A little help on the solution?

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A little help on the solution?

by fambrini » Fri Oct 21, 2016 3:39 pm
What is the value of a� - b�?

1) a² - b² = 16

2) a + b = 8

OA: C

I have trouble while using statement 2. Thanks in advance,
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by ceilidh.erickson » Fri Oct 21, 2016 3:57 pm
The key is to recognize that a� - b� is a DIFFERENCE OF SQUARES. We can factor this expression:
a� - b� = (a² + b²)(a² - b²)

Again, we can factor the difference of squares:
(a² + b²)(a² - b²) = (a² + b²)(a + b)(a - b)

Target question: what is the value of (a² + b²)(a + b)(a - b)?

1) a² - b² = 16

If we factor, this tells us that (a + b)(a - b) = 16. Since we know nothing about (a² + b²), though, we can't solve. Insufficient.

2) a + b = 8

Again, this alone gives us the value of one piece of our target question, but not the whole thing. Insufficient.

(1) & (2) together:
When we put the two together, we can substitute statement 2 into statement 1:
(a + b)(a - b) = 16
(8)(a - b) = 16
a - b = 2

Now, combine the two equations:
a + b = 8
a - b = 2
2a = 10
a = 5
b = 3

Now that we have values for both of our variables, we clearly have sufficient information to answer the question. Sufficient.
The answer is C.

Does that help?
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by fambrini » Fri Oct 21, 2016 4:01 pm
Thanks a lot, I was having trouble with factoring the difference of squares for the second time. Silly one, but thanks!

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by ceilidh.erickson » Fri Oct 21, 2016 4:14 pm
Hey, we all get stuck sometimes, even on silly stuff! I'm happy to help.
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by GMATGuruNY » Sat Oct 22, 2016 3:14 am
fambrini wrote:What is the value of a� - b�?

1) a² - b² = 16

2) a + b = 8
Alternate approach:

Statement 1:
If b=0, then a²=16, implying that a = ±4.
In this case, a� - b� = (±4)� - 0� = 4�.
If b=4, then a²=0, implying that a=0.
In this case, a� - b� = 0� - (±4)� = -(4�).
Since a� - b� can be different values, INSUFFICIENT.

Statement 2:
If b=0, then a=8.
In this case, a� - b� = 8� - 0� = 8�.
If b=8, then a=0.
In this case, a� - b� = 0� - 8� = -(8�).
Since a� - b� can be different values, INSUFFICIENT.

Statements combined:
Statement 1 can be rephrased as follows:
(a+b)(a-b) = 16.
Since a+b=8, a-b = 2.
Since we have two distinct linear equations and two variables, we can solve for both variables.
Thus, the value of a� - b� can be determined.
SUFFICIENT.

The correct answer is C.
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