Question

Hi everyone! Can someone explain to me step by step how to get from -x^2-2x+1 to 2-(x+1)^2 by factoring? I need to understand the steps without any magic! Thanks! :o)

Answer 1

\[-x^2 - 2x + 1\]\[= -x^2 - 2x \color{green}{- 1 + 1} + 1\]\[= -(x^2 + 2x + 1) + 2\]\[= -(x +1)^2 +2\]

Answer 2

Thanks Parth!
maybe I should have asked the question differently...
assume I don't know the answer but rather I just need to make it contain (x+1)^2
this was actually part of a much larger problem where -x^2-2x+1 is the numerator and (x+1)^2 is in the denominator...it's hard to just know right away which direction to go in...can you offer any tips?

Answer 3

To be honest, it just gets to you. There's no real way to "learn" all of this. But you reach a point where looking at an expression like \(-x^2 - 2x \) triggers you and makes you say, "OH! This sorta looks like \(-x^2 - 2x - 1 = -(x+1)^2\)."

Answer 4

I see what you mean...tell me if this makes sense however...

Answer 5

If I want the top to look like the bottom, I simply write the top portion directly above the bottom...
then like you did, with the idea in mind to make it look like the bottom, put parenthesis clearly around (-x^2-2x+1)
then -(x^2+2x-1) and then realize that the quantity now looks like the bottom with the exception that the -1 needs to be a +1...so you simply add +2, then rearrange...is that a good strategy that would work everytime you think? :o/

Answer 6

I forgot to mention that before you do any of what I suggested that you foil the bottom first of course

Answer 7

Not sure what you're asking, but yes, that was my thought process.

Answer 8

Okay, then I think I have the technique down then...thanks!:o)

Answer 9

simply a thought he did not add 2 that would have changed the equation

Answer 10

Yeah, um, that's obvious. Adding 2 would be accompanied by subtracting 2.

Answer 11

just wanted her to know you add 1 and add -1