Question: Which of the following always equals \(\sqrt{9+x^2-6x}\)?

1. x - 3
2. 3 + x
3. |3 – x|
4. |3 + x|
5. 3 - x

Correct Answer: C

Solution and Explanation
Approach Solution (1):

\(\sqrt{9+x^2-6x}=\sqrt{(3-x)^2}=|3-x|\)(by the definition of square root)

Approach Solution (2):

I used the plug-in method

Make x = 2 then \(\sqrt{9+4-12}=\sqrt1\)

I then plugged in the two in place of x in the answer choices

[A] 2 – 3 = -1 NO
[B] 3 + 2 = 5 NO
[C] |3 – 2| = |1| MAYBE
[D] |3 + 2| = |5| NO
[E] 3 – 2 = 1 MAYBE

Approach Solution 3:
So we know √9+x2−6x will give us ONLY POSITIVE VALUE, Then we should make sure that the option also matches this property.

Therefore only |3 - x| and |3+x| are the one that will always give positive value
BUT |3+x| is not a root or solution of √9+x2−6x

So the only option left is |3-x|

“Which of the following always equals ?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

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