contradictory O explanation

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Kancz44: Newbie | Next Rank: 10 Posts; Posts: 3; Joined: Thu May 03, 2012 12:24 pm

contradictory O explanation

by Kancz44 » Tue May 29, 2012 7:29 pm

What is the value of x?

1) SqRt of X^4 = 9
2) SqRt of X^2 = -X

OA is C

Explanation seems counter to what i have learned about square roots so far. SqRt of X^2 should equal |X| which is equal to -X OR X. The explanation given to address 2) states that the SqRt of X^2 is positive, hence -X needs to be positive (which makes X negative).

So confused. What are the exceptions to the rule "SqRt of X^2 should equal |X| which is equal to -X OR X"?

Thanks!

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Source: Beat The GMAT — Data Sufficiency | Tue May 29, 2012 7:29 pm

eagleeye: Legendary Member; Posts: 520; Joined: Sat Apr 28, 2012 9:12 pm; Thanked: 339 times; Followed by:49 members; GMAT Score:770

by eagleeye » Tue May 29, 2012 7:46 pm

Hopefully this answers your doubts.

Concept: sqrt (a^2) = |a| ; this is always true for any real a.

Now: we have to find x. Let's evaluate the options.

1) sqrt(x^4) = 9. We know that sqrt (a^2) = |a| , therefore
sqrt(x^4) = sqrt((x^2)^2) = 9
this means from our identity that |x^2| = 9; which means that x^2 = +9, -9 , now x^2 is always non-negative, therefore, x^2 = 9, which means x = |3|, hence x is either +3 or -3. Two values, Hence insufficient.

2) sqrt(x^2) = -x; we know that sqrt (a^2) = |a| ; therefore
|x| = -x , which means x is non-positive. But we don't have a unique value of x, Hence insufficient.

Now combining 1) and 2) we see that x must be -3. Hence sufficient. Therefore answer is C.

Let me know if this helps

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1947: Master | Next Rank: 500 Posts; Posts: 215; Joined: Sat Jun 14, 2008 4:24 pm; Thanked: 13 times; Followed by:1 members

by 1947 » Tue May 29, 2012 11:01 pm

Hi Eagleeye,
understood your explanation...have a doubt on what is real number...
can you please explain what are real numbers.
would all integers be considered real number.

eagleeye wrote:Hopefully this answers your doubts.

Concept: sqrt (a^2) = |a| ; this is always true for any real a.

Now: we have to find x. Let's evaluate the options.

1) sqrt(x^4) = 9. We know that sqrt (a^2) = |a| , therefore
sqrt(x^4) = sqrt((x^2)^2) = 9
this means from our identity that |x^2| = 9; which means that x^2 = +9, -9 , now x^2 is always non-negative, therefore, x^2 = 9, which means x = |3|, hence x is either +3 or -3. Two values, Hence insufficient.

2) sqrt(x^2) = -x; we know that sqrt (a^2) = |a| ; therefore
|x| = -x , which means x is non-positive. But we don't have a unique value of x, Hence insufficient.

Now combining 1) and 2) we see that x must be -3. Hence sufficient. Therefore answer is C.

Let me know if this helps

If my post helped you- let me know by pushing the thanks button. Thanks

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Anurag@Gurome: GMAT Instructor; Posts: 3835; Joined: Fri Apr 02, 2010 10:00 pm; Location: Milpitas, CA; Thanked: 1854 times; Followed by:523 members; GMAT Score:770

by Anurag@Gurome » Wed May 30, 2012 12:55 am

Kancz44 wrote:SqRt of X^2 should equal |X| which is equal to -X OR X. The explanation given to address 2) states that the SqRt of X^2 is positive, hence -X needs to be positive (which makes X negative).

So confused. What are the exceptions to the rule "SqRt of X^2 should equal |X| which is equal to -X OR X"?

Every positive number x has two square roots: âˆšx, which is positive, and -âˆšx, which is negative. Together, these two roots are denoted Â±âˆšx. The positive square root, âˆšx is known as the principal square root of x.

By definition âˆšx is positive.
Note that the positive or negative sign comes before the âˆš sign. That's your clue.

Whenever the square root sign 'âˆš' is used it is used to mean the principal square root, i.e. the positive square root. Hence, square roots of 4 are Â±âˆš4, i.e. -2 and 2. But âˆš4 is always equal to 2 NOT -2.

To remove ambiguities we use the modulus notation.
We write âˆš(xÂ²) = |x|, so that we always get the principal square root, i.e. the positive square root. Now, |x| is always positive. Hence,

For x > 0 --> âˆš(xÂ²) = |x| = x > 0
For x < 0 --> âˆš(xÂ²) = |x| = -x > 0

Thus we always get the positive square root.

Let's take two examples,
1. For x = 2 = âˆš(xÂ²) = âˆš[(2)Â²] = |2| = 2
2. For x = -2 = âˆš(xÂ²) = âˆš[(-2)Â²] = |-2| = 2

Hope this clears your confusions.

Anurag Mairal, Ph.D., MBA
GMAT Expert, Admissions and Career Guidance
Gurome, Inc.
1-800-566-4043 (USA)

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Anurag@Gurome: GMAT Instructor; Posts: 3835; Joined: Fri Apr 02, 2010 10:00 pm; Location: Milpitas, CA; Thanked: 1854 times; Followed by:523 members; GMAT Score:770

by Anurag@Gurome » Wed May 30, 2012 1:06 am

Kancz44 wrote:What is the value of x?

1) âˆš(x^4) = 9
2) âˆš(x^2) = -x

Statement 1: âˆš(x^4) = |x^2| = 9 --> x = Â±3

Not sufficient

Statement 2: âˆš(x^2) = |x| = -x --> x is negative

Not sufficient

1 & 2 Together: x = 3

Sufficient

The correct answer is C.

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GMAT/MBA Expert

Anurag@Gurome: GMAT Instructor; Posts: 3835; Joined: Fri Apr 02, 2010 10:00 pm; Location: Milpitas, CA; Thanked: 1854 times; Followed by:523 members; GMAT Score:770

by Anurag@Gurome » Wed May 30, 2012 1:23 am

1947 wrote:can you please explain what are real numbers.
would all integers be considered real number.

For GMAT, you don't need to know anything other than real numbers. All numbers used in GMAT are real numbers. Integers, fractions, decimals, irrational numbers... everything.

Numbers other than real numbers are know as imaginary numbers, which are... well, imaginary and hence not part of GMAT syllabus. If you want to know about them, go to this webpage.