question from a student

[RonPurewal]

courtesy of a student:

Hi Ron,

On the Equations, Inequalities, and VICs strategy guide, on Chapter 5, In-Action problem #2, the answer key says I need to give 2 solutions, one for a positive root and one for a negative root. But then, on problem #4, there is also a square root in the equation but the answer key only assumes a positive root. If I plug in a negative root, that would also work. But the answer key is assuming a positive one. What should I do in the GMAT? Assume that square roots have only a positive root or should I give an answer with a plus/minus one?

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what's actually happening here is this: the reason you see two possibilities is precisely BECAUSE the "√" sign must indicate a positive number.

here's a simpler example first --
consider the expression √(x^2).
if x is positive, this is simple enough -- it's x.
however, if x is negative, then squaring and then square rooting gives a number with the same magnitude as the original x, but with the opposite sign -- i.e., -x.
so:
√(x^2) is either x (if x ≥ 0) or -x (if x ≤ 0). note that these quantities are both non-negative at all times, because each only applies to those values of x for which it is positive.

the problem in the strategy guide is doing the same thing -- you have to split this up into two cases, depending on whether (d + 3) is a positive or negative quantity.
if (d + 3) is positive, then √((d + 3)^2) is just d + 3.
however, if (d + 3) is negative, then √((d + 3)^2) is -(d + 3).
both are non-negative.

if you still don't quite see how this works, you should try plugging in some numbers. make sure you plug in numbers that are greater than -3 as well as numbers that are less than -3, so you can see how one or the other interpretation works in each case.