# 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

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

Factor the Difference of Two Squares: a^{2} - b^{2}

## Example(x + 2)(x - 2)

Solution

Solution (Detail)

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

2^{2} = 4

So

x^{2} - 4

is the answer.

## Example103⋅97

You can directly solve 103⋅97.

But let's solve this

by using the (a + b)(a - b) formula.

Solution

Solution (Detail)

Change 103 and 97

to a sum and a difference.

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

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

100^{2} = 10000

3^{2} = 9

10000 - 9 = 9991

So 9991 is the answer.

## Example(x^{2} + 1)(x + 1)(x - 1)

Solution

Solution (Detail)

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

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

-1^{2} = -1

(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.