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) = a2 - b2

The product of a sum (a + b) and a difference (a - b)
is the difference of two squares: a2 - b2.

Example 1

Example

Solution

(x + 2) is the sum.
(x - 2) is the difference.

So the given expression is x2 - 22.

22 = 4.
So x2 - 22 = x2 - 4.

So x2 - 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) = 1002 - 32.

1002 = 10000
32 = 9

So 1002 - 32 = 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) = (x2 - 12).

-12 = -1

(x2 + 1) is the sum.
(x2 - 1) is the difference.
So (x2 + 1)(x2 - 1) = ((x2)2 - 12).

(x2)2 = x2⋅2 = x4

Power of a Power

-12 = -1

So x4 - 1 is the answer.