M19-18

Question

If  b = a + 4 , for which of the following values of  x  is the expression  (x - a)^2 + (x - b)^2  the smallest?

A.  a-1  
B.  a  
C.  a + 2  
D.  a + 3  
E.  a + 5

Bunuel · Answer

Official Solution:

If  b = a + 4 , for which of the following values of  x  is the expression  (x - a)^2 + (x - b)^2  the smallest?

A.  a-1  
B.  a  
C.  a + 2  
D.  a + 3  
E.  a + 5

Given  b = a + 4 , we can rewrite the expression  (x - a)^2 + (x - b)^2  as  (x - a)^2 + (x - a - 4)^2 . To find the smallest value, substitute each answer choice for  x .
 
The smallest expression value is attained at  x = a + 2 :  (x - a)^2 + (x - a - 4)^2  becomes  (a + 2 - a)^2 + (a + 2 - a - 4)^2 = 8 .

Answer: C

Spartan85 · Answer

Answer: C

Expression= (x−a)^2+(x−b)^2 = 2x^2 - 2ax -2bx +a^2 + b^2

For extreme values of expression, d/dx(expression) =0

Differentiating,
d/dx(expression) = 4x -2a -2b = 0

or, 4x= 2a+2b = 4a+8  
Therefore, x= a+2

It can be ascertained that this extreme value is minimum as d^2/dx^2(expression) = 4 (positive)

Note: Differentiation is not tested on the GMAT, but following little information helps

1. d/dx(constant) = 0
2. d/dx(x^n) = n * x^(n-1)
3. d/dx(c*x) = c ; where c = constant
4. Expression has extreme values at d/dx(expression)=0
 ; d2/dx^2(expression) = positive signifies minimum value
 ; d2/dx^2(expression) = negative signifies maximum value

KSANDEEPREDDY · Answer

I think this is a  high-quality  question and I agree with explanation. Assume a = 0
(x)2+(x−4)2

-1 --- 10 
0 --- 16 
2 --- 8 
3 --- 10
5 ---- 26

So the answer is option C

RenanBragion · Answer

I think this is a  high-quality  question and I agree with explanation.

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

rickyric395 · Answer

We are give  (x-a)^2-(x-b)^2 , and since both are squared values, the least possible value for the whole expression can be 0.
Now doing  (x-a)^2-(x-b)^2=0 . Now  (x-a)^2=-(x-b)^2 . 
Ignore this -ve sign(as LHS is square number and RHS can't be -ve), and take square root on both sides.
There can be 2 cases:-
1)  x-a=x-b=0  (this does not give us any value of x, thus ignore this case.)
2)  x-a=b-x  , we get  2x=b+a , replace b with a+4 as given, we get  x=a+2 .

avellanjc · Answer

Hi, Ricky..

How do you get in the last step that x-a=b-x?
Why you only consider the negative root for only 1 side of the equation.

Regards,

Juan C. Avellan

Bunuel · Answer

That solution is not precise, so I'd ignore it.

gchandana · Answer

Although I did substitute the options and solved this during the test, while reviewing, I saw an algebra approach, so if someone is looking for one:

(x - a)^2 + (x - b)^2  
=  (x - a)^2 + (x - a - 4)^2  
=  (x - a)^2 + ((x - a) - 4)^2  
Take x - a = k
=  k^2 + (k - 4)^2  
=  k^2 + k^2 + 16 - 8k  
=  2k^2 - 8k + 16  
=  2(k^2 - 4k + 8)  
=   2(k^2 - 4k + 4 + 4)   
=   2(k - 2)^2 + 8   
Since a square of a term is always greater than or equal to 0, the minimum value of this expression is 8, and it occurs when
  2(k - 2)^2 = 0  , which means k = 2, which means, x - a = 2, and x = a + 2.

Hope it helps.

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