Here's an example of a trinomial that is NOT a "perfect square":
x^{2} 6x+ 7
It's not a perfect square because it cannot be written as (xa)^{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 x^{2} 6x+ 7, we "complete the square" .
Knowing that (xa)^{2}= x^{2} 2ax+ a^{2}, we look at that 6x term and think: if 2ax= 6x then a= 3. We would have to have a^{2}= 9: x^{2}6x+ 9 is a perfect square: it is (x3)^{2}.
x^{2} 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
x^{2} 6x+ 9 2=(x 3)^{2} 2.
A "perfect square" is NEVER negative: 0^{2}= 0 and the square of any other number is positive. Looking at (x3)[sup[2[/sup] 2, we see that if x= 3, then this is 0^{2} 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.
