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.
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 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
Perfect Square Formula
- (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 |
Examples on Perfect Square Formula
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
Similar Reads
Square Root Formula Square root formula is a mathematical expression that calculates the square root of a number. For a non-negative real number xx, the square root y is a number such that y2 = x. A square root is an operation that is used in many formulas and different fields of mathematics. This article is about squa
8 min read
Perfect Square Trinomial Formula Perfect Square Trinomial is when a binomial is multiplied with itself it gives an expression which consists of three terms and this expression. Perfect square trinomials consist of variable as well as constant terms in their algebraic expression. It can be represented in the form of ax2 + bx +c wher
4 min read
Perfect Squares A perfect square or square number is a number that can be expressed as the product of an integer with itself. In other words, a perfect square is n raised to the power 2 where n is an integer. Let’s understand Perfect Squares with the following illustration:Examples: 9 is a perfect square as we can
9 min read
Is 3 a Perfect Square? No, 3 is not a perfect square because there is no integer that, when squared, equals 3. Let’s explore the characteristics of perfect squares and apply them to the number 3 to answer this question.What are Perfect Square Numbers?Perfect square numbers are numbers that can be expressed as the square o
2 min read
Recursive Formula A recursive function is a function that defines each term of a sequence using the previous term i.e., The next term is dependent on one or more known previous terms. Recursive function h(x) is written as,h(x) = a0h(0) + a1h(1) + a2h(2) + ... + ax - 1h(x - 1)where, ai ≥ 0i = 0, 1, 2, 3, ... ,(x - 1)T
4 min read