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GMAT Algebra (Data Sufficiency)
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Author:  questioner [ Wed Apr 17, 2013 7:47 am ]
Post subject:  GMAT Algebra (Data Sufficiency)

What is the value of x?
(1) 4x² – 4x + 1 = 0
(2) 4x² – 1 = 0


A. Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by itself is not.
B. Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by itself is not.
C. Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, even though NEITHER statement BY ITSELF is sufficient.
D. Either statement BY ITSELF is sufficient to answer the question.
E. Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question, meaning that further information would be needed to answer the question.

(A) The both statements are quadratic equations. A quadratic equation can have one solution, two distinct solutions, or no solution. Let’s solve each one.
4x² – 4x + 1 = 0
(2x)² – 2 × (2x) + 1 = 0
Now we can apply the well-known formula and get the square of the difference.
(2x – 1)² = 0
2x – 1 = 0
x = 1/2
We have the only one value as a solution. Therefore statement (1) by itself is sufficient.

4x² – 1 = 0
(2x)² = 1
2x = 1 or 2x = -1
x = 1/2 or x = -1/2
We have two distinct values as solutions. Therefore statement (2) by itself is NOT sufficient.

Statement (1) by itself is sufficient, while statement (2) by itself is not. The correct answer is A.
----------
4x² – 1 = 0 can be solved by itself as there is not written anywhere that it must not be a fraction.

So 4x² – 1 = 0
4x² = 1
x² = 1/4
So x = 1/2

Hence the answer should be D.

Author:  Gennadiy [ Wed Apr 17, 2013 7:48 am ]
Post subject:  Re: GMAT Algebra (Data Sufficiency)

Quote:
x² = 1/4
So x = 1/2
x² = 1/4 yields two solutions: x = 1/2 and x = -1/2 .

Author:  questioner [ Wed Apr 17, 2013 7:49 am ]
Post subject:  Re: GMAT Algebra (Data Sufficiency)

I didn't really understand how the jump was made to (2x – 1)² = 0

Author:  Gennadiy [ Wed Apr 17, 2013 7:49 am ]
Post subject:  Re: GMAT Algebra (Data Sufficiency)

questioner wrote:
I didn't really understand how the jump was made to (2x – 1)² = 0
The well-known formula is (ab)² = a² –2ab + b²

We have (2x)² – 2 × (2x) + 1 = 0
Comparing to the formula, take a = 2x and b = 1:
(2x)² – 2 × (2x) + 1 = 0
a² –2ab + b²

(2x1)² = 0
(ab)² = 0

Author:  questioner [ Wed Apr 17, 2013 7:50 am ]
Post subject:  Re: GMAT Algebra (Data Sufficiency)

Please explain how and why you go from (2x)² – 2 × 2x + 1 = 0 to (2x – 1)² .

Also in the previous step why is it necessary to separate the terms like that?

Author:  Gennadiy [ Wed Apr 17, 2013 7:50 am ]
Post subject:  Re: GMAT Algebra (Data Sufficiency)

dave wrote:
Please explain how and why you go from (2x)² – 2 × 2x + 1 = 0 to (2x – 1)² .
1. Why?
Our goal is to determine what set of values Statement (1) defines. Statement (1) is a quadratic equation and thus we need to solve it. You may use any method that you like. We solved the equation by factoring.

2. How?
We used the formula (ab)² = a² –2ab + b² in reverse.
It comes directly from multiplying the two factors in parentheses:
(ab)(ab) = aabaab + (-b)(-b) = a² –2ab + b²

In case of the equation we have:
(2x – 1)² = (2x)² – 2 × 2x + 1

Quote:
Also in the previous step why is it necessary to separate the terms like that?
We did it to apply the formula, to see what a is and what b is.

Author:  questioner [ Wed Apr 17, 2013 7:50 am ]
Post subject:  Re: GMAT Algebra (Data Sufficiency)

Thank you!

Author:  questioner [ Wed Apr 17, 2013 7:51 am ]
Post subject:  Re: GMAT Algebra (Data Sufficiency)

I'm a confused with statement two since I arrived at only one solution. Please correct me if I am wrong.

4x² - 1 = 0
4x² = 1
x² = 1/4
x² = √0.25
x = 0.5

Thanks in advance.

Author:  Gennadiy [ Wed Apr 17, 2013 7:51 am ]
Post subject:  Re: GMAT Algebra (Data Sufficiency)

Quote:
x² = 1/4
x² = √0.25
x = 0.5
You can square root the both sides, because both x² and 0.25 are not negative. However you do not know if x is negative or positive. So
x² = |x| , NOT just x.

Thus, we get
x² = √0.25
|x| = 0.5
x = 0.5 or x = -0.5


By making the step from √x² = √0.25 to x = 0.5 you imposed additional condition that x is positive.

Plug in both 0.5 and -0.5 into the original equation to see that each one fits.

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