prachich1987 wrote:Sorry But I don't understand why we are considering -ve root here
why don't we simply take x+4 as an only root
Hi prachihi1987!
We are not considering negative roots here. In fact we're considering
only positive roots by taking the square root as |x + 4|. Let's see how?
As x is a variable, we don't know what could be the value of x. Thus we don't know whether (x + 4) is positive or negative. Now if we take √(x + 4)² = (x + 4), then for any x > -4, (x + 4) is positive and the result is okay. But for x < -4, (x + 4) is negative! Thus for any x < -4, we are making a mistake! Let's take an example. Say x = -6 => (x + 4) = -2. This means for x = -6, according to our assumption √(x + 4)² = (x + 4), √(-6 + 4)² = (-6 + 4) = -2 => A
negative square root!
What you mentioned earlier "
on GMAT you should never consider -ve roots & always you have to consider the positive roots" is correct. That's why we take |x| as the value of √(x)² which ensures only non-negative roots for any value of x. Thus in this case we must take √(x + 4)² = |x + 4|.
From your post I think you have a misconception about |x|.
|x| can never be a negative quantity! By definition |x| = x for x ≥ 0 and |x| = -x, for x < 0. Note that although we are using +ve and -ve signs both, we are considering only the positive values! As for positive x, |x| = x =>
positive and for negative x, |x| = -x =>
positive (As putting a minus sign before a negative value makes it positive!)
Thus if we consider |x + 4| as the value of √(x + 4)², we are considering only the positive root! In fact we are ensuring that the root is positive by taking the absolute value.
