What are Algebraic Identities? Simple Explanation for Class 9

If you are a Class 9 student, you have probably seen long algebraic equations that look very confusing. But what if I told you there is a shortcut to solve them? That shortcut is called an Algebraic Identity.

In this article, we will explain algebraic identities in the simplest way possible, share the most important formulas for Class 9, and show you how to use them.

What is an Algebraic Identity? (Simple Definition)

An algebraic identity is a mathematical equation that is true for all values of the variables.

No matter what numbers you put into the equation, the left side will always equal the right side.

Example:
$(a+b{)}^2=a^2+2ab+b^2$
If you replace $a$ with 2 and $b$ with 3, the left side equals the right side. If you replace them with 10 and 5, it still works! That is why it is an identity.

Identity vs. Equation: What is the Difference?

Many students get confused between an equation and an identity. Here is the easiest way to understand:

• Equation: True for some values.

  • Example: $x+2=5$. This is only true if $x=3$.

• Identity: True for all values.

  • Example: $(x+y{)}^2=x^2+2xy+y^2$. This is true for any number you use for $x$ and $y$.

Why are Algebraic Identities Important for Class 9?

In Class 9, you start learning about polynomials and factorization. Algebraic identities help you:

1. Save Time: You can solve long multiplication problems in seconds.

2. Avoid Mistakes: They give you a fixed formula, so you don’t have to do lengthy calculations.

3. Build Strong Basics: These identities are also used in higher classes (Class 10, 11, and 12) for physics and advanced math.

The 8 Standard Algebraic Identities for Class 9

Here is the list of the most important identities you must memorize for your Class 9 exams.

1. Square of a Sum
$(a+b{)}^2=a^2+2ab+b^2$

2. Square of a Difference
$(a-b{)}^2=a^2-2ab+b^2$

3. Difference of Squares
$a^2-b^2=(a+b)(a-b)$

4. Product of Two Binomials
$(x+a)(x+b)=x^2+(a+b)x+ab$

5. Square of a Trinomial
$(a+b+c{)}^2=a^2+b^2+c^2+2ab+2bc+2ca$

6. Cube of a Sum
$(a+b{)}^3=a^3+b^3+3ab(a+b)$

7. Cube of a Difference
$(a-b{)}^3=a^3-b^3-3ab(a-b)$

8. Sum of Cubes Identity
$a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)$

How to Use Algebraic Identities (Solved Example)

Let’s see how to use an identity to solve a math problem quickly.

Question: Find the value of $(105{)}^2$ without multiplying 105 by 105.

Step 1: Break 105 into two easy numbers. $105=100+5$.
Step 2: Use the identity $(a+b{)}^2=a^2+2ab+b^2$.
Here, $a=100$ and $b=5$.
Step 3: Put the numbers into the formula.
$100^2+2(100)(5)+5^2$
$=10000+1000+25$
Answer: $11025$

See how easy that was?

Tips to Remember Algebraic Identities Easily

1. Don’t just memorize: Understand how the formula is made.

2. Look for patterns: The square identities ($a+b{)}^2$ and $(a-b{)}^2$ are almost the same, except for the plus and minus signs.

3. Practice daily: Write the formulas down every morning. Use them in your homework.

4. Use Flashcards: Write the left side on one side of a card and the right side on the other.