Need to understand this simplification

[Jayson76]

Given (-a/-b) \pm (\sqrt(c)/-d)?

Why is it a/b \pm (\sqrt(c)/d)?

Thanks.

[Math Tutor]

There are really two separate steps going on.

1) The first fraction: (-a)/(-b) = a/b, because a negative divided by a negative is positive. Nothing subtle there.

2) The second term: sqrt(c)/(-d) = -sqrt(c)/d. So strictly speaking the expression becomes a/b (minus-or-plus) sqrt(c)/d, i.e. the sign in front gets flipped.

So why is it fine to write a/b (plus-or-minus) sqrt(c)/d? Because the symbol is just shorthand for "take both cases". One version gives you the two values a/b + sqrt(c)/d and a/b - sqrt(c)/d; the flipped version gives a/b - sqrt(c)/d and a/b + sqrt(c)/d. Same pair of values, just written in the other order. Since the plus-or-minus doesn't commit you to which case comes first, the two forms describe exactly the same set of answers, and a/b (plus-or-minus) sqrt(c)/d is the tidier way to write it.

One caveat: this only works because of the plus-or-minus. If the expression had a single fixed sign (only + or only -), then the minus from the -d genuinely does change the answer and you must flip it.

This comes up most often with the quadratic formula, where people simplify things like (-(-b) +/- sqrt(D))/(-2a) and the sign of the radical part looks like it "disappears" - it hasn't, it has just been absorbed into the plus-or-minus.