## Where does this negative sign come from?

Simplificatation, evaluation, linear equations, linear graphs, linear inequalities, basic word problems, etc.

### Where does this negative sign come from?

Change to equivalent fraction:
$1-\frac{2a^2+3}{a^2-2a+3}$
these are the steps I take:
1. multiply the whole number by the denominator $1(a^2-2a+3)$ which just gives me $a^2-2a+3$
2. add this to the numerator part $a^2-2a+3-(2a^2+3)$
3. which gives me $a^2-2a+3-2a^2-3$
4. and put this over the denominator to get $\frac{a^2-2a}{a^2-2a+3}$

BUT the answer is actually $-\frac{a^2-2a}{a^2-2a+3}$

I can't for the life of me see where the negative sign comes from!
Jherek2

Posts: 9
Joined: Sat Aug 04, 2012 2:45 pm

### Re: Where does this negative sign come from?

Jherek2 wrote:2. add this to the numerator part $a^2-2a+3-(2a^2+3)$
3. which gives me $a^2-2a+3-2a^2-3$
4. and put this over the denominator to get $\frac{a^2-2a}{a^2-2a+3}$

It might prove helpful to put the terms in descending order before doing the simplification:

$a^2\,-\,2a^2\,-\,2a\,+\,3\,-\,3$

Note: If the negation sign is, as you have posted, in front of the entire expression, then the sign between the two terms in the numerator should be additive, not subtractive.
nona.m.nona

Posts: 249
Joined: Sun Dec 14, 2008 11:07 pm

### Re: Where does this negative sign come from?

It might prove helpful to put the terms in descending order before doing the simplification:

Ha, of course! I believe that's why 'Doh!' was invented!

Note: If the negation sign is, as you have posted, in front of the entire expression, then the sign between the two terms in the numerator should be additive, not subtractive.

Yes. I couldn't even copy the answer down correctly!

so that leaves me with: $\frac{-a^2-2a}{a^2-2a+3}$ My next question is how and (more importantly) WHY should it be changed to $-\frac{a^2+2a}{a^2-2a+3}$

If I multiply by -1 won't this also change the signs of the denominator? This always confuses me!
Jherek2

Posts: 9
Joined: Sat Aug 04, 2012 2:45 pm

### Re: Where does this negative sign come from?

Jherek2 wrote:so that leaves me with: $\frac{-a^2-2a}{a^2-2a+3}$ My next question is how and (more importantly) WHY should it be changed to $-\frac{a^2+2a}{a^2-2a+3}$

I can think of no particular reason, other than the author's preference. It might be wise to consult with your instructor regarding preferred preference with respect to formatting.

Jherek2 wrote:If I multiply by -1 won't this also change the signs of the denominator? This always confuses me!

Once the -1 has been distributed through the terms of the numerator, there remains to -1 to be distributed elsewhere.
nona.m.nona

Posts: 249
Joined: Sun Dec 14, 2008 11:07 pm

### Re: Where does this negative sign come from?

I can think of no particular reason, other than the author's preference. It might be wise to consult with your instructor regarding preferred preference with respect to formatting.

It's many years since I left school, I am using some old books written in the 1890s (which I found on the internet, not those which I was using at the time!) to try and learn this stuff now. Which is why I am so glad the internet and forums such as this exist. If only they had forty years ago! Thanks for your help.
Jherek2

Posts: 9
Joined: Sat Aug 04, 2012 2:45 pm