Solve square root of (d)^2+6d+9?

Question

Answer to the question should be:
Question: Square root of (d)^2+6d+9
Answer: Square root of (d+3)^2
d+3.

BUT in the book, they have given two answers:
d+3 and -(d+3)

Arent they wrong? if something is under the square root sign we always consider positive answer only, don't we? Please clarify this doubt.

If someone has Manhattan Prep 6th Edition Algebra book, this is on page 182 and the explanation is page 71.

Reactzz · Answer

If you take the square roof of a number there are always 2 possible solutions:
E.g.: 
(-x)^2 = x^2
x^2 = x^2
Hope this helps

Posted from my mobile device

aj3001 · Answer

I know there are 2 possible solutions but when the square root sign is there, there is only one positive solution as mentioned in the book. Please see my doubt.

rahul16singh28 · Answer

As I understand, anything under the Square Root sign is considered as a positive.

Bunuel · Answer

\sqrt{d^2+6d+9}=\sqrt{(d+3)^2}=|d+3| . So, the answer is d + 3 or -(d + 3). Notice that -(d + 3) is not necessarily negative, for example consider d = -4.

aj3001 · Answer

Thank you so much for your explanation. I got confused since in the book they have mentioned that if a square root sign is used in the GMAT, only the positive value is considered. e.g..  \sqrt{16}  then answer should be only 4 and not both 4 and -4.

Bunuel · Answer

The book is correct.

Check again:  \sqrt{d^2+6d+9}=\sqrt{(d+3)^2}=|d+3| . So, we got that the result is |d + 3| (the absolute value of d + 3). Absolute value is always non-negative, so the result we got (|d + 3|) cannot be negative, it could be positive or 0.

More about the even roots on the GMAT:

\sqrt{...}  is the square root sign, a function (called the principal square root function), which cannot give negative result. So,  this sign ( \sqrt{...} ) always means non-negative square root .

https://gmatclub.com/forum/download/file.php?id=39502 
 The graph of the function f(x) = √x

Notice that it's defined for non-negative numbers and is producing non-negative results.

TO SUMMARIZE:
  When the GMAT provides the square root sign for an even root, such as a square root, fourth root, etc. then the only accepted answer is the non-negative root.   That is:

\sqrt{9} = 3 , NOT +3 or -3;
 \sqrt {16} = 2 , NOT +2 or -2;

Notice that in contrast, the equation  x^2 = 9  has TWO solutions, +3 and -3. Because  x^2 = 9  means that  x =-\sqrt{9}=-3  or  x=\sqrt{9}=3 .

aj3001 · Answer

So the book is right,   When the GMAT provides the square root sign for an even root, such as a square root, fourth root, etc. then the only accepted answer is the non-negative root.  .
 
But if  \sqrt{9} = 3 , NOT +3 or -3
then why is  \sqrt{d^2+6d+9}=\sqrt{(d+3)^2}=|d+3| . i.e (d+3) and -(d+3). Shouldn't it be d+3 ONLY?

Sorry I am really confused.

Bunuel · Answer

No. Absolute value sign, ||, ensures that the result is not negative: |d + 3| cannot be negative. If d >=-3, then |d + 3| = d + 3 (notice that if d >= -3, then d + 3 is NOT negative). If d < -3, then |d + 3| = -(d + 3) (notice that if d < -3, then -(d + 3) is NOT negative). So, as you can see the result is non-negative in all cases. Try plugging numbers to check.

aj3001 · Answer

Got it, thank you so much!

Reactzz · Answer

If you take the square roof of a number there are always 2 possible solutions:
E.g.:
(-x)^2 = x^2
x^2 = x^2
Hope this helps

Posted from my mobile device

ccooley · Answer

That's true, but not technically true on the GMAT. A lot of people find this counterintuitive, so here's an article about the square root situation: https://www.manhattanprep.com/gmat/blog ... -the-gmat/ For the original question, Bunuel's answer is right. -(d+3) isn't necessarily a negative number - for instance, if d = -100, then -(d+3) is positive. So, -(d+3) could actually be the positive square root!

woohoo921 · Answer

MartyTargetTestPrep

If you have time, I would be so appreciative for your thoughts on this thread.
I am still a bit confused and do not understand how if the square root is already in the equation why you have to include the negative version as an option --> -(d+3)

Thank you

GmatKnightTutor · Answer

Might be easier to recognize if you replace d+ 3 with another value such as "z".

-z^2 = z^2
z^2 = z^2

If you Square Root either of these, the value will be z.

MartyTargetTestPrep · Answer

The reason is that we don't know whether (d + 3) is positive or negative.

The square root of (d + 3)^2 must be a positive number or 0.

So, if (d + 3) is positive or 0, then the square root of (d + 3)^2 is (d + 3).

However if (d + 3) is negative, then the square root of (d + 3)^2 is -(d + 3). After all, if (d + 3) is negative, then -(d + 3) is positive.

Solve square root of (d)^2+6d+9?

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards