator mainly solves the following operations.

• Addition: In addition operation in algebra, two or more expressions are separated by a plus (+) sign between them.
• Subtraction: In subtraction operation in algebra, two or more expressions are separated by a minus (-) sign between them.
• Multiplication: In the multiplication operation in algebra, two or more expressions are separated by a multiplication (×) sign between them.
• Division: In division operation in algebra, two or more expressions are separated by a “/” sign between them.

Algebra Expressions

The combination of numbers and letters connected by any of the basic mathematical operations like addition (+), subtraction (-), multiplication (×), and division (÷) forms an algebraic expression. The algebraic expressions are used in various applications to find the values of unknown quantities. Examples of algebraic expressions are 4x + 2y + 5, 2y + 3, 2x + 3y + z, etc. Based on the degree of algebraic expressions they are of the following three types:

Algebra Calculator

In Algebra calculator, the algebraic expressions and algebraic equations are used to calculate the values of the given expression. The methodology used to solve the questions is based on the type of questions, several unknowns, several variables, etc. The students can get familiar with practice only to solve the questions based on algebra.

Algebra Questions

The important and useful questions based on algebra are mentioned below. These algebra questions will be helpful for students to ace their preparation for exams and crack the exam with a good score. Here are some examples of Algebra Questions.

1. Evaluate (x-1)² = [4√(x-4)]²

Solution: x²-2x+1 = 16(x-4)

x²-2x+1 = 16x-64

x²-18x+65 = 0

(x-13) (x-5) = 0

Hence, x = 13 and x = 5

2. Solve the expression, 4x + 5 when x = 3.

Solution: Given, 4x + 5

Now putting the value of x=3, we get;

4 (3) + 5 = 12 + 5 = 17

3. There are 47 boys in the class. This is three more than four times the number of girls. How many girls are there in the class?

Solution: Let the number of girls be x

As per the given statement,

4 x + 3 = 47

4x = 47 – 3

x = 44/4

x = 11

4. The sum of two consecutive numbers is 41. What are the numbers?

Solution: Let one of the numbers be x.

Then the other number will x+1

Now, as per the given questions,

x + x + 1 = 41

2x + 1 = 41

2x = 40

x = 20

So, the first number is 20 and the second number is 20+1 = 21

5. A number is increased by 2 and then multiplied by 3. The result is 24. What is this number?

Solution:

Let the number be x:

This number is increased by 2: x + 2

Then, it is multiplied by 3: 3(x + 2)

The result is 24: 3(x + 2) = 24 … Solving this linear equation, we obtain the value of the number:

3(x + 2)/3 = 24 /3   Divide both sides by 3 and cancel  out

x + 2 = 24 / 3

x + 2 = 8

x = 8 – 2   – 2 crosses the equal sign so negative

x = 6

6. Solve (2x+y)²

Solution: Using the identity: (a+b)² = a² + b² + 2 abs, we get;

(2x+y) = (2x)² + y² + 2.2x.y = 4x² + y² + 4xy

7. Solve (99)² using the algebraic identity.

Solution: We can write, 99 = 100 -1

Therefore, (100 – 1 )²

= 100² + 1² – 2 x 100 x 1  [By identity: (a -b)² = a² + b² – 2ab

= 10000 + 1 – 200

= 9801

8. Solve 5 (- 3 x – 2) – (x – 3) = – 4 (4 x + 5) + 13

Solution: 
5 (- 3 x – 2) – (x – 3) = – 4 (4 x + 5) + 13

On simplify
-15 x – 10 – x + 3 = – 16 x – 20 + 13
Grouping the above terms
– 16 x – 7 = – 16 x – 7

Add 16x + 7 to both sides of the equation then the equation will be

0 = 0

The above statement is true for all the values of x and therefore all real numbers can be a solution to the given equation.

9. Solve 2 (a -3) + 4 b – 2 (a – b – 3) + 5
Solution: Given an algebraic expression
2 (a -3) + 4 b – 2 (a – b – 3) + 5
On multiplying factors
= 2 a – 6 + 4 b – 2 a + 2 b + 6 + 5
On grouping the like terms
= 6 b + 5

10. Solve the expression 4x + 2 (3+6) = 0

Solution: Given an algebraic expression

4x + 2 (3+6) = 0

On solving

4x + 18 = 0

4x = -18

x = -18/4 = -4.5

Formula of Algebra: FAQ

Q. What are the 4 main operations of algebra?

Ans: The four main operations of algebra are addition, subtraction, multiplication and division.

Q. What are the Different Branches or Types of Algebra?

Ans: There are five different branches or types of algebra which are elementary algebra, abstract algebra, advanced algebra, commutative algebra, and linear algebra.