What is the purpose of a perfect square trinomial?

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In summary, a perfect square trinomial is the product of two identical binomials, and it is called "perfect" because it is a perfect square number. It has practical applications in finding extreme values of functions, but its main purpose is to prepare students for more advanced math concepts.
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Quasaire
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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?
 
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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.
 
  • #3
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.
 

What is a perfect square trinomial?

A perfect square trinomial is a polynomial with three terms that can be factored into the square of a binomial.

What is the general form of a perfect square trinomial?

The general form of a perfect square trinomial is ax2 + bx + c, where a is the coefficient of the squared term, b is the coefficient of the linear term, and c is the constant term.

How can you determine if a trinomial is a perfect square trinomial?

A trinomial is a perfect square trinomial if the first and last terms are perfect squares and the middle term is twice the product of the square roots of the first and last terms.

What are the steps to factor a perfect square trinomial?

The steps to factor a perfect square trinomial are:
1. Identify if the trinomial is a perfect square trinomial.
2. Take the square root of the first and last terms.
3. Double the square root of the first term and write it as the middle term.
4. Write the perfect square trinomial in factored form using the square roots of the first and last terms.
5. Simplify the expression if possible.

Why are perfect square trinomials important in mathematics?

Perfect square trinomials are important in mathematics because they have many applications in algebra, geometry, and physics. They can be used to solve equations, find the vertex of a parabola, and simplify complex expressions. They also provide a foundation for understanding more advanced concepts in mathematics such as completing the square and the quadratic formula.

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