Question

How do you factor u^2-v^2+2v-1?

Please explain how, not just give the answer.

Answer 1

Well, can you factor first

v^2 - 2v + 1
?

This should look familiar to you.

Answer 2

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

Answer 3

For the record, you will need later the equation for the difference of squares:

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

Answer 4

And since I'm not hearing from you, I'll assume that you're stuck. But a useful special case of the first formula I gave you is when b=1

(a+b)^2 = (a+1)^2 = a^2 + 2a + 1

If you still stuck what happens when b= -1?

(a+b)^2 = ...

Answer 5

I've never used specific formulas before. But ok, i factored down to (v-1)^2, im just not sure what to do with the remaining u^2.. or would it be -u^2?

Answer 6

Right

So now we have

u^2 - v^2 + 2v - 1 = u^2 - ( v^2 - 2v + 1)
= u^2 - (v-1)^2

a difference of squares.

So now you need to factor this ....

Answer 7

Ok thanks, I got (v-1+u)(v-1+u),is that correct?

Answer 8

No, not quite

Answer 9

both of those terms are the same, but you're factoring now a difference of squares, and the factors look like (a-b)(a+b)

Answer 10

Do I need to flip the signs back when I factor in the u^2?

Answer 11

You are factoring now

u^2 - (v-1)^2

Hint: write for now w = v-1. Then you need to factorize

u^2 - w^2

Answer 12

What are the factors of u^2 - w^2 ?

Answer 13

Just give me one sec to figure it out

Answer 14

Look above: I gave you the formula already in terms of a and b.

Answer 15

(u-w)(u+w)

Answer 16

Yes.

Answer 17

Now get rid of the w in your expression so you just have something in terms of the variables of the problem: u and v.

Answer 18

(u-v-1)(u+v-1)

Answer 19

No, be very careful with signs.

Answer 20

(u-v+1)(u+v-1)

Answer 21

Correct.

Answer 22

Ok thank you.