Determining trinomial squares?

Question

What does it mean to determine a trinomial squares and how do you do it?

These practice problems are from algebrahelp.com but I don't understand the way they explain it.

Determine whether each of the following is a trinomial square.

1. x^2-14x + 49

2. x^2-10x - 25

3. 36x^2-24x+16

4. 16x^2-40xy+25y^2

5. 9x^2 - 36x + 24

Answer 1

to find out if it is a trinomial square, you see if it is factorable.

example : #1 - (x-7)^2

#2 - cannot be factored

#3 - cannot be completely factored - 4(9x^2 - 6x + 4)

#4 - same as #3 - 16x^2 + 5 (5y^2 - 8xy)

#5 - same as #3 - 3(3x^2 -12x + 8)

Answer 2

You are basically trying to figure out if you can have

(x+a)^2

as the trinomial's factors.

1. I like to look at the first term (the x^2) first:

The square root of x^2 is what? (Or, in other words, what times itself gives x^2?) The answer is x. So I would write on paper something that looks a bit like this:

(x + ___)(x + ___)

knowing that the SAME number has to go on those lines for it to be a square.

Then I look at the last term: 49. What's the square root of 49? 7 or -7. If I use 7, will I get -14 in the middle? No. But if I use -7, I will. So I'll put that in my blanks:

(x - 7)(x - 7), or (x-7)^2

Then I expand to check my answer. x^2 -7x - 7x + 49 = x^2 -14x + 49.

It is therefore a trinomial square.

2. Looking at the last term, right away I can tell that this is NOT a trinomial square. Why not? Because there is no square root of -25. I don't even need to work anything else out. Both the first and the last terms MUST have a square root, and then they have to work accordingly to get the middle term.

3. Both 36x^2 and 16 have square roots: 6x and 4 or -4. Since my middle term is -24, I'll use -4 to see if it works.

(6x - 4)(6x - 4)

Then expand:

36x^2 -24x -24x +16

=36x^2 -48x +16

It doesn't match with the original trinomial. The given trinomial (36x^2 - 24x + 16) is not a trinomial square.

4. Square root of the first term (16x^2) is 4x. Square root of the last term (25y^2) is 5y or -5y. Since the middle term is negative, we'll use -5y to see:

(4x - 5y)(4x-5y)

Expand: 16x^2 -20xy -20xy +25y^2

=16x^2 -40xy + 25y^2

This was our original trinomial. It is therefore a trinomial square since (4x-5y)^2 produces it.

5. There is no whole number square root for 24, so it can not be a trinomial square.