why x^2-3^2 is not equal to (x-3)^2

Question

i had been confused, sometime the equation come to be x^2-3^2 then i always wrtie it into (x-3)^2 but the product is not same as (x-3)^2 is equal to x^2-6x+9  thus i realize that x^2-3^2=x^2-9 but how about (x-3)(x+3) can it simply into single form like (x-3)^2 or (x+3)^2?

Answer 1

The difference of two squares:

a² - b² = (a-b)(a+b)

 

Proof:  (a-b)(a+b) = a(a) + ab + (-b)(a) + (-b)(b)

                           = a² + ab - ab - b(b)

                           = a² - b²

 

(x-3)(x+3) = x² - (3²) = x² - 9

 

(a - b)² = a² - 2ab + b²

Proof:  (a-b)(a-b) = a(a) + a(-b) + (-b)(a) + (-b)(-b)

                          = a² - ab - ab + b²

                          = a² - 2ab + b²

 

(x-3)² = x² - 6x + 9

 

(x-3)(x+3) = x² - (3)² = x² - 9

 

As you can see, these are not equal.