perfect square trinomial - how are they factored?

Question

I just need a brief explanation

Answer 1

ok
"unsquare" the first term
do the same for the last
put those answers in your (here & here)

"unsquare" is just a silly way to say square root

Answer 2

You have to find 2 perfects squares

For instance:

4x^2 + 4x + 1

4x^2 is the square of 2x

1 is the square of 1

So, this trinomial could be the square of (2x+1),

To be sure, you have to find the middle term

A perfect square is:

(a+b)^2 = a^2 + 2ab + b^2

We have already found a^2 and b^2

So, in this example a = 2x and b = 1

2 ab = 2 *(2x)* (1) = 4x

So, 4x^2 + 4x + 1 is really (2x+1)^2

What about 4x^2 + 5x + 1? The squares are the same, but the middle term is not the right one

What about 4x^2 - 4x + 1?

The square are the same, but 2ab is 4x and we want to get -4x. But, if we consider (2x-1), and since (-1)^2 = 1 too, then the middle term will be

2 (2x)(-1) = -4x.

So, 4x^2 - 4x + 1 = (2x-1)^2

Lets practice with another example so that things are clearer

4x^2 + 20 x + 25

What are the squares now?

a = 2x and b = 5

The middle term is: 2 (2x)(5) = 10

So, 4x^2 + 20x + 25 = (2x+5)^2

And 4x^2 - 20x + 25 = (2x - 5)^2 since b is -5 now and 2ab = 2 (2x)(-5) = -20x

To find the squares is not difficult, remember than the order of the terms is unimportant

For instance:

10x + x^2 + 25. The squares are x^2 and 25

Only the middle term can be negative, so

-4x^2 + 4x + 1 cant be a square, because (2x)^2 = 4x^2 and (-2x)^2 = 4x^2 too

Ana

Answer 3

For example, to factor: 9x^2-30x+25,

you just take the square root of the first term (3x), the sign of the second term (-) and the square root of the third term (5), hence:
9x^2-30x+25 = (3x-5)^2

Answer 4

If it's a perfect square, then (in 2 variables) it's of the form
(ax + by)² = a²x² + 2abxy + b²y².
If it's only in one variable, then it's of the form
(ax + b)² = a²x² + 2abx + b².
Mostly it's all about 'seeing' the perfect squares and the sums of them.

Doug

Answer 5

simply you square root!