How come sometimes (a+b)^2= a^2+b^2. While other times it equals (a+b)(a+b)

Question

Kinda silly question probably, but I know sometimes

(a+b)² = a²+b². While other times (a+b)² = (a+b)(a+b).

I had to make (x+3)² = (x+3)(x+3). I started to wonder why it wouldn’t be x² + 9 since when I look at the rules of exponents it almost always shows the a² + b² rule.

Hope this makes sense, but thanks in advance!

Answer 1

(a+b)^2 = a^2 + 2ab + b^2 which only equals a^2+b^2 if ab=0, that requires either a or b =0

(0+b)^2 = 0^2 + b^2 = 0 + b^2 = b^2 or

(a+0)^2 = a^2 + 0^2 = a^2 + 0 = a^2

(x+3)^2 doesn't equal x^2+9. exponent rules do make (3x)^2 = (3^2)(x^2) = 9x^2

You can distribute the exponent 2 over a product, but not over a sum

(ab)^2 = a^2b^2 but (a+b)^2 does not equal a^2+b^2 unless either a=0 or b=0