a^2-b^2 Formula

The a2 - b2 formula is also known as "the difference of squares formula". The a square minus b square is used to find the difference between the two squares without actually calculating the squares.

What is a^2-b^2 Formula?

The a2 - b2 formula is given as: a2 - b2 = (a - b) (a + b)

If you would like to verify this, you can just multiply (a - b) (a + b) and see whether you get a2 - b2. 

Verification of a2 - b2 Formula 

Let us see the proof of a square minus b square formula. To verify that a2 - b2 = (a - b) (a + b) we need to prove LHS = RHS. Let us try to solve the equation:

a2 - b2 = (a - b) (a + b)
Multiply (a - b) and (a + b) we get
=a(a+b) -b(a + b)
=a2 + ab - ba - b2
=a2 + 0 + b2
=a2 - b2 
Hence Verified
a2 - b2 = (a - b) (a + b)

You can understand the a2 - b2 formula geometrically using the following figure:

a square minus b square formula

Proof of a^2 - b^2 Formula

The proof that the value a2 - b2 is (a + b)(a - b). Let us consider the above figure. Take the two squares of sides a units and b units respectively.  This can also be represented as the sum of are areas of two rectangles as presented in the below figure.

One rectangle has a length of a unit and a breadth of (a - b) units on the other side the second rectangle has a length of (a - b) and a breadth of b units. Now add the areas of the two rectangles to obtain the resultant values. The respective areas of the two rectangles are (a - b) × a = a(a - b) , and (a - b) × b = b(a - b). The sum of the areas of rectangles is the actual obtained resultant expression i.e., a(a + b) + b(a - b) = (a + b)(a - b). Again re-arranging the individual rectangles and squares, we get: (a+b)(a−b)=a2−b2

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Examples on a^2-b^2 Formula

Let us solve some interesting problems using the a^2-b^2 formula.

Example 1: Using a2 - b2 formula find the value of 1062 - 62.

Solution: To find: 1002 - 62.

Let us assume that a = 100 and b = 6.
We will substitute these in the a2 - b2 formula.
a2 - b2 = (a - b) (a + b)
1062 - 62 = (106 - 6) (106 + 6)
= (100) (112)
= 11200

Answer: 1062 - 62 = 11200.

Example 2: Factorize the expression 25x2 - 64.

Solution: To factorize: 25x2 - 64.
We will use the a2 - b2 formula to factorize this.
We can write the given expression as
25x2 - 64 = (5x)2 - 82
We will substitute a = 5x and b = 8 in the formula of a2 - b2. 
a2 - b2 = (a - b) (a + b)
(5x)2 - 82 = (5x - 8) (5x + 8)

Answer: 25x2 - 64 = (5x - 8) (5x + 8)

Example 3: Simplify 102 - 52 using a2 - b2 formula 

Solution: To find 102 - 52

Let us assume  a = 10 and b = 5
Using formula a2 - b2 = (a - b) (a + b)
102-52 = (10 - 5) (10 + 5)
= 10(10 +5) - 5(10 + 5)
= 10(15) - 5(15)
= 150-75 = 75
Answer: 102 - 52 = 75.

FAQs on a^2 - b^2 Formula 

What Is the Expansion of a2 - b2 Formula?

a2 - b2 formula is read as a square minus b square. Its expansion is expressed as a2 - b2 = (a - b) (a + b) 

What Is the a2 - b2 Formula in Algebra?

The a2 - b2 formula is also known as one of the important algebraic identity. It is read as a square minus b square. Its a2 - b2 formula is expressed as a2 - b2 = (a - b) (a + b)

How To Simplify Numbers Using the a2 - b2 Formula?

Let us understand the use of the a2 - b2 formula with the help of the following example.
Example: Find the value of 102 - 22 using the a2 - b2 formula.
To find: 102 - 22
Let us assume that a = 10 and b = 2.
We will substitute these in the formula of a2 - b2. 
a2 - b2 = (a - b) (a + b)
102-22 = (10-2)(10 + 2)
= 10 (10 + 2) - 2 (10 + 2)
= 10(12) - 2(12)
=120 - 24 = 96
Answer: 102 - 22 = 96.

How To Use the a2 - b2 Formula Give Steps?

The following steps are followed while using a2 - b2 formula.

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