Here's an example of a trinomial that is NOT a "perfect square":
x2- 6x+ 7
It's not a perfect square because it cannot be written as (x-a)2 for any number a.
One application of the idea of perfect squares is finding largest or smallest possible values for a function (a very important specfic application of mathematics).
In order to find the smallest possible value of x2- 6x+ 7, we "complete the square" .
Knowing that (x-a)2= x2- 2ax+ a2, we look at that -6x term and think: if -2ax= -6x then a= 3. We would have to have a2= 9: x2-6x+ 9 is a perfect square: it is (x-3)2.
x2- 6x+ 7 is NOT a perfect square because it has that 7 instead of 9. But 7= 9- 2 so we can rewrite this as
x2- 6x+ 9- 2=(x- 3)2- 2.
A "perfect square" is NEVER negative: 02= 0 and the square of any other number is positive. Looking at (x-3)[sup[2[/sup]- 2, we see that if x= 3, then this is 02- 2= -2 while for any other value of x it is -2 plus a positive number: larger than -2.
The smallest possible value of this function is -2 and it happens when x= 3.