(a+b)^2 Formula

The (a + b)2 formula is used to find the square of a binomial. This formula is also used to factorize some special types of trinomials. This formula is one of the algebraic identities. The (a + b)2 formula is the formula for the square of the sum of two terms. The (a + b)2 formula is widely used to factorize the binomial. The (a + b)2 formula is explained below along with solved examples in the following section.

What is (a+b)^2 Formula?

The (a + b)2 formula is the algebraic identity used to find the square of the sum of two numbers. To find the formula of the binomial in the form (a + b)2, we will just multiply (a + b) (a + b).

(a + b)2= (a + b)(a + b)

= a2 + ab + ba + b2

= a2 + 2ab + b2 

Therefore, (a + b)2 formula is: (a + b)2 = a2 + 2ab + b2

You can understand this formula geometrically using the following figure:

proof of (a+b) whole square formula

From the above figure (a+b)2 can be written as (a+b) × (a+b). Consider a square whose sides are (a+b) and the area is (a+b)2. The square with a side of (a + b) can be considered as four areas of a2, ab, ab, b2. The sum of these areas is (a2 + ab + ab + b2) gives the area of the square (a+b)2. Rearranging the terms to find area of the square (a+b)2 =  a2 + ab + ab + b2 proves the algebraic identity, (a + b)2 formula is: (a + b)2 = a2 + 2ab + b2

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Examples on (a+b)^2 Formula

Let us consider few illustrations based on the (a+b)^2 formula in this solved examples section.

Example 1: Find the value of (3x + 2y)2 using (a + b)2 formula.

Solution: 

To find: The value of (3x + 2y)2.
Let us assume that a = 3x and b = 2y.
We will substitute these values in (a + b)2 formula:
(a + b)2 = a2 + 2ab + b2
(3x+2y)2 =(3x)2 + 2(3x)(2y) +(2y)2
= 9x2+12xy+4y2

Answer: (3x + 2y)2 = 9x2 + 12xy + 4y2.

Example 2: Factorize x2 + 4xy + 4y2 using (a + b)2 formula.

Solution: 

To factorize: x2 + 4xy + 4y2.
We can write the given expression as: (x)2 + 2 (x) (2y) + (2y)2.
Using (a + b)2 formula:
a2 + 2ab + b2 = (a + b)2 
Substitute a = x and b = 2y in this formula:
(x)2 + 2 (x) (2y) + (2y)2. = (x + 2y)2

Answer: x2 + 4xy + 4y2 = (x + 2y)2.

Example 3: Simplify the following using (a+b)2 formula.

(7x + 4y)2

Solution:

a = 7x and b = 4y
Using formula (a + b)2  =  a2  + 2ab + b2
(7x)2 + 2(7x)(4y) + (4y)2
49x2 + 56xy + 16y2

Answer: (7x + 4y)2 = 49x2 + 56xy + 16y2.

FAQs on (a + b)2 Formula

What Is the Expansion of (a + b)2 Formula?

(a + b)2 formula is read as a plus b whole square. Its expansion is expressed as (a + b)2  =  a2 + 2ab + b2

What Is the (a + b)2 Formula in Algebra?

The (a + b)2 formula is also known as one of the important algebraic identities. It is read as a plus b whole square. Its (a + b)2 formula is expressed as (a + b)2  =  a2 + 2ab + b2

How To Simplify Numbers Using the (a + b)2 Formula?

Let us understand the use of the (a + b)2 formula with the help of the following example.
Example: Find the value of (20 + 5)2 using the (a + b)2 formula.
To find: (20+5)2
Let us assume that a = 20 and b = 5.
We will substitute these in the formula of (a + b)2. 
(a + b)2  =  a2 + 2ab + b2
(20+5)2 = 202 + 2(20)(5) + 52
=400 + 200 + 25 
=625
Answer: (20 + 5)2 = 625.

How To Use the (a + b)2 Formula?

The following steps are followed while using (a + b)2 formula.

  • Firstly observe the pattern of the numbers whether the numbers have whole ^2 as power or not.
  • Write down the formula of (a + b)2
  • (a + b)2  =  a2 + 2ab + b2
  • substitute the values of a and b in the (a + b)2 formula and simplify.
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