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Algebraic Identities

Algebraic identities are helpful strategies in algebra that accurately resolve calculations and aid in problem-solving. These are said as a constant relation and true throughout with different variable estimates. Their usage in mathematics, such as in algebra, helps notice relationships in math, allows estimation, and improves efficiency. Anyone who works with algebra becomes familiar with facing problems involving terms containing letters and numbers. 

Expanding, simplifying, and resolving these expressions can be mastered with the knowledge of basic algebraic identities, which makes the exerciser's work effortless. For anyone wishing to know all the concepts in algebra or revise their work, through this post, all will be able to grasp what algebraic identities entail. Examples will be provided alongside formulas and their applicability in daily practice.

 

Table of Contents

 

What Are Algebraic Identities?

Algebraic identities are special equations that are true for all values of the variables involved. These are different from algebraic equations, which may be true only for specific values. An identity shows a relationship that always holds.

For example, the expression
(a + b)² = a² + 2ab + b²
It is an identity. No matter what values you choose for a and b, both sides of the equation will always be equal.

 

Importance of Algebraic Identities

Learning algebraic identities is very important because they:

  • Help to simplify algebraic expressions

  • Make calculations faster and easier

  • Support factoring and expansion

  • Help solve complex equations

  • They are used in geometry, physics, and higher-level math

Understanding algebraic identities also builds a strong foundation for more advanced topics in algebra, such as quadratic equations, polynomials, and calculus.

 

Basic Algebraic Identities

Here are some of the most commonly used algebraic identities. Memorising these will help you with a wide range of math problems.

  • (a + b)² = a² + 2ab + b²

  • (a − b)² = a² − 2ab + b²

  • a² − b² = (a + b)(a − b)

  • (x + a)(x + b) = x² + (a + b)x + ab

  • (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca

  • (x + y)³ = x³ + y³ + 3xy(x + y)

  • (x − y)³ = x³ − y³ − 3xy(x − y)

  • x³ + y³ = (x + y)(x² − xy + y²)

  • x³ − y³ = (x − y)(x² + xy + y²)

These identities of algebra are essential for simplifying and solving expressions, especially when working with squares and cubes of binomials or trinomials.

 

Algebraic Identities Formula

The algebraic identities formula set is helpful when solving math problems quickly. Let's look at how these math identities are formed and used.

Example 1:

Use (a + b)² = a² + 2ab + b²

If a = 3 and b = 2

(3 + 2)² = 25

3² + 2×3×2 + 2² = 9 + 12 + 4 = 25

Example 2:

Use a² − b² = (a + b)(a − b)

Let a = 5, b = 1

a² − b² = 25 − 1 = 24

(5 + 1)(5 − 1) = 6 × 4 = 24

As you can see, using the algebraic identities formula makes these calculations quick and accurate.

 

How to Use Algebraic Identities

To use algebraic identities in a problem, first check the structure of the expression. If the expression matches one of the common identity patterns, substitute it directly.

Step-by-step guide:

  1. Identify the structure of the expression

  2. Match it with the correct identity

  3. Replace it with the simplified identity

  4. Perform final calculations if needed

Example:

Simplify (x + 4)(x + 3)

This matches identity (x + a)(x + b) = x² + (a + b)x + ab

Here, a = 4, b = 3

x² + (4 + 3)x + 4×3 = x² + 7x + 12

 

Applications of Algebraic Identities

Algebraic identities are not only useful in exams but also in real life. Here are a few practical applications:

  • Geometry: Used to calculate areas and expand equations in coordinate geometry

  • Physics: Used in formulas involving force, energy, and speed

  • Engineering is important in designing structures, circuits, and machines

  • Programming: Help optimise formulas used in algorithms and computer code

  • Banking and Finance: Used to calculate interests, growth rates, and percentages

Understanding these math identities can help you in a variety of real-world situations.

 

Real-Life Example Using Algebraic Identities

Suppose you want to square a large number like 101.

You can write 101 = 100 + 1

Now apply (a + b)² = a² + 2ab + b²

(100 + 1)² = 100² + 2×100×1 + 1² = 10000 + 200 + 1 = 10201

Much faster than multiplying 101 × 101 manually.

 

Factorisation Using Algebraic Identities

Identities also help in factorising expressions, which is useful in solving equations.

Example:

Factor x² − 9

This is a² − b² = (a + b)(a − b)

So, x² − 9 = (x + 3)(x − 3)

This process of breaking down expressions into products of simpler expressions is called factorisation.

 

Common Mistakes in Algebraic Identities

Many students make mistakes while using these identities. Here are a few to avoid:

  • Confusing a²-b² with (a-b)²

  • Forgetting to apply the complete identity

  • Ignoring signs (positive or negative)

  • Incorrect multiplication of terms

  • Not matching the identity properly

Always double-check whether the identity you are applying is the correct match for the given expression.

 

Practice Questions on Algebraic Identities

  1. Simplify: (x + 5)²

  2. Expand: (x − 3)²

  3. Factor: x² − 16

  4. Use identity to solve: (2x + 3)(2x − 3)

  5. Find cube of (a + b) using identity

These questions will help you gain confidence in applying the algebraic identities formula.

 

Real-World Word Problem Using Algebraic Identity

Problem:

A square garden has a side length of (x + 5) meters. What is the area of the garden?

Solution:

Area = side × side = (x + 5)²

Use identity: a² + 2ab + b² = x² + 10x + 25 square meters

This is how algebraic identities help solve practical geometry problems too.

 

Algebraic Identities for Cubes

Cubes are also important in algebra. Memorizing cube identities helps a lot.

  • (a + b)³ = a³ + 3a²b + 3ab² + b³

  • (a − b)³ = a³ − 3a²b + 3ab² − b³

  • a³ + b³ = (a + b)(a² − ab + b²)

  • a³ − b³ = (a − b)(a² + ab + b²)

Use these identities when you see cube expressions in equations or problems.

 

Algebraic Identities in Class 8, 9, 10 and Beyond

In schools, algebraic identities are usually introduced around class 8 and are deeply used in class 9 and 10. These identities are the foundation for future learning in algebra, trigonometry, and calculus.

Higher classes introduce advanced identities such as:

  • Trigonometric identities

  • Polynomial identities

  • Logarithmic identities

But all of them are based on the basic algebraic identities you learn now.

 

Conclusion

Mathematical tools called algebraic identities assist you in solving and simplifying problems at different levels. They significantly reduce the time needed to complete tasks. Understanding algebraic formulas allows you to deconstruct intricate equations methodically. Whether tackling academic math exercises or using equations in daily activities, algebraic identities are always beneficial. Regular practice with the algebraic identity formula helps reinforce confidence to take on complex algebra problems, sharpening overall problem-solving capabilities.

 

Related Topics

Addition - Build your confidence in solving math problems by mastering the basics of addition. Start exploring more fun examples today!

Subtraction - Simply understand subtraction and apply it in real-life math challenges. Practice and improve your accuracy now!

Multiplication - Learn multiplication the easy way with smart techniques and real-life examples. Strengthen your math skills with daily practice!

 

Frequently Asked Questions on Algebraic Identities

What are the 12 algebraic identities?

The 12 common algebraic identities are:

  1. (a + b)² = a² + 2ab + b²

  2. (a - b)² = a² - 2ab + b²

  3. a² - b² = (a + b)(a - b)

  4. (x + a)(x + b) = x² + (a + b)x + ab

  5. (x + a)³ = x³ + 3ax² + 3a²x + a³

  6. (x - a)³ = x³ - 3ax² + 3a²x - a³

  7. x³ + y³ = (x + y)(x² - xy + y²)

  8. x³ - y³ = (x - y)(x² + xy + y²)

  9. (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca

  10. (a + b)³ = a³ + b³ + 3ab(a + b)

  11. (a - b)³ = a³ - b³ - 3ab(a - b)

  12. x³ + y³ + z³ - 3xyz = (x + y + z)(x² + y² + z² - xy - yz - zx)

 

What are the first 7 algebraic identities?

The first 7 algebraic identities are:

  1. (a + b)² = a² + 2ab + b²

  2. (a - b)² = a² - 2ab + b²

  3. a² - b² = (a + b)(a - b)

  4. (x + a)(x + b) = x² + (a + b)x + ab

  5. (x + a)³ = x³ + 3ax² + 3a²x + a³

  6. (x - a)³ = x³ - 3ax² + 3a²x - a³

  7. x³ + y³ = (x + y)(x² - xy + y²)

 

Who is the father of algebraic identity?

Muhammad ibn Musa al-Khwarizmi is considered the father of algebra. He introduced early algebraic methods in his book “Al-Kitab al-Mukhtasar fi Hisab al-Jabr wal-Muqabala.”

 

What are the 10 algebraic formulas?

Here are 10 important algebraic identities:

  1. (a + b)² = a² + 2ab + b²

  2. (a - b)² = a² - 2ab + b²

  3. a² - b² = (a + b)(a - b)

  4. (x + a)(x + b) = x² + (a + b)x + ab

  5. (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca

  6. x³ + y³ = (x + y)(x² - xy + y²)

  7. x³ - y³ = (x - y)(x² + xy + y²)

  8. (a + b)³ = a³ + b³ + 3ab(a + b)

  9. (a - b)³ = a³ - b³ - 3ab(a - b)

  10. x³ + y³ + z³ - 3xyz = (x + y + z)(x² + y² + z² - xy - yz - zx)

 

What are the 8 types of identities?

The 8 commonly used algebraic identities are:

  1. (a + b)²

  2. (a - b)²

  3. a² - b²

  4. (a + b)³

  5. (a - b)³

  6. x³ + y³

  7. x³ - y³

  8. (a + b + c)²


Learn more at Orchids The International School, where math becomes fun, clear, and easy to understand.

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