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 + b22. 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 is an algebraic identity in Maths?
An algebraic identity is an equality that holds true for all possible values of its variables. Unlike an equation, which is only true for specific values, an identity represents a universal mathematical relationship. For example, the expression (a+b)² = a² + 2ab + b² is an identity because it is valid for any numbers substituted for 'a' and 'b'.
2. What is the main difference between an algebraic identity and an algebraic equation?
The fundamental difference lies in their validity. An algebraic identity is true for all values of the variables involved. In contrast, an algebraic equation is only true for specific values of its variables, which are called the 'solutions' or 'roots' of the equation. For instance, x + 5 = 8 is an equation (only true for x=3), while (x+5)² = x² + 10x + 25 is an identity (true for any value of x).
3. What are the three basic algebraic identities every student must know?
The three most fundamental algebraic identities, often introduced in CBSE Class 8, are:
- Square of a Sum: (a + b)² = a² + 2ab + b²
- Square of a Difference: (a - b)² = a² - 2ab + b²
- Difference of Squares: a² - b² = (a + b)(a - b)
These form the foundation for factorising and simplifying many algebraic expressions.
4. How do you prove the identity (a + b)² = a² + 2ab + b²?
To prove this identity, we start with the Left-Hand Side (L.H.S.) and expand it until it matches the Right-Hand Side (R.H.S.).
- L.H.S. = (a + b)²
- This can be written as (a + b)(a + b).
- Using the distributive property, we multiply each term in the first bracket by each term in the second: a(a + b) + b(a + b).
- This expands to a² + ab + ba + b².
- Since multiplication is commutative (ab = ba), we combine the like terms: a² + 2ab + b².
- Thus, L.H.S. = R.H.S., and the identity is proven.
5. Can algebraic identities be proved using geometric shapes?
Yes, a geometric proof provides a powerful visual understanding. For example, to prove (a + b)² = a² + 2ab + b², you can visualise a large square with side length (a + b). The area of this square is (a + b)². This large square can be divided into four smaller pieces: one square of side 'a' (area a²), one square of side 'b' (area b²), and two rectangles of sides 'a' and 'b' (each with area 'ab'). The sum of the areas of these smaller pieces (a² + b² + ab + ab) equals the area of the large square. Therefore, (a + b)² = a² + 2ab + b².
6. How are algebraic identities used to simplify complex expressions?
Algebraic identities offer a shortcut for expanding or factorising expressions, avoiding lengthy calculations. For example, to calculate 102², instead of multiplying 102 by 102, we can use the identity (a + b)² = a² + 2ab + b². We can write 102 as (100 + 2). Applying the identity, we get:
(100 + 2)² = 100² + 2(100)(2) + 2² = 10000 + 400 + 4 = 10404. This makes the calculation much faster and less prone to error.
7. What are the key cubic identities used in higher classes like Class 9?
As per the NCERT syllabus for Class 9, several cubic identities become important for solving polynomials. The key ones include:
- Cube of a Sum: (a + b)³ = a³ + b³ + 3ab(a + b)
- Cube of a Difference: (a - b)³ = a³ - b³ - 3ab(a - b)
- Sum of Cubes: a³ + b³ = (a + b)(a² - ab + b²)
- Difference of Cubes: a³ - b³ = (a - b)(a² + ab + b²)
8. Why is (a+b)² not equal to a² + b²? What is the common misconception?
A very common mistake is to assume that the square of a sum is the sum of the squares. However, (a+b)² is not equal to a² + b². This is because (a+b)² means (a+b) multiplied by (a+b). When you expand this using the distributive property, you get a² + ab + ba + b², which simplifies to a² + 2ab + b². The middle term, 2ab, is missing if you simply write a² + b². This middle term represents the area of the two rectangles in the geometric proof and is crucial for the identity to hold true.
9. Where are algebraic identities applied in real-world scenarios?
Algebraic identities are fundamental in many fields. In engineering and physics, they are used to simplify complex formulas for calculating areas, volumes, and trajectories. In computer graphics, they help in algorithms that render images and animations. In finance, they can model and simplify calculations for compound interest and investment growth. They provide a foundational language for expressing relationships in science and technology.
10. What is the importance of the identity a² - b² = (a+b)(a-b)?
The 'difference of squares' identity is extremely important for factorisation. It allows us to break down a complex expression (a² - b²) into simpler, linear factors ((a+b) and (a-b)). This is crucial for simplifying fractions, solving quadratic equations, and understanding the roots of polynomials. For example, to solve x² - 9 = 0, we can factorise it to (x+3)(x-3) = 0, which immediately gives the solutions x = -3 and x = 3.
11. How do identities help in factorising polynomials that are not immediately obvious?
Identities help us recognise patterns within complex polynomials. For example, a polynomial like 8x³ + 27y³ + 36x²y + 54xy² might look intimidating. However, a student familiar with the identity (a+b)³ = a³ + b³ + 3a²b + 3ab² can recognise that 8x³ is (2x)³ and 27y³ is (3y)³. By checking the other terms, they can confirm this polynomial is simply the expanded form of (2x + 3y)³. This transforms a complicated expression into a simple, factored form.
12. What common mistakes should be avoided when using the sum and difference of cubes identities?
When using the sum and difference of cubes identities (a³ ± b³), the most common errors occur in the signs of the trinomial factor. A helpful way to remember is the acronym SOAP:
- Same: The sign in the binomial factor is the same as the original expression. (e.g., a³ + b³ = (a + b)...)
- Opposite: The first sign in the trinomial factor is opposite to the original. (e.g., ... (a² - ab + b²))
- Always Positive: The last sign in the trinomial factor is always positive. (e.g., ... + b²)
Forgetting this pattern, especially the opposite sign for the 'ab' term, is a frequent mistake.
