a^2 - b^2 Formula - Examples | A Square - B Square Formula (2024)

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.

  • It is one of the algebraic identities.
  • It is used to factorize the binomials of 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) and (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 the binomials (a - b) and (a + b) we get

(a - b) (a + b)

=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^2 - b^2 Formula - Examples | A Square - B Square Formula (1)

Also Check:

  • (a-b)^2 Formula
  • (a+b)^2 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. They can be arranged such that two rectangles are also formed as shown in the above 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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a^2 - b^2 Formula - Examples | A Square - B Square Formula (2)

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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 identities. It is read as a square - b square. a2 - b2 formula is expressed as a2 - b2 = (a - b) (a + b). It says, the difference of squares of two numbers is the product of their sum and the difference.

What is the Difference Between a square - b square and a - b Whole Square?

These two formulas are completely different:

  • a2 - b2 = (a - b) (a + b)
  • (a - b)2 = a2 + b2 - 2ab

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 a square - b square formula.

  • First, observe the pattern of the numbers and 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.

Is a Square Minus b Square Same as a Square Plus b Square?

No, these two have different formulas:

  • a2 - b2 = (a - b) (a + b)
  • a2 + b2 = (a + b)2 - 2ab
a^2 - b^2 Formula - Examples | A Square - B Square Formula (2024)
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