pre-factored equations that equal 0

[secondtimesacharm]

I'm not a big math person (working on it) but I have a question that has puzzled me for a while now. How is it possible for an equation at a certain number x to be unsolvable (0 denominator) yet after you factor the equation out a bit, all of a sudden it has an answer. This seems to violate some basic rule of an equation should always equal the same thing. For instance

x^2-x-6
________
x-3

at x=3 you get 0 / 0 which obviously does not exist.

However, if you factor it to

(x-3)(x+2)
________
x-3

and remove the x-3's then you have a perfectly solvable x+2 or (5).

How is this possible? How can I change the answer of the equation just by simplifying it?

sorry for the noob question, but I really would like to understand this.

[FWT]

You're right about the change. After you take out the zero from the denominator you have a different domain. To be fully right you should keep track of that so the domains match. You would put the answer as "=x+2, with x not equal to 3". You can see examples here (go down to the bottom of the page) and here.

[secondtimesacharm]

hi FWT, thanks for the reply. What does it mean to 'change the domain'? How can a simplification of an equation change what it equals?

thanks!

[FWT]

What does it mean to 'change the domain'?

The domain is the numbers you can put in for the variable. When you canceled off the factor and got rid of a zero you put that zero back into the allowed values. To be the same as you started with you have to take that value back out so you have the same numbers you can put in as you started with.

How can a simplification of an equation change what it equals?

Just like the links showed. That's why you have to put the restrictions back in (take the zeros back out) so you don't change what the expression is.