# Product of a Sum and a Difference (a + b)(a - b)

How to solve the product of a sum and a difference (a + b)(a - b): formula, 3 examples, and their solutions.

## Formula

### Formula

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

The product of a sum (a + b) and a difference (a - b)

is the difference of two squares: a^{2} - b^{2}.

## Example 1

### Example

### Solution

(x + 2) is the sum.

(x - 2) is the difference.

So the given expression is x^{2} - 2^{2}.

2^{2} = 4.

So x^{2} - 2^{2} = x^{2} - 4.

So x^{2} - 4 is the answer.

## Example 2

### Example

You can directly solve 103*97.

But let's solve this by using the (a + b)(a - b) formula.

### Solution

Change 103 and 97 to a sum and a difference.

Then 103*97 = (100 + 3)(100 - 3).

(100 + 3) is the sum.

And (100 - 3) is the difference.

Use the (a + b)(a - b) formula.

Then (100 + 3)(100 - 3) = 100^{2} - 3^{2}.

100^{2} = 10000

3^{2} = 9

So 100^{2} - 3^{2} = 10000 - 9.

10000 - 9 = 9991

So 9991 is the answer.

## Example 3

### Example

### Solution

First see (x + 1)(x - 1).

(x + 1) is the sum.

(x - 1) is the difference.

So (x + 1)(x - 1) = (x^{2} - 1^{2}).

-1^{2} = -1

(x^{2} + 1) is the sum.

(x^{2} - 1) is the difference.

So (x^{2} + 1)(x^{2} - 1) = ((x^{2})^{2} - 1^{2}).

(x^{2})^{2} = x^{2⋅2} = x^{4}

Power of a Power

-1^{2} = -1

So x^{4} - 1 is the answer.