Equations, Inequalities, & VICs: Chapter 5, Question 2

[guy29]

Hi, I'm hoping someone can clarify this for me, question 2 states:

g(x) = 3x + sqrt(x), what is the value of g(d^2 + 6d + 9)?

The answer gives 2 solutions:

3d^2 + 19d + 30 and 3d^2 + 17d + 24

The reasoning behind having two answers is that the square root of the equation d^2 + 6d + 9 is both d+3 and -(d+3). But since the question shows a square root symbol, shouldn't we assume it's asking for the positive square root? A positive square root would only produce one answer: 3d^2 + 19d + 30.

[kylo]

i think u have to consider both the cases.


Thanks!

[guy29]

Why would you consider both cases? For instance, if a question asked what is sqrt(4)? You wouldn't have two answers, since you have to assume that it's asking you for the positive square root. Why would you do so if the question asks for a square root of a formula?

[esledge]

When you take the square root of a plain-old number, not an expression, the GMAT always means the positive root. For example, sqrt(3^2) = sqrt(9) = 3, NOT -3. Similarly, sqrt[(-3)^2] = sqrt[9] = 3, NOT -3, even though -3 was the original base we squared inside the radical. More generally, sqrt(x^2) = |x|.

In this problem, the question is what to do with sqrt[(d+3)^2]. Following the general rule above, sqrt[(d+3)^2] = |d+3|, which could be (d+3) or -(d+3), depending on the sign of (d+3).

Going back to your original question, guy29:

The reasoning behind having two answers is that the square root of the equation d^2 + 6d + 9 is both d+3 and -(d+3). But since the question shows a square root symbol, shouldn't we assume it's asking for the positive square root?

You are right, you want the positive square root. But without knowing the value of d, how can you know which one that is? When unknowns are found under the square root sign, you must consider both cases.

[guy29]

I think I get it now. Thanks Emily!

[JonathanSchneider]

: )