Early notable figures

Common special cases

Certain products involving two sets of brackets occur very frequently. Rather than re-deriving the double distributive property each time, we can use ready-made formulas known as remarkable identities.

The square of a sum

(a+b)^2 = a^2 + 2ab + b^2

Example: (x+3)^2 = x^2 + 2×x×3 + 3^2 = x^2 + 6x + 9

The square of a difference

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

Example: (x-2)^2 = x^2 - 2×x×2 + 2^2 = x^2 - 4x + 4

The product of a sum and a difference

(a+b)(a-b) = a² - b²

Example: (x+5)(x-5) = x² - 5² = x² - 25

Summary table

Identity	Formula	Example
Square of a sum	(a+b)^2 = a^2+2ab+b^2	(x+1)^2 = x^2+2x+1
Square of a difference	(a-b)² = a² - 2ab + b²	(x-1)² = x² - 2x + 1
Sum and difference	(a+b)(a-b) = a² - b²	(x+1)(x-1) = x² - 1

Common pitfall

The most common mistake is to write (a+b)² = a² + b², completely forgetting the middle term 2ab. This shortcut is incorrect! Always check using the double distributive property if you’re in any doubt: (a+b)(a+b) does indeed equal a² + 2ab + b², and never just a² + b².