Question

Factor 4z^2-4z+1 I know that (a+b)^2=a2+2ab+b2 and I know that the answer is (2z-1)^2. I just can't understand how to factor it.

Answer 1

@tkhunny

Answer 2

For me, I would give it the suspicion test.

4z^2 is a perfect square
+1 is a perfect square.

The very first thing I would try is: (2z + 1)^2 Does that work?
Oh, wait. The second thing I would try, noting the negative middle term, is (2z-1)^2 Does that work?

You can ALWAYS use the quadratic formula.

Answer 3

I could use (a+b)^2=a2+2ab+b2 right?

Answer 4

that's what my teacher wants me to use @tkhunny

Answer 5

...or (a-b)^2 = a^2 - 2ab + b^2

Answer 6

oh yeah oops, how would I use it though ?

Answer 7

Essentially what I did in the Suspicion Method. Establish the correct pattern and apply the principle.

Answer 8

So a=4z_ and b=1_

Answer 9

@tkhunny

Answer 10

That's what I did first, too. Now, figure out that we need a negative and change b.

Answer 11

I got (4z-1)^2 when I plug in

Answer 12

I don't understand how I can get from their to (2z-1)^2 . Do you mind teaching me? @tkhunny

Answer 13

Seriously, you just have to recognize the pattern.

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

It works both ways.

Answer 14

but let's say they didn't give me the answer: (2z-1)^2. How would I do it?

Answer 15

We have been over this. My suspicion method IS how I would do it. Recognize the pattern and implement the relationship.

Alternatively, use the quadratic formula.

If you were to solve, 4z^2-4z+1 = 0, how would you do it?

Answer 16

why don't you treat 2z as big X
then it is clear that \(X^2-2X+1=(X-1)^2\)

Answer 17

my teacher tells me not to use the quadratic formula @tkhunny

@xapproachesinfinity but really I don't have 2z. I got the answer cuz my teacher when over it but I am suppose to solve it without knowing that it equals (2z-1)^2

Answer 18

i find it particularly odd to tell you not to use tools you are taught.
Substitution is a very effective method. Try it.

Answer 19

well you have to look for 2z
there
you have 4z^2=(2z)^2
you need to look for perfect square as @tkhunny said
and 1 is already good

Answer 20

Essentially, what you are asking is how to solve the problem without using any known method. You need to see that this is an impossible assignment.

Recognize the form.
Make a substitution.
Use the quadratic formula.
Ask a friend.

So far, everything you suggest it immediately ruled out. You must do SOMETHING or you cannot solve the problem. There is not a method called "Fall Out of the Sky". That doesn't work at all.

Answer 21

see this example:
9c^2+12c+4

Answer 22

how would you think about this?

Answer 23

i would have to agree with @tkhunny , you must use one way or another to solve this
otherwise it is impossible

Answer 24

that's making me feel dumb @tkhunny lol

ok so for that example I would use (a+b)^2=a2+2ab+b2 am I right?

Answer 25

right! that the essential of the essentials to recognize perfect square
actually it is not essential is the heart of this haha

Answer 26

No need to feel dumb. That is a voluntary feeling. Just don't do it.

There is room in this world for just "recognize" the pattern.

Answer 27

^_^! if you don't get stuck you don't learn!

Answer 28

haha @xapproachesinfinity :)

Answer 29

What would i do next?

Answer 30

now you need to recognize two perfect squares

Answer 31

9c^2=?
4=?

Answer 32

3c and 2

Answer 33

correct! so we could have (3c+2)^2
or (3c-2)^2

Answer 34

what you need to check is if the middle term is 2ab

Answer 35

12c

Answer 36

so it is good! now we have +12c
which one these (3c+2)^2 , (3c-2)^2
would give us +12c in the middle
recall (a+b)^2=a^2+2ab+b^2
(a-b)^2=a^2-2ab+b^2

Answer 37

(3c+2)^2

Answer 38

correct!

Answer 39

yayy! thanks you so much! :D

Answer 40

yw! with pratice you went need to go over all of this
right away you see what you need to do! it is brain training lol

Answer 41

practice*

Answer 42

won't*

Answer 43

yes so true!! lol :)

Answer 44

darn it! i'm pretty bad a typing haha