Algebraic Equations

An algebraic equation shows the connection between two quantities when one or both of the values are unknown.

Given below are the different types of Algebraic equations, based on the degree of the variable:

1. Linear Equations
2. Quadratic Equations
3. Cubic Equations

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• Algebraic Formula

Linear Equation

A linear equation is an equation in which the highest power of a variable is 1. They are also known as first-order equations.

• A linear equation is an equation for a straight line when seen in the coordinate system.

• Equation of Straight Line is written as y = mx + b, where m denotes the line’s slope and b denotes the y-intercept. 

Below are some of the important topics covered in Linear equations.

1. Standard Algebraic Identities
2. Algebraic expressions
3. Like and Unlike Algebraic Terms
4. Mathematical Operations on Algebraic Expressions
5. Standard Algebraic Identities
6. Factorization
7. Introduction to factorization
8. Division of Algebraic Expressions
9. Linear Equations in One Variable
10. Solve Linear Equations with Variables on Both Sides
11. Solving Equations that have Linear Expressions on one Side and Numbers on the other Side
12. Reducing Equations to Simpler Form
13. Linear Equations and their solutions
14. Graph of Linear Equations in Two Variables
15. Equations of Lines Parallel to the X-axis and Y-axis
16. Pair of Linear Equations in Two Variables
17. Number of Solutions to a System of Equations Algebraically
18. Graphical Methods of Solving a Pair of Linear Equations
19. Algebraic Methods of Solving a Pair of Linear Equations
20. Equation Reducible to a Pair of linear equations in two variables

Quadratic Equation

A quadratic equation is a type of algebraic equation that contains one or more terms in which the variable is raised to the power of 2 (i.e., a quadratic term). It is an equation of the form ax² + bx + c = 0, where a, b, and c are constants and x is the variable.

Here are the topics that discuss quadratic equations thoroughly:

1. Quadratic Equations
2. Solution of a Quadratic equation by different methods
3. Roots of a Quadratic Equation
4. Complex Numbers
5. Algebra of Real Functions
6. Algebraic Operations on Complex Numbers
7. Argand plane and polar representation
8. Absolute Value of a Complex Number
9. Imaginary Numbers
10. Complex Conjugate
11. Compound Inequalities
12. Algebraic Solutions of Linear Inequalities in One Variable and Their Graphical Representation
13. Graphical Solution of Linear Inequalities in Two Variables
14. Solving Linear Inequalities Word Problems
15. Fundamental Principle of Counting

Cubic Equation

A three-degree equation, or a cubic equation, has a variable whose maximum power is 3. A cubic equation has the general form ax³ + bx² + cx + d = 0.

where x is a variable and a, b, c, and d are constants.

• A cubic equation can have one, two, or three solutions that are real or complex numbers, depending on the coefficients in the equation.

Polynomial

Polynomials are made by variables and coefficients, which are Algebraic expressions. Indeterminate is another name for variables. For polynomials, we can perform addition, subtraction, multiplication, and positive integer exponents, but not division by variable. For ex: 3x³-5x+8.

Following are the topics that discuss polynomials deeply.

1. Polynomials in One Variable
2. Zeroes of a polynomial
3. Remainder Theorem
4. Types of Polynomials
5. Multiplying Polynomials
6. Algebraic Identities of Polynomials
7. Geometrical meaning of the Zeroes of polynomial
8. Relationship between Zeroes and coefficients of a polynomial
9. Division Algorithm for polynomials
10. Division Algorithm Problems and Solutions

Sequence and Series

An ordered collection of numbers or other elements of mathematics that follow a pattern or rule is called a sequence.

• The position of each term of a sequence within the sequence is marked by its index or subscript.
• The series of even numbers, for example, can be written as 2, 4, 6, 8, 10, and so on.
• The total of a sequence’s terms forms a series.
• For instance, the series 2 + 4 + 6 +… + 2n gives the sum of the first n terms of the even number sequence.
• Series may be finite or infinite.
• Sequence and Series can be classified into two major categories – Arithmetic Progression and Geometric Progression. 

Arithmetic Progression 

Arithmetic Progression(A.P) is a series of numbers where each term is obtained by adding a constant /fixed value to the previous term. This continuous difference in the terms is denoted by ‘d’.

General format of an A.P is:

a, a + d, a + 2d, a + 3d, …, a + nd

where,

• a is First Term
• d is Common Difference
• n is Number of Terms

Geometric Progression

Geometric Progression(G.P) is a series of numbers where each term is obtained by multiplying the preceding term by a fixed constant value called the common ratio, denoted by r.

General form of a geometric progression is:

a, ar, ar², ar³, …, arⁿ

where,

• a is First Term,
• r is Common Ratio
• n is Number of Terms

Given below is the list of topics that will give you a better understanding of sequence and series:

1. Common difference and Nth term
2. A sum of First n Terms
3. Binomial Theorem for positive integral indices
4. Pascal’s Triangle
5. Introduction to Sequences and Series
6. General and Middle Terms – Binomial Theorem
7. Arithmetic Series
8. Arithmetic Sequences
9. Geometric Sequence and Series
10. Geometric Series
11. Arithmetic Progression and Geometric Progression
12. Special Series

Exponents 

Exponent is a mathematical operation, written as aⁿ where a is the base and n is the power or the exponent. Exponents help us simplify Algebraic expressions. Exponent can be represented in the form 

aⁿ = a*a*a*….n times. 

Logarithms 

Algebraic opposite of exponents is the logarithm. It is practical to simplify complicated Algebraic formulas using logarithms. Exponential form denoted by the formula a^x = n can be converted to logarithmic form by using formula log_an = x. 

In 1614, John Napier discovered logarithms. Nowadays, logarithms are a crucial component of modern mathematics.

Set Theory

Set theory is a branch of mathematical logic that investigates sets, which are arrays of objects informally.

• Term “set” refers to a well-defined group of unique items that are used to express Algebraic variables.
• Sets are used to depict the collection of important elements in a group.
• Sets can be expressed in set-builder or roster form.
• Sets are usually denoted by curly braces;{} for example, A = {1,2,3,4} is a collection.

Let’s learn more about the sets in the following articles:

1. Sets and their representations
2. Different kinds of Sets
3. Subsets, Power Sets, and Universal Sets
4. Venn Diagrams
5. Operations on Sets
6. Union and Intersection of sets
7. Cartesian Product of Sets

Vectors

A vector is a two-dimensional object of both magnitude and direction. It is normally represented by an arrow with a length (→) that indicates the magnitude and direction. It is denoted by the letter V.

• One of the most important aspects of Algebra is vector Algebra.
• It is a course that focuses on the Algebra of vector quantities.
• As we all know, there are two kinds of measurable quantities: scalars and vectors.
• The magnitude of a scalar quantity is the only thing that matters, while the magnitude and direction of a vector quantity are also essential.
• The vector’s magnitude is denoted by the letter |V|.

Let’s discuss the vector and its Algebra in the following articles:

1. Vector Algebra
2. Dot and Cross Product of two vectors
3. How to Find the Angle Between Two Vectors?
4. Section Formula

Relations and Functions

Relations and functions are two distinct terms that have different mathematical interpretations. One might be puzzled by the differences between them.

• Before we go even further, let’s look at a clear example of the differences between the two.
• An ordered pair is represented as (INPUT, OUTPUT): The relation shows the relationship between INPUT and OUTPUT. Whereas, a function is a relation that derives one OUTPUT for each given INPUT.

Let’s discuss more of the topic in the following articles:

1. Relations and functions
2. Types of Functions
3. Composite functions
4. Invertible Functions
5. Composition of Functions
6. Inverse Functions
7. Verifying Inverse Functions by Composition
8. Introduction to Domain and Range
9. Piecewise Function
10. Range of a Function

Matrices and Determinants

In linear Algebra, determinants, and matrices are used to solve linear equations by applying Cramer’s law to a series of non-homogeneous linear equations.

• Only square matrices are used to measure determinants. While a matrix’s determinant is empty, it’s known as a singular determinant, and when its determinant is one, it’s known as unimodular.
• The determinant of the matrix must be nonsingular, that is, its value must be nonzero, for the set of equations to have a unique solution.

Let us look at the definitions of determinants and matrices, as well as the various forms of matrices and their properties, using examples in the following articles:

1. Matrices and their Types
2. Mathematical Operations on Matrices
3. Properties of Matrix Addition and Scalar Multiplication
4. How to Multiply Matrices
5. Transpose of a matrix
6. Symmetric and Skew Symmetric Matrices
7. Elementary Operations on Matrices
8. Inverse of a Matrix by Elementary Operations
9. Invertible Matrices
10. Determinants
11. Properties of Determinants
12. Area of a Triangle using Determinants
13. Minors and Cofactors
14. Adjoint of a Matrix
15. Application of Determinants and Matrices

Permutations and Combinations

Permutation and Combination are methods for representing a collection of objects by choosing them from a list and dividing them into subsets.

• It specifies the different methods for organizing a set of data.
• Permutations are used to choose data or events from a group, while combinations are used to represent the order in which they are represented.

Mathematical Induction

For every natural number n, mathematical induction is a technique for proving a proposition, hypothesis, or formula that is assumed to be valid. The ‘Principle of Mathematical Induction‘ is a generalization of this that we can use to prove any mathematical statement.

Algebra is the branch of mathematics that deals with symbols and the rules for manipulating these symbols. It is a unifying thread of almost all of mathematics and includes everything from solving elementary equations to studying abstractions such as groups, rings, and fields.

It helps represent problems or situations in the form of mathematical expressions. It is different from Arithmetic as Arithmetic deals with specific numbers and simple operations such as addition and subtraction. Algebra, on the other hand, introduces more complex operations and includes the use of variables, equations, and functions.