I don't understand how to solve this equation properly by simplifying -4+a/16-a^2. The answer for the equation is -1/ 4+a. Please show me how you get to the solution.

Question

I don't understand how to solve this equation properly by simplifying  -4+a/16-a^2. The answer for the equation is -1/ 4+a. Please show me how you get to the solution.

help!!!! · Answer

The answer I got was -1 /-1(a+4) or 1/(a+4) , which is clearly wrong

larseighner · Answer

Okay.  You obviously caught on to factoring the denominator).  Now multiply in the denominator by -1 times -1. Do not mess with the numerator.

help!!!! · Answer

Why -1*-1?

larseighner · Answer

$$ \large {{-4+a} \over {16-a^2}} $$

You factored the denominator:

$$ \large {{-4+a} \over {(4-a)(4+a)}} $$

Now
$$ \large {{-4+a} \over {(-1)(-1)(4-a)(4+a)}} $$

What happens when you absorb one of the -1s into the first factor in the denominator  (after the -1s)

help!!!! · Answer

They cancel out but why not -1(-a+4) (a+4) for the denominator

help!!!! · Answer

But why -1*-1? Doesn't -1(-a+4) (a+4) work too for the denominator?

larseighner · Answer

Because multiplying by -1(-1) is the same as multiplying by 1. I want you to see the common factor in the denominator without doing anything to the numerator. If you just multiply the denominator by -1, you would have to multiply the numerator too.

larseighner · Answer

What you were doing was getting an extra -1 somewhere.  This assures they are under control.