Perfect Square Formula

A polynomial or number which when multiplied by itself is called a perfect square. The perfect square is calculated by two algebraic expressions that include: (a + b)² = a² + 2ab + b² and (a – b)² = a² – 2ab + b².

In this article, we have covered the perfect square definition, how to identify perfect square, perfect square formulas, and other related topics in detail.

What is Perfect Square?

A perfect square is an integer which is the square of some other integer or we can say that it is a second exponent of an integer. We can take the example below and understand it.

The perfect square formula in mathematics is shown in the image below:

Let us take 25 and find out if it is a perfect square or not. So factors of 25 are 5×5 = (5)². So 25 is a perfect square as it is a square of 5. 

How to Identify Perfect Square

There are three rules that we need to check to find if a number is a perfect square:

Rule 1:

• There should be 1, 4, 5, 6, 9 or 0 at one’s (last) digit space of the number to be checked.

Example:

(i) 49 = (7)²

(ii) 121 = (11)²

Rule 2:

• (i) If 1, 4 or 9 is at one’s (last) digit space. Then the digit at ten’s (second last) place should be an even number or 0.

Example:

(i) 81 = (9)²

(ii) 169 = (13)²

• (ii) If 6 is at one’s (last) digit space. Then digit at ten’s (second last) place should be an odd number.

Example:

(i) 196 = (14)²

(ii) 36 = (6)²

(iii) If 5 is at one’s(last) digit place. Then digit at ten’s (second last) place should be 2.

Example:

(i) 25 = (5)²

(ii) 625 = (25)²

Rule 3:

• The digit sum of a perfect square should be an odd number or 4.

Example:

(i) 49

= 4 + 9 = 13 = 1 + 3 = 4 

So, digital sum of 49 is 4. So it is a perfect square.

(ii) 196

= 1 + 9 + 6 = 16 = 1 + 6 = 7

So, the digital sum of 196 is an odd number. So, it is a perfect square.  

Note: If all three conditions are satisfied then only a number is said to be a perfect square.

Perfect Square Formula

Perfect Square formula is used to the square of sum/subtraction of two terms i.e (a+b)² or (a-b)². The expansion of the perfect formula is expressed as

• (a + b)² = a² + 2 × a × b + b²

•  (a – b)² =  a²  – 2 × a × b + b²

Proof of Perfect Square Formula

(i) Proof of (a + b)²

⇒ (a + b)² = (a + b) × (a + b)

⇒(a + b)² = a × (a + b) + b × (a + b)

⇒(a + b)² = a² + ab + ba + b²

⇒(a + b)² = a² + ab + ab + b²  (ba = ab because of commutative law)

⇒(a + b)² = a² + 2ab + b²

Hence Proved

(ii) Proof of (a – b)²

⇒(a – b)² = (a – b) × (a – b)

⇒(a – b)² = a × (a – b) – b × (a – b)

⇒(a – b)² = a² – ab – ba + (-b) × (-b)

⇒(a – b)² = a² – ab – ba + b²

⇒(a – b)² = a² – ab – ab + b² (ba=ab because of commutative law)

⇒(a – b)² = a² – 2ab + b²

Hence Proved

Perfect Squares from 1 to 100

Perfect squares from 1 to 100 is added in the table below,

Perfect Square Numbers From 1 to 100
1	=	1 × 1	=	12
4	=	2 × 2	=	22
9	=	3 × 3	=	32
16	=	4 × 4	=	42
25	=	5 × 5	=	52
36	=	6 × 6	=	62
49	=	7 × 7	=	72
64	=	8 × 8	=	82
81	=	9 × 9	=	92
100	=	10 × 10	=	102

Perfect Square Examples

Example 1: Find square of (2x + y) using perfect formula

Solution: 

Given (2x + y)²

Using perfect square formula

(a + b)² = a² + 2ab + b²

a = 2x and b = y

Put the values

(2x + y)² = ((2x)² + 2 × (2x) × (y) + (y)²)

(2x + y)² = (4x² + 4xy + y²)

Square of (2x + y) is 4x² + 4xy + y².

Example 2: Simplify (5x+2y)² using the perfect square formula.

Solution: 

Using perfect square formula

(a + b)² = a² + 2ab + b²

a = 5x and b = 2y

Put the values

(5x + 2y)² = ((5x)² + 2 × (5x) × (2y) + (2y)²)                                    

So, (5x + 2y)² = 25x² + 20xy + 4y²

Example 3: Find if x² + 4y² + 4xy is perfect square or not.

Solution: 

Given x² + 4y² + 4xy

Now rearranging the given expression;

x² + 4xy + 4y²

On expanding the above equation we get

((x) × (x)) + 2 × (x) × (2y) + ((2y) × (2y))

On comparing with perfect square formula, we get

(a + b)²= a² + 2ab + b²

On comparing values we get

a = x and b = 2y

So, x² + 4y² + 4xy = (x + 2y)²

Hence, x² + 4y² + 4xy is perfect square.

Example 4: Evaluate: (99)²

Solution: 

So, it can also be written as:

(100 – 1)²

Using perfect square formula:

(a – b)² = a² + 2ab + b²

a = 100 and b = 1

(100 – 1)² = ((100)² – 2 × (100) × (1) + (1)²

(100 – 1)² = (10000 – 200 + 1)

(100 – 1)² = (10001 – 200)

(100 – 1)² = 9801

So (99)² = 9801

Example 5: Find if x² + 4 – 4x is perfect square or not.

Solution: 

Given x² + 4 – 4x

Rearranging the above expression;

x² – 4x + 4    

On expanding, we get

((x) × (x)) – 2 × (x) × (2) + ((2) × (2))

On comparing with perfect square formula

(a – b)² = a² – 2ab + b²

On comparing values we get

a = x and b = 2

So, x² + 4 – 4x = (x – 2)²

Hence, x² + 4 – 4x is a perfect square

Perfect Squares FAQs

What are perfect squares?

A number or polynomial which when multiplied by itself results in a perfect square.

What are examples of perfect squares?

Some examples of perfect square includes:

• 4 = 2²
• 9 = 3²
• 16 = 4²
• 25 = 5²

What is the perfect square rule?

Perfect squares are numbers or expressions that are product of the number multiplied by itself. Perfect square rule is a number which when multiplied by itself result in a perfect square.

How do you find the perfect square?

As, we known perfect numbers are numbers which are formed when a number is multiplied by itself. Thus, prime factorization is used to find perfect squares.

What are the first 100 perfect squares?

Perfect squares up to 100 include 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100.