The difference between two edges

The third remarkable identity

There is a third, widely used formula which relates a sum and a difference: (a+b)(a-b) = a² - b²

This result is known as the difference of two squares.

Where does this formula come from?

By expanding (a+b)(a-b) term by term: aa - ab + ba - bb = a^2 - ab + ab - b^2 = a^2 - b^2

The two middle terms (-ab and +ab) cancel each other out, leaving only the squares.

A practical example

Let’s calculate (x+5)(x-5) with a=x and b=5: (x+5)(x-5) = x² - 5² = x² - 25

This formula also allows for quick mental calculations. For example, 21 × 19 = (20 + 1)(20 – 1) = 20² – 1² = 400 – 1 = 399.

Summary table

Identity	Formula
Square of a sum	(a+b)² = a² + 2ab + b²
Square of a difference	(a–b)² = a² – 2ab + b²
Sum times difference	(a+b)(a–b) = a² – b²

Pitfalls to avoid

Do not confuse (a-b)² (which gives a² - 2ab + b², always positive or zero) with a² - b² (which can be negative). These are not the same expression at all, even though they look similar.