# a^2-b^2 Formula

The **a ^{2} - b^{2} 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 a^{2} - b^{2} formula is given as: **a ^{2} - b^{2} = (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 a^{2} - b^{2}.

### Verification of a^{2} - b^{2} Formula

Let us see the proof of a square minus b square formula. To verify that a^{2} - b^{2} = (a - b) (a + b) we need to prove LHS = RHS. Let us try to solve the equation:

a^{2} - b^{2} = (a - b) (a + b)

Multiply the binomials (a - b) and (a + b) we get

(a - b) (a + b)

=a (a + b) - b (a + b)

=a^{2} + ab - ba - b^{2}

=a^{2} + 0 + b^{2}

=a^{2} - b^{2}

Hence Verified

a^{2} - b^{2} = (a - b) (a + b)

You can understand the a^{2} - b^{2} formula geometrically using the following figure:

☛ **Also Check:**

## Proof of a^2 - b^2 Formula

The proof that the value a^{2} - b^{2} 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) = a^{2} − b^{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 a^{2} - b^{2} formula find the value of 106^{2} - 6^{2}.

**Solution:** To find: 100^{2} - 6^{2}.

Let us assume that a = 100 and b = 6.

We will substitute these in the a^{2} - b^{2} formula.

a^{2} - b^{2} = (a - b) (a + b)

106^{2}- 6^{2} = (106 - 6) (106 + 6)

= (100) (112)

= 11200

**Answer:** 106^{2} - 6^{2} = 11200.

**Example 2:** Factorize the expression 25x^{2} - 64.

**Solution:** To factorize: 25x^{2} - 64.

We will use the a^{2} - b^{2} formula to factorize this.

We can write the given expression as

25x^{2} - 64 = (5x)^{2} - 8^{2}

We will substitute a = 5x and b = 8 in the formula of a^{2} - b^{2}.

a^{2} - b^{2} = (a - b) (a + b)

(5x)^{2} - 8^{2} = (5x - 8) (5x + 8)

**Answer:** 25x^{2} - 64 = (5x - 8) (5x + 8)

**Example 3:** Simplify 10^{2} - 5^{2} using a^{2} - b^{2} formula

**Solution:** To find 10^{2} - 5^{2}

Let us assume a = 10 and b = 5

Using formula a^{2} - b^{2} = (a - b) (a + b)

10^{2} - 5^{2} = (10 - 5) (10 + 5)

= 10(10 +5) - 5(10 + 5)

= 10(15) - 5(15)

= 150-75 = 75

**Answer:** 10^{2} - 5^{2} = 75.

## FAQs on a^2 - b^2 Formula

### What is the Expansion of a^{2} - b^{2} Formula?

**a ^{2} - b^{2} formula** is read as a square minus b square. Its expansion is expressed as a

^{2}- b

^{2}= (a - b) (a + b).

### What is the a^{2} - b^{2} Formula in Algebra?

The a^{2} - b^{2} formula is also known as one of the important algebraic identities. It is read as a square - b square. a^{2} - b^{2} formula is expressed as a^{2} - b^{2} = (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:

- a
^{2}- b^{2}= (a - b) (a + b) - (a - b)
^{2}= a^{2}+ b^{2}- 2ab

### How to Simplify Numbers Using the a^{2} - b^{2} Formula?

Let us understand the use of the a^{2} - b^{2} formula with the help of the following example.

**Example:** Find the value of 10^{2} - 2^{2} using the a^{2} - b^{2} formula.

To find: 10^{2} - 2^{2}

Let us assume that a = 10 and b = 2.

We will substitute these in the formula of a^{2} - b^{2}.

a^{2} - b^{2} = (a - b) (a + b)

10^{2}-2^{2} = (10 - 2)(10 + 2)

= 10 (10 + 2) - 2 (10 + 2)

= 10(12) - 2(12)

=120 - 24 = 96

**Answer:** 10^{2} - 2^{2} = 96.

### How To Use the a^{2} - b^{2} 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 a
^{2}- b^{2} - a
^{2}- b^{2}= (a - b) (a + b) - Substitute the values of a and b in the a
^{2}- b^{2}formula and simplify.

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

No, these two have different formulas:

- a
^{2}- b^{2}= (a - b) (a + b) - a
^{2}+ b^{2}= (a + b)^{2}- 2ab

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