SOLUTION: So these are 2 answers from A question I worked out : a=-1/(1-b^2) a=1/(b^2-1) Can you please tell me how these 2 are the same exact thing ?(if I am not mistaken) Thanks i

Algebra ->  Equations -> SOLUTION: So these are 2 answers from A question I worked out : a=-1/(1-b^2) a=1/(b^2-1) Can you please tell me how these 2 are the same exact thing ?(if I am not mistaken) Thanks i      Log On


   

Question 1205626: So these are 2 answers from A question I worked out :
a=-1/(1-b^2)
a=1/(b^2-1)
Can you please tell me how these 2 are the same exact thing ?(if I am not mistaken)
Thanks in advance
Found 4 solutions by josgarithmetic, math_tutor2020, MathLover1, ikleyn:
Answer by josgarithmetic(39616) About Me  (Show Source):
Answer by math_tutor2020(3816) About Me  (Show Source):
You can put this solution on YOUR website!

Yes they are equivalent.

Scratch work
a=-1/(1-b^2)
a=-1/(-1(-1+b^2))
a=1/(-1+b^2)
a=1/(b^2-1)

Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!
a=-1%2F%281-b%5E2%29
prove that
a=1%2F%28b%5E2-1%29 is same
a=1%2F%28b%5E2-1%29 ...we can write %28b%5E2-1%29 as -%28-b%5E2%2B1%29, and rearange terms inside parentheses -%281-b%5E2%29
so, we have
a=1%2F%28-%281-b%5E2%29%29 which is
a=-1%2F%281-b%5E2%29

Answer by ikleyn(52776) About Me  (Show Source):
You can put this solution on YOUR website!
.
So these are 2 answers from A question I worked out :
a = -1/(1-b^2)
a = 1/(b^2-1)
Can you please tell me how these 2 are the same exact thing ?(if I am not mistaken)
Thanks in advance
~~~~~~~~~~~~~~~~~~~~~~ 

First, I explain it to you in a formal way as a sequence of steps


(1)  a = %28-1%29%2F%281-b%5E2%29     given.


(2)  multiply both sides by 1

     a = a*1 = %28%28-1%29%2F%281-b%5E2%29%29%2A1.


(3)  Express 1 as the ratio  1 = %28-1%29%2F%28-1%29.  Then you will get

     a = a*1 = %28%28-1%29%2F%281-b%5E2%29%29%2A%28%28-1%29%2F%28-1%29%29.


(4)  Multiply in the numerator and in the denominator separately.

     Multiplying by (-1) means "change the sign"

     a = a*1 = %28%28-1%29%2A%28-1%29%29%2F%28%281-b%5E2%29%2A%28-1%29%29 = 1%2F%28b%5E2-1%29.


(5)  Thus you get

     %28-1%29%2F%281-b%5E2%29 = 1%2F%28b%5E2-1%29.

Q.E.D., and your question is answered.

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Informally, your two ratios have opposite signs in the numerator and in the denominator -
therefore, these two ratios are equal.