Courses

Courses for Kids

Free study material

Store

Talk to our experts

1800-120-456-456

Maths

Algebraic Identities in Maths: Formulas, Proofs & Examples

Algebraic Identities in Maths: Formulas, Proofs & Examples

Q: 1. What are the 12 algebraic identities?

The 12 algebraic identities commonly taught in secondary mathematics are fundamental formulas used to simplify expressions and solve equations. These are:$ (a + b)^2 = a^2 + 2ab + b^2 $$ (a - b)^2 = a^2 - 2ab + b^2 $$ a^2 - b^2 = (a + b)(a - b) $$ (a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca $$ (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 $$ (a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3 $$ a^3 + b^3 = (a + b)(a^2 - ab + b^2) $$ a^3 - b^3 = (a - b)(a^2 + ab + b^2) $$ (x + a)(x + b) = x^2 + (a + b)x + ab $$ (a + b)^2 - (a - b)^2 = 4ab $$ (a + b + c)^3 = a^3 + b^3 + c^3 + 3(a + b)(b + c)(c + a) $$ (x + y + z)^2 = x^2 + y^2 + z^2 + 2(xy + yz + zx) $Mastering these algebraic identities with Vedantu’s resources helps students solve algebraic problems efficiently.

Q: 2. What are the 20 algebraic identities class 9?

For Class 9, students often study an expanded list of 20 algebraic identities, which covers standard identities plus additional formulas useful for higher algebra. These include all basic identities and their rearrangements such as:$ (a + b + c)^2 $ and $ (a + b + c)^3 $$ (x + a)(x + b) $$ a^3 + b^3 $ and $ a^3 - b^3 $Product of binomials, trinomial squares, cubesDifference of squares, cubes, and mixed expansionsVedantu’s online classes and study materials provide comprehensive guidance on memorizing, understanding, and applying all 20 algebraic identities prescribed in the curriculum.

Q: 3. What is an example of an algebraic identity?

An algebraic identity is an equation that holds true for all permitted values of the variables. An example is:$ (a + b)^2 = a^2 + 2ab + b^2 $For instance, taking $ a = 2 $ and $ b = 3 $:Left side: $ (2 + 3)^2 = 25 $Right side: $ 2^2 + 2 \times 2 \times 3 + 3^2 = 4 + 12 + 9 = 25 $This proves the validity of the algebraic identity. Vedantu’s faculty uses such examples extensively in classes to help students grasp concepts clearly.

Q: 4. What are the 5 basic rules of algebra?

The 5 basic rules of algebra are foundational properties that govern algebraic operations. These include:Commutative Law: $ a + b = b + a $Associative Law: $ (a + b) + c = a + (b + c) $Distributive Law: $ a(b + c) = ab + ac $Additive Identity: $ a + 0 = a $Multiplicative Identity: $ a \times 1 = a $Mastering these rules with Vedantu’s guidance helps students easily simplify and solve algebraic expressions and equations.

Q: 5. What is the importance of algebraic identities in mathematics?

Algebraic identities are crucial in mathematics as they:Enable the quick simplification of complex algebraic expressionsHelp in expanding and factoring polynomialsLay the foundation for advanced topics in algebra and geometryReduce calculation effort in solving equationsVedantu’s interactive online classes demonstrate real-world applications and provide problem-solving practice to strengthen students’ understanding of these identities.

Q: 6. How can students memorize algebraic identities effectively?

To memorize algebraic identities efficiently, students should:Break down each identity into its components and understand the patternRegularly practice problems that involve applying each identityCreate flashcards for quick revisionUse real-life examples to connect concepts with applicationVedantu offers interactive quizzes, flashcards, and revision sessions tailored to helping learners retain and recall all important algebraic identities.

Q: 7. What are real-life applications of algebraic identities?

Algebraic identities have several real-life applications, such as:Engineering calculations and design modellingFinancial analysis and computation of compound interestComputer programming for developing algorithmsPhysics for equations of motion and energy calculationsVedantu’s curriculum demonstrates how algebraic identities are used in various careers, helping students appreciate their practical importance.

Q: 8. What are some tips to solve problems involving algebraic identities in exams?

For exam success with algebraic identities:Identify which identity fits the given expressionPractice using shortcuts and patterns regularlySolve previous years’ questions for familiarityCheck steps for accuracy and completenessVedantu provides targeted practice sets, mock tests, and strategy sessions to help students excel in exams involving algebraic identities.

Q: 9. How does understanding algebraic identities help with higher-level mathematics?

Understanding algebraic identities forms the basis for advanced topics, such as:Polynomial factorization and equation solvingQuadratic and cubic equationsBinomial expansions and calculus foundationsSimplification in trigonometry and geometryVedantu’s step-wise teaching approach ensures that students gain clarity on identities for a smooth transition to college-level algebra.

Q: 10. Are all algebraic identities applicable for every value of variables?

Most algebraic identities are universally valid for all real numbers; however, some identities might exclude values where a variable causes division by zero or is undefined. It is important to understand the domain of variables when applying algebraic identities. Vedantu’s instructors clarify such exceptions and help students use identities accurately.

Reviewed by:

Rama Sharma

List of Important Algebraic Identities with Explanation

The concept of algebraic identities plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Whether you’re simplifying expressions, factorising polynomials, or solving equations for school or competitive exams, understanding algebraic identities saves time and reduces mistakes.

What Is Algebraic Identities?

An algebraic identity is a mathematical equation that remains true for all values of the variables involved. This means both sides of the equation are always equal, no matter which numbers you substitute for the letters. You’ll find this concept applied in areas such as expansion of expressions, factorisation, and solving simultaneous equations.

Key Formula for Algebraic Identities

Here are some of the most important algebraic identities used in maths:

Identity Name	Formula	Usage
Square of Sum	(a + b)² = a² + 2ab + b²	Expanding binomials
Square of Difference	(a - b)² = a² - 2ab + b²	Expanding, shortcuts
Product of Sum and Difference	(a + b)(a - b) = a² - b²	Factorisation
Expansion (x + a)(x + b)	(x + a)(x + b) = x² + x(a + b) + ab	Quadratic expansions
Cube of Sum	(a + b)³ = a³ + 3a²b + 3ab² + b³	Advanced algebra
Cube of Difference	(a - b)³ = a³ - 3a²b + 3ab² - b³	Advanced algebra
Sum of Cubes	a³ + b³ = (a + b)(a² - ab + b²)	Factorisation
Difference of Cubes	a³ - b³ = (a - b)(a² + ab + b²)	Factorisation

Cross-Disciplinary Usage

Algebraic identities are not only useful in Maths but also play an important role in Physics for equations of motion, in Computer Science for algorithm design, and in logical reasoning during competitive exams. Students preparing for JEE, NTSE or Olympiads will find these formulas repeating across complex problem-solving.

Step-by-Step Illustration

Let’s see how to use an algebraic identity in a real problem:

Example: Expand (x + 2)² using algebraic identities.

1. Identify the formula: (a + b)² = a² + 2ab + b²

2. Set a = x, b = 2

3. Substitute: (x + 2)² = x² + 2 × x × 2 + 2²

4. Simplify: = x² + 4x + 4

Final expansion: x² + 4x + 4

Speed Trick or Vedic Shortcut

Here’s a quick shortcut that helps solve problems faster with algebraic identities, especially in mental maths.

Example Trick: To calculate 104² in seconds:

Use (a + b)² identity where a = 100, b = 4
Compute: 100² = 10000
Compute: 2ab = 2 × 100 × 4 = 800
Compute: b² = 16
Add: 10000 + 800 + 16 = 10816

Thus, 104² = 10816 instantly, without traditional multiplication! Vedantu’s live classes include many such mental maths tricks using algebraic identities for school and exams.

Try These Yourself

Write the value of (a − b)² for a = 6, b = 2.
Factorise a² − 25 using a suitable identity.
Evaluate (3x + 1)(3x − 1) by identity.
Expand (y − 4)² stepwise.

Frequent Errors and Misunderstandings

Confusing algebraic identities with actual equations or expressions.
Forgetting double signs, e.g., using –2ab instead of +2ab in (a − b)².
Applying an identity to non-matching patterns (e.g., using (a + b)² when there is a – sign).

Relation to Other Concepts

The idea of algebraic identities connects closely with algebraic expressions and identities and factorization of algebraic expressions. Mastering these helps in learning quadratic equations and polynomial identities in later chapters.

Classroom Tip

A quick way to remember algebraic identities is to visualize the formula as an “expansion pattern”. For example, in (a + b)² = a² + 2ab + b², think: “square the first, double the product, square the last.” Vedantu’s teachers often illustrate this using color and boxes for each term, making the pattern memorable.

We explored algebraic identities—from definition, formula, examples, common mistakes, and their use across maths and science. Continue practicing with Vedantu to become confident in solving problems using this concept. For more tips and printable formula sheets, explore algebraic expressions, polynomial identities, and algebraic equations on Vedantu’s website.

FAQs on Algebraic Identities in Maths: Formulas, Proofs & Examples

1. What are the 12 algebraic identities?

The 12 algebraic identities commonly taught in secondary mathematics are fundamental formulas used to simplify expressions and solve equations. These are:

$ (a + b)^2 = a^2 + 2ab + b^2 $
$ (a - b)^2 = a^2 - 2ab + b^2 $
$ a^2 - b^2 = (a + b)(a - b) $
$ (a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca $
$ (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 $
$ (a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3 $
$ a^3 + b^3 = (a + b)(a^2 - ab + b^2) $
$ a^3 - b^3 = (a - b)(a^2 + ab + b^2) $
$ (x + a)(x + b) = x^2 + (a + b)x + ab $
$ (a + b)^2 - (a - b)^2 = 4ab $
$ (a + b + c)^3 = a^3 + b^3 + c^3 + 3(a + b)(b + c)(c + a) $
$ (x + y + z)^2 = x^2 + y^2 + z^2 + 2(xy + yz + zx) $

Mastering these algebraic identities with Vedantu’s resources helps students solve algebraic problems efficiently.

2. What are the 20 algebraic identities class 9?

For Class 9, students often study an expanded list of 20 algebraic identities, which covers standard identities plus additional formulas useful for higher algebra. These include all basic identities and their rearrangements such as:

$ (a + b + c)^2 $ and $ (a + b + c)^3 $
$ (x + a)(x + b) $
$ a^3 + b^3 $ and $ a^3 - b^3 $
Product of binomials, trinomial squares, cubes
Difference of squares, cubes, and mixed expansions

Vedantu’s online classes and study materials provide comprehensive guidance on memorizing, understanding, and applying all 20 algebraic identities prescribed in the curriculum.

3. What is an example of an algebraic identity?

An algebraic identity is an equation that holds true for all permitted values of the variables. An example is:

$ (a + b)^2 = a^2 + 2ab + b^2 $

For instance, taking $ a = 2 $ and $ b = 3 $:
Left side: $ (2 + 3)^2 = 25 $
Right side: $ 2^2 + 2 \times 2 \times 3 + 3^2 = 4 + 12 + 9 = 25 $
This proves the validity of the algebraic identity. Vedantu’s faculty uses such examples extensively in classes to help students grasp concepts clearly.

4. What are the 5 basic rules of algebra?

The 5 basic rules of algebra are foundational properties that govern algebraic operations. These include:

Commutative Law: $ a + b = b + a $
Associative Law: $ (a + b) + c = a + (b + c) $
Distributive Law: $ a(b + c) = ab + ac $
Additive Identity: $ a + 0 = a $
Multiplicative Identity: $ a \times 1 = a $

Mastering these rules with Vedantu’s guidance helps students easily simplify and solve algebraic expressions and equations.

5. What is the importance of algebraic identities in mathematics?

Algebraic identities are crucial in mathematics as they:

Enable the quick simplification of complex algebraic expressions
Help in expanding and factoring polynomials
Lay the foundation for advanced topics in algebra and geometry
Reduce calculation effort in solving equations

Vedantu’s interactive online classes demonstrate real-world applications and provide problem-solving practice to strengthen students’ understanding of these identities.

6. How can students memorize algebraic identities effectively?

To memorize algebraic identities efficiently, students should:

Break down each identity into its components and understand the pattern
Regularly practice problems that involve applying each identity
Create flashcards for quick revision
Use real-life examples to connect concepts with application

Vedantu offers interactive quizzes, flashcards, and revision sessions tailored to helping learners retain and recall all important algebraic identities.

7. What are real-life applications of algebraic identities?

Algebraic identities have several real-life applications, such as:

Engineering calculations and design modelling
Financial analysis and computation of compound interest
Computer programming for developing algorithms
Physics for equations of motion and energy calculations

Vedantu’s curriculum demonstrates how algebraic identities are used in various careers, helping students appreciate their practical importance.

8. What are some tips to solve problems involving algebraic identities in exams?

For exam success with algebraic identities:

Identify which identity fits the given expression
Practice using shortcuts and patterns regularly
Solve previous years’ questions for familiarity
Check steps for accuracy and completeness

Vedantu provides targeted practice sets, mock tests, and strategy sessions to help students excel in exams involving algebraic identities.

9. How does understanding algebraic identities help with higher-level mathematics?

Understanding algebraic identities forms the basis for advanced topics, such as:

Polynomial factorization and equation solving
Quadratic and cubic equations
Binomial expansions and calculus foundations
Simplification in trigonometry and geometry

Vedantu’s step-wise teaching approach ensures that students gain clarity on identities for a smooth transition to college-level algebra.

10. Are all algebraic identities applicable for every value of variables?

Most algebraic identities are universally valid for all real numbers; however, some identities might exclude values where a variable causes division by zero or is undefined. It is important to understand the domain of variables when applying algebraic identities. Vedantu’s instructors clarify such exceptions and help students use identities accurately.