What is the purpose of a perfect square trinomial?

[Quasaire]

Okay so you want to find the square of this binomial (8x + 2)^2. The square of it is what they call a perfect square trinomial which is:
(8x + 2)^2 = (binomial right now)
(8x + 2)(8x + 2) =
(8x * 8x) + (8x * 2) + (8x * 2) + (2 * 2) =
64x^2 + 16x + 16x + 4 =
(64x^2 + 32x + 4) = the perfect square trinomial

I just simply want to know what is the purpose of this? I know why they call it a square trinomial but why do they call it "perfect" or is my algebra book unique in calling it perfect? What practical applications or technological systems use perfect square trinomials or are they just something to exercise your brain with for enculturation into more difficult mathematics?

 

[maximus]

your book is definitely not unique in calling it perfect square trinomials. that's their name!


and it is called a perfect square trinomial becasuse as you can see with your calculations, it is the product of two binomials (the same), and thus a perfect number (as in perfect squrare).

they have some application to life, but mostly, (as with all math) they are mainly for getting you ready for higher math levels, which are highly applicable.

 

[HallsofIvy]

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.