What are the 4 main operations of algebra?

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

What are the Different Branches or Types of Algebra?

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

Describe the steps to solve a linear equation and a quadratic equation?

To solve a linear equation, isolate the variable on one side of the equation and perform inverse operations to find the value of the variable. To solve a quadratic equation, use the quadratic formula or factorization methods to find the roots.

What is the difference between a monomial, binomial, and polynomial expression?

A monomial is an algebraic expression with only one term, a binomial has two terms, and a polynomial has more than two terms.

Algebra Formula, Types, Expressions, Identities and Operations

Table of Contents

Algebra is a fundamental branch of mathematics that allows us to represent and solve problems using mathematical expressions. It involves the use of variables, such as x, y, and z, along with various mathematical operations like addition, subtraction, multiplication, and division. The power of algebra lies in its ability to transform real-world situations into mathematical expressions that can be manipulated and solved.

This makes it a crucial skill that is used in many other areas of mathematics, including trigonometry, calculus, and coordinate geometry. One simple example of an algebraic expression is 2x + 4 = 8, where the variable ‘x’ is used to represent an unknown quantity, and the operations are used to create a meaningful mathematical relationship.

Algebra isn’t just for solving equations, though. It’s a way of thinking about and representing mathematical relationships. The key is to focus on understanding the concepts, not just memorizing formulas. Algebra is also an important topic for competitive exams like SSC CGL, Railways, and many other State-level examinations. The article will help you understand the basics 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 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 help 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:

Linear Equations: Linear equations has only one degree of help in representing the relationship between variables such as x, y, and z. In these linear equations, we use algebra, starting from the basics such as the addition and subtraction of algebraic expressions.
Quadratic Equations: A quadratic equation has a degree of two and can be written in the standard form as ax²+ bx + c = 0, where a, b, and c are constants and x is the variable. The values of x that satisfy the equation are called solutions of the equation, and a quadratic equation has at most two solutions.
Cubic Equations: The algebraic equations having variables with power 3 are referred to as cubic equations. A generalized form of a cubic equation is ax³ + bx² + cx + d = 0. A cubic equation has numerous applications in calculus and three-dimensional geometry.

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