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A and B are integers and B = A + 4 . Which of the following represents integer X for which expression (X-A)^2 + (X-B)^2 is the smallest?

A-1
A
A+2
A+3
B+1

I don't even know where to start.



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bigfernhead
A and B are integers and B = A + 4 . Which of the following represents integer X for which expression (X-A)^2 + (X-B)^2 is the smallest?

A-1
A
A+2
A+3
B+1

I don't even know where to start.
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quite simple (I think :wink: ) Re arrange the equation as this : (X-A)^2 + (X-A-4)^2 ... meaning replace B with A+4..
Now starting with the first one replace X with each of the options one by one and calculate to check which of the options yields the lowest value for this expression. The last option B+1 is simply A+5... As per my quick, rough calculations "A+2" yields '8' which is the lowest... do recheck the arithmetic... but I calculated A+2 to be the answer.
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o m g, you're right. :| the way the question was written threw me off.

thanks!
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The answer is C (A+2)

There are two ways to do, one is "cheating" since GMAC doesn't expect you to know calculus to take the GMAT.
The easiest way is to plug value in, since it's also the fastest.

When X = A-1, (X-A)^2 + (X-B)^2 = (-1)^2 + (-5)^2 = 26
When X = A, (X-A)^2 + (X-B)^2 = (0)^2 + (4)^2 = 16
When X = A+2, (X-A)^2 + (X-B)^2 = (2)^2 + (-2)^2 = 8
When X = A+3, (X-A)^2 + (X-B)^2 = (3)^2 + (-1)^2 = 10
When X = B+1, (X-A)^2 + (X-B)^2 = (5)^2 + (1)^2 = 26

So the answer is C, X = A+2

Now the "cheating"/calculus way.
Differentiate the expression.

Let y = (X-A)^2 + (X-B)^2
Therefore dy/dX = 2(X-A) + (X-B)
At a minima, dy/dX = 0, so 2(X-A) + (X-B) = 0
Hence, X = (A+B)/2 = A+2
Technically you should also check the second derivative but since the question asks for a minima, you can safely assume this solution gives you a minimum.
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Great explanation as well. It's been awhile since i've done calculus :-)

xenok
The answer is C (A+2)

There are two ways to do, one is "cheating" since GMAC doesn't expect you to know calculus to take the GMAT.
The easiest way is to plug value in, since it's also the fastest.

When X = A-1, (X-A)^2 + (X-B)^2 = (-1)^2 + (-5)^2 = 26
When X = A, (X-A)^2 + (X-B)^2 = (0)^2 + (4)^2 = 16
When X = A+2, (X-A)^2 + (X-B)^2 = (2)^2 + (-2)^2 = 8
When X = A+3, (X-A)^2 + (X-B)^2 = (3)^2 + (-1)^2 = 10
When X = B+1, (X-A)^2 + (X-B)^2 = (5)^2 + (1)^2 = 26

So the answer is C, X = A+2

Now the "cheating"/calculus way.
Differentiate the expression.

Let y = (X-A)^2 + (X-B)^2
Therefore dy/dX = 2(X-A) + (X-B)
At a minima, dy/dX = 0, so 2(X-A) + (X-B) = 0
Hence, X = (A+B)/2 = A+2
Technically you should also check the second derivative but since the question asks for a minima, you can safely assume this solution gives you a minimum.
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(X-A)^2+(X-B)^2= (X-A+X-B)^2-2*(X-A)*(X-B) min <=> X-A=B-X
=> 2X=2A+4 => X=A+2
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DestinyChild
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C

cheat or no cheat :) :)
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Jgroeten
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MJ2009
bigfernhead
A and B are integers and B = A + 4 . Which of the following represents integer X for which expression (X-A)^2 + (X-B)^2 is the smallest?

A-1
A
A+2
A+3
B+1

I don't even know where to start.

quite simple (I think :wink: ) Re arrange the equation as this : (X-A)^2 + (X-A-4)^2 ... meaning replace B with A+4..
Now starting with the first one replace X with each of the options one by one and calculate to check which of the options yields the lowest value for this expression. The last option B+1 is simply A+5... As per my quick, rough calculations "A+2" yields '8' which is the lowest... do recheck the arithmetic... but I calculated A+2 to be the answer.
Show more

A function is minimum when its first derivative equals 0. Thus,

2*(X-A) + 2*(X-B) = 0 --> 2x-2a-2a-8 = 0 --> x=a+2



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Hi there,
This topic has been closed and archived due to inactivity or violation of community quality standards. No more replies are possible here.
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