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10 Feb 2021, 15:28
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D
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
55%
(hard)
Question Stats:
69% (02:11) correct
31%
(02:12)
wrong
based on 705
sessions
History
Date
Time
Result
Not Attempted Yet
Which of the following is not a factor of the solution to the equation \(x-3\sqrt{x}-18=0\)?
(A) 3
(B) 4
(C) 6
(D) 8
(E) 18
(A) 3
(B) 4
(C) 6
(D) 8
(E) 18
Show SpoilerSource
Self-made question.
10 Feb 2021, 18:31
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Rearranging, we get \(x - 18 = 3\sqrt{x}\)
Squaring both sides, \(x^2 - 36x + 324 = 9x\).
Rearranging, \(x^2 - 45x + 324 = 0\)
\(x^2 - 36x - 9x + 324 = 0 \)
x(x - 36) - 9(x - 36) = 0
(x - 36) (x - 9)
The solutions are 36 and 9 and the factors are 1,2,3,4,6,9,12,18 and 36. 8 is not a factor of the solutions.
Option D
Arun Kumar
Squaring both sides, \(x^2 - 36x + 324 = 9x\).
Rearranging, \(x^2 - 45x + 324 = 0\)
\(x^2 - 36x - 9x + 324 = 0 \)
x(x - 36) - 9(x - 36) = 0
(x - 36) (x - 9)
The solutions are 36 and 9 and the factors are 1,2,3,4,6,9,12,18 and 36. 8 is not a factor of the solutions.
Option D
Arun Kumar
11 Feb 2021, 04:58
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AndrewN
Hello, everyone. I came up with this question because I like to encourage creativity in solving quant questions. In addition to the solution outlined above by Arun Kumar of CrackVerbalGMAT, I thought I would offer two of my own.
Method #1: Factor the expression on the left-hand side just as you would a quadratic.
\((\sqrt{x}-6)(\sqrt{x}+3)=0\)
\(\sqrt{x}-6=0\) or
\(\sqrt{x}+3=0\)
The first will lead to a real solution:
\(\sqrt{x}=6\)
\(x=36\)
The second will lead to an extraneous solution, since we cannot invoke the complex i, and the √ sign indicates the positive square root only:
\(\sqrt{x}=-3\)
Thus, x must equal 36. Looking at the answer choices, only 8 is not a factor of 36. The answer must be (D).
Method #2: Use logic and work with the answers.
1) We know that x - 3√x has to equal 18, since subtracting 18 leads to 0. We can deduce that x must be a perfect square, since any other type of number will lead to a non-integer when the square root is taken.
2) Come up with some reasonable targets, starting from 1: 1, 4, 9, 16, 25, 36, 49...
3) Test a reasonable value, knowing that, again, subtracting 18 from x - 3√x will equal 0. How about 16?
\(16-3\sqrt{16}-18=0\)
\(16-3(4)-18=0\)
\(16-12-18=0\)
\(4-18=0\)
\(-14≠0\)
The outcome was too low. Test a higher value instead. Why not 36 this time?
\(36-3\sqrt{36}-18=0\)
\(36-3(6)-18=0\)
\(36-18-18=0\)
\(36-36=0\)
\(0=0\)
This time, we know we have the right value for x. All we need to do is examine the answers for non-factors of 36. The answer must be (D).
Either method above should lead to the answer in under a minute. I am guessing that Experts such as Arun Kumar could also use the method in the above post to crack this one in under a minute. There you have it: three solutions to a fairly intimidating question.
I hope you had fun with this one. As always, good luck with your studies.
- Andrew
