Strange products » Square of difference
The rule:
(a – b)2 = a2 – 2ab + b2
Why is this true?
The normal rule is:
(a – b)(c – d) = ac – ad – bc + bd
When we choose a and b also for c and d, we get:
(a – b)(a – b) = a2 – ab – ab + b2 = a2 – 2ab + b2
See also removing brackets.
Examples
1. (4x – 5)2 = (4x)2 – 2 · 4x · 5 + 52 = 16x2 – 40x + 25
2. (3z – 7)2 = (3z)2 – 2 · 3z · 7 + 72 = 9z2 – 42z + 49
3. (7x – 6y)2 = (7x)2 – 2 · 7x · 6y + (6y)2 = 49x2 – 84xy + 36y2