If a = 4x^2 + 4xy and b = 4y^2 + 4xy, which of the following is

Question

If a = 4x^2 + 4xy and b = 4y^2 + 4xy, which of the following is equivalent to x + y?

A.  \sqrt{a+b}

B.  2\sqrt{ab}

C.  \frac{a+b}{\sqrt{2}}

D.  2\sqrt{a}-2\sqrt{b}

E.  \frac{\sqrt{a+b}}{2}

GMATinsight · Answer

Given : a = 4x^2 + 4xy and b = 4y^2 + 4xy

Let's add the two a and b because their addition resembles with the property (a+b)^2

i.e. (a+b) = 4x^2 + 4xy + 4y^2 + 4xy

i.e. (a+b) = 4x^2 + 8xy + 4y^2 = 4*(x^2 + 2xy + y^2)

i.e.  (a+b) = 4*(x + y)^2

i.e.  \sqrt{\frac{(a+b)}{4}} = (x+y)

i.e.  \sqrt{(a+b)}/2 = (x+y)

Answer: Option  E

EMPOWERgmatRichC · Answer

Hi All,

While this question is certainly complex-looking, it can be be solved by TESTing VALUES and taking advantage of how the answer choices are written (to avoid some of the work).

We're told that...
A = 4(X^2) + 4XY
B = 4(Y^2) + 4XY

We're asked for the value of X+Y.

IF....
X=1
Y=2
A = 4(1^2) + 4(1)(2) = 4+8 = 12
B = 4(2^2) + 4(1)(2) = 16+8 = 24

The value of X+Y = 1+2 = 3, so we're looking for an answer that equals 3 when A=12 and B=24.

Answer A: root(12+24) = root(36) = 6. NOT a match

Answer B: 2  = BIGGER than 3. Not a match

Answer C: (12+24)/Root2 = 36/Root2 = BIGGER than 3. Not a match.

Answer D: 2(root12) - 2(root24) = NEGATIVE. Not a match.

Answer E: root(12+24)/2 = 6/2 = 3. This IS a MATCH.

Final Answer:  E

GMAT assassins aren't born, they're made,
Rich

KS15 · Answer

a+b=4x^2 + 4xy+4y^2 + 4xy=4(x^2+y^2+2xy)
a+b=4(x+y)^2
a+b/4=(x+y)^2
Taking square root on both sides
\sqrt{a+b}/2=x+y
Answer E

Bunuel · Answer

MANHATTAN GMAT OFFICIAL SOLUTION:

If we add a and b, we get a + b = 4x^2 + 8xy + 4y^2 = 4(x^2 + 2xy + y^2).

The right side is of the “square of a sum,” so we can factor and solve:
a + b = 4(x + y)^2.
 \frac{\sqrt{a+b}}{2}=x+y

Note that we could safely take the square root of both sides, since we know any square is nonnegative.

The correct answer is E.

wolfof6thstreet · Answer

Can you explain how we can assume that taking the square root of (x+y)^2 results in (x+y) and not |(x+y)|?

I thought when you take the square root of a variable, you assume the negative and positive, but when you take the square of a number, you only consider the positive. So in this case, we are doing it to variables...

Bunuel · Answer

You are correct, the question should either ask about the value of |x + y| or should say that x and y are non-negative numbers. For example, of x = 0 and y = -1, x + y = -1, however  \frac{\sqrt{a+b}}{2}  would be equal to 1.

You are correct; the question should either ask about the value of |x + y| or specify that x and y are non-negative numbers. For example, if x = 0 and y = -1, then x + y = -1. However,  \frac{\sqrt{a+b}}{2}  would be equal to 1 becasue a = 0 and b = 4..

luisdicampo · Answer

Deconstructing the Question  
We are given  a=4x^2+4xy  and  b=4y^2+4xy .
We want an expression in  a,b  that is equivalent to  x+y .
Key idea: add  a+b  to create  (x+y)^2 , then take a square root.

Step-by-step  
Add:
 a+b=(4x^2+4xy)+(4y^2+4xy)=4x^2+8xy+4y^2

Factor a perfect square:
 a+b=4(x^2+2xy+y^2)=4(x+y)^2

Square root:
 \sqrt{a+b}=\sqrt{4(x+y)^2}=2|x+y|

Divide by 2:
 \frac{\sqrt{a+b}}{2}=|x+y|

This matches choice  \frac{\sqrt{a+b}}{2} .

Answer: E

If a = 4x^2 + 4xy and b = 4y^2 + 4xy, which of the following is

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If a = 4x^2 + 4xy and b = 4y^2 + 4xy, which of the following is

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