The square of a sum and a difference

A new way of expanding

When expanding (a+b)², we could write (a+b)(a+b) and multiply term by term. But there is a very useful shortcut, known as the remarkable identity, which gives the result directly without having to use the full distributive property.

The two formulas you need to know

Expression	Expansion
(a+b)²	a² + 2ab + b²
(a-b)²	a² - 2ab + b²

These two formulas are similar: only the sign of the middle term changes.

Step-by-step example

Let’s calculate (x+3)^2 with a = x and b = 3: (x+3)^2 = x^2 + 2x3 + 3^2 = x^2 + 6x + 9

Another example: (2x-1)^2 with a = 2x and b = 1: (2x-1)^2 = (2x)^2 - 22x1 + 1^2 = 4x^2 - 4x + 1

Common pitfall

Note: (a+b)² is NOT equal to a² + b²! The middle term, 2ab, is missing. Let’s check with a = 2 and b = 3: (2+3)² = 5² = 25, whereas 2² + 3² = 4 + 9 = 13. These are not the same values: the error is easy to spot by testing simple numbers before drawing any conclusions.