Question

What is the difference between perfect square and difference of squares?

Answer 1

A perfect square can be factored as `(p+q)^2`
The difference of squares can be factored as `(p+q)(p-q)`

Explanation:

I assume here that we are dealing with polynomials that can be factored into binomials.

For trinomials of the form
`color(white)("XXXX")` `ax^2+bx+c`

`ax^2+bx+c` is a perfect square if
`color(white)("XXXX")` `EE_p | p^2 = ax^2`
`color(white)("XXXX")` `EE_q | q^2 = c`
and
`color(white)("XXXX")` `p*q = bx`

Since the difference of squares factors as
`color(white)("XXXX")` `(p+q)(p-q)`
`color(white)("XXXX")` `color(white)("XXXX")` `=p^2 - q^2`
an obvious requirement for `ax^2+bx+c` to be the difference of squares is that `b=0`