Negative solutions to square roots?

[NaomiD519]

I understand the mathematical convention - I was always taught this in school. However, based on this question, I have a follow-up from the Algebra Strategy Guide (6th ed). For Chapter 12, question #7, the solution provides a positive and negative root.

Question: If g(x)=3x+√x what is the value of g(d^2+6d+9)?

Solution in MGMAT Guide:

3d^2+19d+30 (assuming positive root)

3d^2+17d+24 (assuming negative root)

Why can't we just assume the root is positive in this case? I.e. Why can't I assume √(d+3)^2=d+3 instead of taking into account the case where √(d+3)^2=d+3 and where
√(d+3)^2=-(d+3)?

[RonPurewal]

if d is less than -3, then d + 3 is the negative root, and -(d + 3) is the positive root.

if that doesn't make immediate sense, just plug in a couple of values (-4, -5, -1000, etc.) and it should make sense instantly.

[NaomiD519]

Ron - The plugging numbers in part is fine. Understanding it theoretically, here's what I've come up with: x in the original function cannot be negative because it's under a square root, but the perfect square is never negative (because it's a perfect square), so d can be positive or negative, even though x can only be 0 or positive. Is this correct?

[RonPurewal]

yes.